Introduction to Topology and Differential

GeometryMAT 389.60 - L. M. Ionescu,Textbook by S. Stahl, 2005.

Ch. 1 Informal Topology

Topological Spaces and Mappings- Euclid's Elements and the five Platonic solids (fig.)- Ambient space X, piece-wise linear figure Y < X and properties:- Geometric, involving length, area etc.- Topological, involving incidence and hierarchic structure, e.g. Euler characteristic:

E(Y)= Vertices - Edges + Faces, X: 3D-space.

- Examples: cube, octahedron, tetrahedron, dodecahedron, icosahedron.

- A mapping between spaces is a function compatible with the topological structure; it transforms the subspaces in a continuous manner, preserving the relative relations of "vicinity" between figures (subspaces).

What is Topology?- Compared to geometry , which is much more familiar, topology does not assumes a distance is defined, and considers many more transformations then geometry does.

- "Definition": Topology is the study of the properties of subspaces of a space, preserved (invariant) under continuous deformations (transformations).

- To define a topological space, we need to know the open (closed) sets. The functions preserving them are the continuous functions.- Geometry "a la" Klein is the study of the properties of a set under the action of a compact group of transformations. Topology relaxes the compactness requirement; intuitively:

Topology <-> "rubber-sheet geometry".

Homeomorphisms

The equivalence relation between topological spaces is generated by homeomorphisms.

- Definition. A homeomorphism f:X->Y is a continuous invertible function, whose inverse is also invertible (topological isomorphism in categorical language).

- Examples

Isotopy- Embedding of a space X into Y is a continuous 1:1 mapping; - Examples: X=point, curve, surface, Y=R^n- Sometimes two embeddings can be continuously deformed into one another.

- Definition: an isotopy of Y1 into Y2 (as embeddings) is a continuous family of embeddings, deforming Y1 into Y2, without tearing or juxtaposition. Then Y1 and Y2 are called isotopic.- Example of isotopies: translations and rotations from geometry; scaling.

Equivalence Relations

- Equivalence relation: 1) reflexive, 2) symmetric and 3) transitive.

- Theorem: Homeomorphism and isotopy are equivalence relations.

- Exercise: Prove the above theorem.

Graduate credit: See Ch. 10 for additional details regarding Topology.

Summary

- Topological space, continuous mappings, homeomorphism

- Embeddings, isotopy

- Homeomorphic and isotopic are equivalence relations

Ch.2 Graphs

2.1 Nodes and Arcs2.2 Transversality2.3 Colorings2.4 Planarity2.5 Graph Homeomorphisms

2.1 Nodes and Graphs- Terminology: open arc = line segment, closed arc or loop = circle (where "=" means "homeomorphic with"); arc: open or closed;- Definition: a graph G=(N,A) is a set of nodes N(G) and a set of arcs A(G) connecting the nodes (multiple arcs connecting the same vertices are allowed: "multi-graph").The number of nodes is p=card(N(G)), and the number of arcs is denoted by q=card(A(G)).- Adjacent nodes; degree of a vertex, denoted deg(v).- Examples (graphical representation).

- The term graph, originally viewed as a topological space, will refer to the equivalence class of graphs under homeomorphism (For more on Graph Theory see Wikipedia).

The Adjacency Matrix

- Alternatively, a graph can be described as a combinatorial object, by its adjacency matrix: G(ij)=number of arcs connecting the vertex i from vertex j (a symmetric matrix, for non-oriented graphs).

- Example: the adjacency matrix of a triangle.

- Parallel arcs: arcs with the same ends (multiple arcs).

The Degree Sequence of a Graph- Listing the vertices in a decreasing order of their order:

N(G)=(v1,v2,...,vn), deg(v1)>=deg(v2)>=...then the sequence of their degrees is called the degree sequence of the graph:

(deg(v1), deg(v2), ..., deg(vn)).- Example fig.2.1- Proposition In any graph Sum deg(v)=2 q(G) proof. Each arc is counted exactly twice, when the index equals its two ends.- The proposition provides a necessary condition for a sequence of numbers to be a degree sequence of a graph; example.

Classes of Graphs- Simple graphs: no parallel arcs, without loops (one can always adjoin or discard the loops), but closed circuits are allowed.

- The Complete Graph Kn: - Every pair of vertices is connected by exactly one arc. - The degree of each vertex is n-1. - p=n, q=n(n-1)/2 - Degree sequence: (n-1, ..., n-1) n-vector.

- The Complete Bipartite Graph K(n,m): - Vertices: [n] v [m]; - Arcs: (i,j) with i from [n] and j from [m].

- The Complete k-Partite Graph K(n1, ...,nk)

Subgraphs

- A subgraph H of G is a graph with vertices and arcs belonging to G:

N(H)<N(G), A(H)<A(G).- The concept of "complement" of a subgraph has two possible definitions. Here we will use the edge-complement H' of a subgraph H of a graph G:

N(H')=N(G), A(H')=A(G)-A(H).Its nodes are all the nodes of G, but its edges form the complement of the edges of H in the original graph G. - Drawback of the definition: the complement of the complement is the node-completion of the original graph (H<= H'').

- Exercise: determine the (edge) complement of G in G.

Supplements

- A graph, representing objects (points) and relations (arcs), define a category (the paths of the graph - to be defined later).

- In turn, graphs form the objects of a category of mathematical objects, and what is left to define are the "functions", called morphisms, from a graph to another graph (later).

2.2 Traversability in a Graph- A u-v walk of length n between two vertices u & v of a graph G is an alternating sequence (u, a1, v1, ..., an, v) consisting of composable arcs ai, starting at u and ending at v.- A walk is closed if u=v.

- If the arcs of a walk are distinct, it is called a trail.- A closed trail is a circuit.

- A circuit with distinct nodes is a cycle (no self-intersections).- A trail with distinct nodes is a path.

Side Remark: The parallel with category theory is this: points correspond to objects and arcs to morphisms between objects (under representations).

Connected Graphs and Components

- Def. u & v are connectable iff there is a path joining them.

- Def. A graph is connected iff any two nodes can be joined by a path.- Problem 1: Check that this is the same as being connected by a trail.

- Def. The connected components of a graph are the maximally connected subgraphs.

- Problem 2: Explain the 1:1 correspondence between connected components and the partition of nodes defined by the equivalence relation "connectable".

Eulerian Graphs- Eulerian circuit: a circuit containing all the arcs (only once) .- Eulerian graph: has an Eulerian circuit.- Example/Exercise 2: K3, K5 are Eulerian, but K4 is not.- Theorem (An early result of Graph Theory: Euler 1736)A connected non-trivial graph is Eulerian iff all the nodes have even degrees. Proof:"=>": Eulerian circuit can be represented on a circle (has distinct arcs). Each occurrence of v <-> 2 arcs.Step 1: Remove loops (changes deg by 2).Step 2: Distinct occurrences of v have disjoint pairs of arcs."<=" Exercise 3: Read the proof & outline the steps, using the - Lemma (Cycle decomposition of a graph)An even degree graph has a decomposition in cycles.Proof: by induction.

The Koenigsberg Bridges Problem

- The graph associated to the city of Koenigsberg: contract "land regions" to points, and join them by arcs representing the seven bridges, as in fig. 1.5.

- Exercise 2.2/8: Solve the Koenigsberg bridges problem.

Hamiltonian Graphs

Arcs and Nodes are dual concepts, so ...

- Hamiltonian cycle is a circuit (distinct arcs) containing all the nodes once (distinct nodes).- Hamiltonian graph: having a Hamiltonian cycle.

- Example: K2 not Hamiltonian (can't close the trail), but Kp, p>2 is Hamiltonian (v1,v2...vp,v1) (1:1 when mapped on the circle).

- Being Hamiltonian is not a topological property of a graph: fig.2.5. Efficient methods to determine the property are required in many applications.

A Hamiltonian Test

- Adjoining an arcs uv to a graph G: E(G+)=E(G)+ uv.

- Theorem (O. Ore 1960)If G is a simple graph with N(G)>2 and for all non-adjacent nodes u & v, deg(u)+deg(v)>=p,then G is Hamiltonian.

Side Remark: Being Hamiltonian (all nodes once without edge repetitions) is not dual to being Euler (all edges once, with possible repetitions of nodes).

2.3 Colorings- Def. G is k-colorable if there is a k-coloring (labeling) with at most k colors, with no adjacent nodes have same color.- Remark: G k-colorable iff Hom(G, Kn) is not empty.- Examples 1) An even cycle Z_2n is 2-colorable (Z2n->Z2 morphism)2) An odd cycle Z_(2n+1) is not 2-colorable.2) Kn is k-colorable only if k>=n3) Graphs with loops are never colorable (by def.)- Theorem G is 2-colorable iff there are no odd cycles.Proof. "=>" obvious; "<=" Each component of G is colorable (Idea: walk the "spanning tree" and label the nodes "left/right").

K-Degenerate Graphs

- Def. G is k-degenerate iff there is a listing of its nodes v1,v2...,vp such that vi has most k neighbors before itself, in the list v1,v2...v(i-1). - Theorem: G loopless k-degenerate => (k+1)-colorable.- Example: Fig 2.13 is a 4-colorable graph.

- Def. the maximum degree of a graph G is D(G)=max deg(v).

- Exercise 3: Prove that any loopless graph is D(G)+1 colorable.

- Proposition: If G contains K(k+1) => not k-colorable.

2.4 Planarity- Def. plane graph: embedding of G in R^2; planar graph: there is an embedding of G in the plane (otherwise: nonplanar graph).- Example: K5 is the smaller non-planar graph.

- Def. region, exterior region and perimeter of a plane graph (embedding!). - The Jordan Curve TheoremEvery plane loop divides the plane into two regions such that any arc between them intersects the curve. - Theorem (Euler’s Equation, 1758)If G is a connected planar graph: nodes - arcs + regions=2.Proof. By induction: p=1; p>1 contract an open arc & count.

Linking Distance

- Def. linking distance l(u,v); length of a walk, path, circuit etc., circumference of a region - LemmaFor any plane graph with q arcs and r(k)=number of regions with perimeter k:

2q=Sum_k k.r(k). Proof: Each arc belongs to two regions. - Corollary G connected simple planar graph (p nodes, q arcs) 1) q<=3p-6, 2) G 2-colorable => q<=2p-4. - Proposition: K5 and K(3,3) are not planar.

Characterization of Non-Planar Graphs- Theorem (Kuratowski 1930) G is non-planar <=> it contains a K5 or K(3,3) subgraph.- Def. plane map: planar graph; k-colorable map: regions are colored using k colors or less (For a k-colorable plane map, the associated planar graph is not necessarily k-colorable).

- The Four Color Theorem: Every plane map is 4-colorable.- Lemma: A simple plane graph is 5-degenerate and 6-colorable.- The 6-Color Theorem: Every plane map is 6-colorable. - Proposition: An Eulerian plane map is 2-colorable.Proof: decompose in disjoint cycles & count "depth" mod 2;C1 C2=frontier in unique cycle: one "in", the other "out".

2.5 Graph Homeomorphisms(When are two graphs equivalent?)- Lemma f:G->H homeo => preserves nodes of deg>2 and their degrees.- Def. reduced degree sequence: deg(v)<>2.- Homeomorphic => same reduced degree sequence. - Proposition: f:G->H homeomorphism => G+uv ~ H+f(u)f(v) - Def. topologically reduced form: reduce degree 2 vertices. - Def. adjacency matrix of G: a(i,j) = |Hom(i,j)|; the adjacency matrix depends on the ordering of the nodes.

Equivalent Graphs(Classifying Graphs Modulo Isomorphism)

Theorem (Adjacency operators classify graphs)Two graphs are homeomorphic iff their topological reduced forms share an adjacency matrix.Proof: First reduce the graph. "=>" by the Lemma; "<=" Arcs are homeomorphic to [0,1]. Intrinsic Description of Graphs & Equivalence- Graphs as generalizations of set relations (categories).- Mappings between relations (functors).- Graph homomorphisms (not "homeOmorphisms"):

f:(N(G),A(G))->(N(G'),A(G')).- Graph isomorphisms <-> graph homeomorphisms.

Applications to ChemistryChirality of Spatial Graphs- All graphs can be embedded in space (3D).- The mirror image of a graph: G*- Two graphs are isotopic if R^3 can be deformed by a homeomorphism to map one graph onto the other.- If G is equivalent to G*, then it is called achiral; otherwise it is chiral.

Molecules and Graphs- Molecules can be modeled as spatial graphs- There are chiral molecules such that the L/R-forms have totally different properties.- Synthesis of molecules having molecular structure a spatial graph (e.g. H. Simmons & A. Paquette, 1981).

The Mobius Band & Molecules- The Mobius band with n-bridges Mn: the graph obtained by connecting the boundary with n transverse and parallel segments.

- D. M. Walba: synthesized a Mobius band M3 molecule.

- Chemistry found that M3 & M3* do not mutually change into one another ... => Conjecture: M3 is chiral!

- It can be proved using knot theoretical techniques.

- Theorem: Mn, n>=4 is chiral; so it cannot be rigidly achiral.

- Theorem: M3 is not isotopic to M3*, if preserving the bridges.

Ch. 3 Surfaces

More on Surfacesfrom Wikipedia

Introduction

- Graphs: 1-dim objects (nodes & arcs: 0/1-D disks)- Surfaces: 2-dim (topological) objects- Greeks studied metric properties, - 18th century – study of topological properties- Abel, Jacobi, Riemann studied them in connection with elliptic integrals, easier to deal with if viewed as - Integrals in complex variables & Riemann surfaces - Topological classification: Euler, Gauss.

- The surfaces considered will be compact: closed & bounded.

Polygonal Presentations- Plane (non-compact), the disk or a topological space homeomorphic with one (2D with a 1-dim boundary), sphere (closed and bounded, border or boundary), torus, double torus- Flattening by cuts: cylinder, sphere, torus, double torus- Pasting: reversal of a cut (gluing).- Def. polygonal representation P: polygon together with a pasting scheme (labels showing how to pair edges to be glued)- Examples: rectangle -> cylinder, rectangle -> torus- P may have several components: 1) Torus (fig. 3.5), 2) Double torus (fig. 3.6)- Def. skeleton G(P) of a polygonal presentation P is a graph consisting of the vertices and pasted edges (after gluing) of P.- Examples

Assambling Surfaces

- Graphs are built out of points and arcs (0/1-disks). Surfaces are constructed from polygonal presentations, which include regions (2D-complexes).

- Reverse problem: identify the surface from a polygonal presentation (construct the 3D embedded surface)- Examples: figs 3.7, 3.8, 3.9

- Example 3.10 (sphere),

- Example: Moebus band (surface with boundary).

- Surfaces are of two types: closed or with boundary.

Properties of Presentations

1) Reversal of two arrows that carry the same label does not affect the resulting surface

2) The union of two edges sharing a degree two vertex (composition) throughout a presentation, does not change the resulting surface (see Fig.3.7 right).

3) The graph G(P) is connected iff the pairing of the labeled edges connect all the polygons of the presentation P.

Non-orientable Surfaces- Klein bottle (fig 3.11): no 3D embedding, but it has a 4D embedding with time as a 4th coordinate- Projective plane R3/R = S2/{+1,-1} (Fig. 3.14): - No 3D embedding, - 4D-embedding - Def. S embeddable in T iff there is f:S->T embedding- Examples: - Planar graphs are embeddable in the plane - Projective plane and Klein bottle: embeddable in 4D - Moebus band and cylinder: embeddable in 3D - Conclusion: every polygonal presentation represents a surface, embeddable or not in R3.

3.2 Closed Surfaces

- Closed surfaces have no boundary (edge-like frontier); they are obtained by presentations such that all edges were glued in pairs.- Goal: classify polygonal presentations up to homeomorphisms.- Def. homeomorphism of surfaces: there is a continuous function, having a continuous inverse.- Homeomorphic surfaces have identical presentations.- Analyze presentations - Vertices and nodes: before and after gluing (fig. 3.21) - Equivalent vertices are mapped to the same node - Interior angles (of P) and angles of G(P) on S (fig. 3.21) - Arcs and their inverses (fig. 3.22)

Skeleton Presentations

- Example 3.2.1: Given a 1-polygon presentation, determine the nodes and their symbolic representation: the cyclic list of arcs outgoing from the node.

- Example 3.2.2: Given the presentation P, determine the nodes and their lists of arcs. Construct the associated skeleton, the graph G(P).

- Def. the list of nodes, together with the corresponding cyclic list of arcs forms the presentation of the skeleton G(P) of the presentation P.

- Goal: show that presentations are classified by the Euler characteristic and the geometric orientability character.

Surface Invariants- Euler characteristic X(P) = nodes – arcs + polygons- Examples: 3.4 (torus), 3.5, 3.11, 3.23- Orientability character: sense of traversing the perimeter (2) - Edge consistent with the orientation - Orientation of P: orientations of the components- Examples: 3.25, - Coherent orientations: each edge occurs twice with opposite orientations - Orientable presentation: there is a coherent orientation- Otherwise they are called non-orientable; ex. 3.27- Orientable surfaces have two faces; embedded, they separate the space: interior & exterior.- Non-orientable surfaces have "one face" (when embedded).- Examples: 3.2.4-3.2.7

Equivalent Presentations- Modifications (moves or transformations, and their inverses) of presentations, which yield homeomorphic surfaces: 1) Transverse cut & disjoint pasting (3.35)2) Partial cut & self-pasting (3.36)3) Subdivision & unification (3.37)- Def. P & P’ are cut-and-paste equivalent iff one can be obtained from the other using the moves 1-3 - Proposition: S~S’ iff they have cut-and-paste equivalent presentations. - Lemma: Moves 1-3 preserve: connectedness, the Euler characteristic and the orientability character.

Normal Presentations- Commutators [a,b]=aba^(-1) b^(-1)- Normal presentation: encode polygones by their cyclic sequence of their edges relative to either orientation: Orientable: Pn=Prod [ai,bi], n=(2-X(P))/2, Non-orientable: P'n=Prod ai^2. - Theorem (Normal presentations 1:1 homeo classes of surfaces)Every P is cut-and-paste equivalent to exactly one normal presentation.- Corollary (Classification of Surfaces): Every surface is homeomorphic to one of S0, Sn or Sn’, corresponding to presentations Pn or P'n.- Corollary: S1~S2 <=> Same Euler & orientability character.- Corollary: X(S)<=2 with equality iff S=S0.

3.3 Operations on Surfaces

- Objectives: 1) Describe surfaces in R3 (orientable) or R4, i.e. classify embeddings of surfaces in R^n.

2) Describe operations on surfaces, which will produce new surfaces: surfaces with boundary. They correspond to presentations where some of the edges are not labeled, i.e. are not supposed to be glued.

3) Understand how the operations on surfaces relate to corresponding operations on presentations.

Cutting and Gluing Surfaces

Excision- Cutting along a closed cycle, which determines on the surface two disjoint regions (Jordan curve on the surface).- The resulting two regions are surfaces with boundary, the cutting cycle (two distinct homeomorphic copies).- Successive excisions can be performed, as long as the cutting cycle is interior, i.e. does not intersect the boundary.

Gluing- Identifying (pasting) boundary cycles, or union of cycles when several excisions are performe, of two distinct surfaces with boundary.

Adding Surfaces: The Connected Sum

- Definition: The connected sum of two surfaces 1) Excise one disk from each surface (basic 2D-blocks), 2) Glue (identify) the borders obtained by excision.- Example: connected sum S1+S2 (the index=genus).- The isomorphism class of the resulting surface is independent of the size or the position of the cut. - Surface addition is “tube connection” of two distinct surfaces.

Properties of the Connected Sum- Proposition (Relation to invariants) 1) X(S1+S2)= X(S1)+ X(S2) - 2, 2) S1+S2 orientable iff S1 and S2 orientable. Proof: 1) Consider presentations such that the excision cycle is part of the presentations, consisting in n nodes and arcs. - Corollary: a) S(m)+S(n) ~ S(m+n), b) S(n) ~ sum of n tori S(1), so S(n) can be embedded in R3. - Corollary a) S(m)+S’(n)=S’(2m+n), b) S’(n) can be embedded in R4. c) S’(n)+S’(m)=S’(n+m)

- Conclusion: A surface is orientable iff it can be embedded in R3.

Handle Addition- Excision of disks produces new boundary components; identifying boundary components (circles) reduces their number. - Alternatively, this operation can be viewed as gluing handles (h), i.e. tube connection on one surface: S+h if circles have opposite orientations on the sphere (handle addition) S+h’ if circles have the same orientations (cross-handle addition).- Definition: the genus of a surface g(S) is the number of holes / handles, or the number "n" of labels "a" in the definition of the normal presentation.

Properties of Handle Addition

- Propositiona) g(S+h)= g(S)+1,

b) S orientable iff S+h orientable c) g(S+h’)=2g(S)+2,

d) S orientable => S+h’ non-orientable

e) If S’ non-orientable => 1) S’+h and S’+h’ non-orientable,2) Their Euler characteristic is X(S)+2.

Fattening a Graph

- Replacing the constitutive elements of a graph, the nodes and arcs with spheres and cilinders is called fattening the graph.

- Example: fattening K4. Removing 3 handles, without disconnecting a sphere, results in a sphere. So, the surface is a sphere with 3 handles: g=3.

- Fattening is an important operation with applications to Quantum Field Theory, allowing to pass from Feynman graphs to Riemann surfaces.

Cross-cap Addition

- Def. cross-cap addition is excision followed by the reverse gluing of the resulting cycle, i.e. divide it into two labeled arcs with the same orientation, and then glue them as for the projective plane: S'=S+c.- Propositiona) S+c is non-orientable,b) S orientable => g(S+c)=2g(S)+1,c) S non-orientable => g(S+c)= g(S)+1. - Corollary The non-orientable surface of genus n is a sphere with n caps:

S’(n)=S(0)+n c.

Movies - Examples of SurfacesThe Real Projective Plane- Authors: Bothmer & OliverLabs- Steiner's surfaces (~ Immersed RP2)- Boy's Surface- Roman's Surface- "Visualizing" projective plane: if it is 2D and satisfies the projective geometry axioms, it is an RP2!

Moebus Strip = Cross-cap- Moebus band + disk (capping the M-band) = RP2.

Klein Bottle: Bothmer & liverLabs

Other: Gluing a torus, Stereographic projection

3.4 Bordered Surfaces- Bordered surfaces have bordered polygonal presentations such that all arcs are labeled, and a label occurs at most twice; in other words, only some of the edges are glued together. It is assumed that the pasting edges connect all the polygons.- Border arcs: appear once; their vertices: border nodes.- The skeleton of a bordered presentation G(P) is the graph determined by the pasted nodes and arcs.- The arcs remaining exposed after the pasting procedure form the border cycles, which can be identified by the node-computing technique (determining the nodes on the resulting surface).- Number of border cycles: b(P).- Examples 3.4.1, 3.4.2.- Note: angle chains at border nodes are linear lists, beginning and ending with border edges.

Closure of a Bordered Surface- Closure of a bordered surface C(P): cap the border holes.- Proposition: a) X(C(P))= X(P)+ b(P), b) C(P) orientable iff P orientable.- The bordered surface (surface with boundary) of a bordered presentation: S(P).- A closed surface with b(P) disks excised/removed (punctures or perforations) is a bordered surface.- Examples: the disk S(0,1)=S(0)-disk, the open cylinder S(0,2) is a sphere with two disks excised: S(0)-2 disks.

- In general: S(n,b)=S(n)-b disks, S'(n,b)=S'(n)-b disks.

Isomorphic Surfaces

- Isomorphic surfaces have the same Euler characteristic and orientability character: X(S(n,b))=2-2g-b c(S’(n,b))=2-g-b- Theorem (X, O, b is a complete set of invariants) S1~S2 iff they have the same X, O and b. - Genus of a bordered surface = genus of Sc, i.e. the closed surface obtained by capping the border disks:

g(S(n,b))=n, g(S'(n,b))=n.

- Example: The disk g=0 & b=1; cap => sphere g=0.

Twisted Bands

- B(n): the band with n twists, embedded in R3 (Non-isotopic embeddings).

- B(n) is homeomorphic to B(n mod 2), i.e. with the cylinder S(0,2) when n is even, or with the Moebus band S'(1,1) for n odd (Once perforated projective plane).

- Proposition If S < R3 and B(2k+1)<S => S is non-orientable.

- Example 3.4.10: Bordered surface presented as a ribbon graph.

Excisions on Bordered Surfaces- Excisions on a bordered surface: 1) Separating excision: it disconnects the surface. 2) Border-to-border (B2B): both endpoints lie on borders. 3) Border-to-interior (B2I): open arc with only one border point.- Examples: Fig. 3.70.

- Proposition a) B2I excision does not change the type of S; b) B2B joining different border cycles are non-separating, and X(S’)= X(S)+1. If S orientable => S' is and g(S)=g(S').- Propositiona) Separating B2B excisions begin and end on the same border: c(S)= c(S’)+ c(S’’)-1. b) If S is orientable => S’ & S’’ are orientable and g(S)=g(S’)+g(S’’).

3.5 Riemann Surfaces- Calculus of complex variables: pioneered by Cauchy, Abel and Riemann; developed in the 19th century.

- Complex curves: w=f(z) the graph of a function, or implicitly defined by a polynomial function P(z,w)=0, is a 2D-real surface.

- Def. A Riemann surface (RS) is the complex curve P(z,w)=0, where P is a complex polynomial in two variables.

- "The study of Riemann Surfaces is a very rich and important area of mathematics ."

- RS can be presented as polygonal presentations

Example 1 - The Cylinder

1) Start with 2 disks with one excision each, with arcs labeled.2) Identify the arcs: a1+ = a2-, a2+=a1-.3) Cap the disks to obtain a polygonal presentation: - Compute genus: g=0 => C(RS)=S(0): sphere; - The resulting RS (b=2) is S(0,2): cylinder; - Compute the node lists of arcs.

4) The construction and interpretation has historic value: - Visualize the disks stacked: S1 on top of S2; - Identify A1 and B1, A2 and B2; - Imagine the excision path a1 on top of a2. - The crossing of the the bands joining the two sheets can avoid intersection in 4-dimensions (e.g. time).

Example 2 - The Pair of Pants

1) Start with 3 disks with one excision each, with arcs labeled.

2) Identify the arcs cyclically: a1+ = a2-, a2+=a3-, a3+=a1-.

3) Cap the disks to obtain a polygonal presentation: - Compute Euler char & genus: g=0 => C(RS)=S(0): sphere; - The resulting RS (b=3) is S(0,3): a "pair of pants"; - Compute the node lists of arcs.

4) It can be interpreted as a genericbinary operations, with 2 inputs andone output. In quantum physics it modelsinteractions: TQFT, String Theory, CFT.

Example 3: n Sheeted RS with 1 Cut1) Start with a stack of n disks S1 ... Sn with one cut each2) Paste cyclically: a1+=a2-, a2+=a3- ... an+=a1-.3) Compute the invariants of the resulting orientable surface: - Euler characteristic: 2-n+n=2, so genus is again g=0; - Boundary components: b=3;4) The surface is S(0,3), i.e. again a punctured sphere.5) The pasting rule is encoded as a permutation P:

aj+=aP(j)-.6) A permutation is a product of commuting cycles. If there is more then one cycle, then the polygonal presentation would be disconnected, resulting several RS; therefore one may assume the permutation is one cycle (RS it is connected). 7) Renumbering the disks, results in the cyclic permutation:

P=(123...n), i.e. 1->2, 2->3 ... n->1.

More Cuts - Higher Genus RS

- More RS (g<>0) are obtained using more complex excisions.

- Example 4: 3 Sheets and 3 cuts 1) Start with 3 disks, with 3 radial cuts each, from the center: Disks: Si, cuts: ai, bi, ci (+ & -), i=1,2,3. 2) Identify the cuts cyclically: aj+=a(j+1)-, bj+=b(j+1)-, cj+=c(j+1)- (+ mod 3). 3) Determine the invariants of the resulting capped surface: Orientable: yes (orientability of the disk & pasting rule) Euler characteristic: A(Ai), B(Bi), C(Ci), Oi(Oij) ( there are 9 points Oij on the polygonal presentation, mapped to 3 on S) (3+3) points -9 arcs + 3 polygons => X(S)=0 => g=1. 4) The punctured RS is S(1,3).

The General Riemann Surface- Def. A Riemann Surface is determined by a set of k excision paths ai=OAi on the disk, starting radially from the center O, together with a set of k permutations Pi of {1..n}, corresponding to the k excisions, such that: 1) P1 P2 ... Pk=Id, 2) {Pi} act transitively on {1..n} (Any two indexes 1<=j1,j2<=n are connected via a sequence of permutations Pi).The RS results from the following construction (presentation):1) Start with n disks Sj, each with k cuts aij (think of the capped disk with cuts as the unit disk with the k-th roots of unity on the Riemann sphere);2) Identify: aij+ with aiP(j)- (Think of <A, PA> as transitions & propagator).

The Skeleton of a Riemann Surface

- View the disks Sj and the cuts aij stacked (S and ai).- The Nodes of the skeleton G(RS) on the RS: 1) O contributes with n-nodes Oj on the RS; lists of arcs: Oj: a1j-, akPk(j)-, a(k-1)P(k-1)P(k)(j), ..., a1P1...Pk(j)-=a1j-. Each permutation P(j) acts on the stack of - cuts a(j+1), maping them to the stack a(j): P(j):a(j+1)->a(j). Then the stack of points Oi (Identifying Oij+ with OiP(j)-) is mapped to the stack O(i+1) by the permutation P(i). 2) The points Aij (ends of the arcs aij+/-) of a stack Ai, are related by a cycle c=(j1 ...js) of the permutation P(i):Ai,c: (aij-)^{-1}, (aiPi(j)-)^{-1}, (aiPi^2(j)-)^{-1} ..., (aiPi^s(j)-)^{-1},and contributes to one point on the surface.

The Euler Characteristic of the RS- So, to each permutation Pi, corresponds a number of nodes on the RS equal to the number of its cycles ||Pi||: the "volume" of an orbit of the permutation (degree or length <-> the "space-like" width, commutative, as opposed to its order - the "time-like length").

- The capped surface has Euler characteristic: X(RS)=(n+Sum ||Pi||) Points - (k n) Arcs + n (Disks),

therefore proving the following theorem.

- Theorem (Characterization of presentations of RS) The RS(P1,...,Pk) has genus (2 + (k-2)n - Sum ||Pi||)/2.

- The relation with Galois Theory will be explored in MAT 407.

Example 4 Revisited

- Adapt the notation to the general case: 1) Start with 3 disks Sj, with 3 cuts each: aij 2) Pasting scheme: aij+=ai(j+1)-; Pi(j)=j+1, P1=P2=P3=(123).

- The nodes (The the inverse ^(-1) is omitted for Ai): O1: a11-, a32-, a23-, a11- A1,c: a11-, a12-, a13-, a11- O2: a21-, a12-, a33-, a21- A2,c: a21-, a22-, a23-, a21- O3: a31-, a22-, a13-, a31- A3,c: a31-, a32-, a33-, a31-

- The "norms" are ||Pi||=1, X(RS)=(3+3)-3.3+3=0, g=(2-X)/2=1.

- The RS is a torus with 3 punctures.

Example 5: a RS with 2 Sheets

1) Start with two disks Sj (n=2) and two cuts Ai (k=2), away from the center O.2) Paste cyclically: aij+=ai(j+1)-, Pi(j)=j+1, P1=P2=(12).3) The nodes (omit -, inverses, and last point: represent as cyclic lists, since there are only interior nodes): O1: a11, a22; A1,c: a11, a12; O2: a12, a21; A2,c: a21, a22.

4) X=(2+2)-2.2+2=2 => g=0.It is the Riemann Sphere with two punctures (N & S poles).5) Figure: w^2=z.

Example 6: RS with 5 Sheets & 4 Cuts

1) Start with n=5 disks Sj, with 4 cuts each Ai.2) Paste corresponding to the permutations: P1=(143)(2)(5), P2=(125)(34), P3=(13)(24)(5), P4=(123)(45) a11+=a14-, a12+=a12-, a13+=a11-, a14+=a13-, a15+=a15-; etc.3) Nodes: 5 Os, 3A1, 2A2, 3A3, 2A44) Euler characteristic: (5+3+2+3+2)-4.5+5=0, g=(2-X)/2=15) RS is a torus with 5 punctures.

RS and Algebraic Equations

- A typical algebraic analytic function is f(z)=z^n, which wraps the disk onto itself n-times. It is a cover map except at the origin, where it has a branching point z=0.

- The corresponding surface is a RS obtained from n-stacked disks, or sheets, and associated to the cyclic group Zn, generated by the cyclic permutation (1...n).

- More general complex functions, the Laurent polynomials:... a(-2)/z^2+a(-1)/z+a0+a1z+a2z^2...

have zeros and poles, with various multiplicities. How many (independent) types of such functions are there, is determined via the Riemann-Roch Theorem.

More on Riemann Surfaces

- More on this topic in MAT 407 (see also MAT 447).

- Optional reading (handout): Galois Theory and Riemann surfaces, by Eric Reyssat, From Number Theory to Physics , M. Waldschmidt, L. Izykson (eds.), 1995.

Ch. 4 Graphs and Surfaces

The interplay between graphs and surfaces.

4.1 Embeddings and Their Regions

- Def. A graph embedding on a surface is a mapping from the graph to the surface which is a homeomorphism onto its image.

- Example: any planar graph can be embedded on a disk.

- Proposition: A graph can be embedded on a sphere iff it is planar. Proof. Use the stereographic projection.

- Goal: given a graph G and a surface S, can the graph be embedded on the surface?

The Genus of a Graph- Proposition Every graph embeds on some orientable surface. Proof. Fatten the graph into a surface; embed the surface in R3, and project the graph on the surface in a fixed direction.

- Proposition Every graph G has an integer g(G), such that G is embedable on Sn iff n>=g(G). Proof. If G<Sn and m>n then G<Sm by adding handles on Sn avoiding the embedded graph (compact set).

- Def. g(G) is called the genus of the graph.

- Examples: g(K4)=0, g(K5)=g(K,3,3)=1 (not planar).

Regions of the Embeddings

- Given an embedding of a graph G on a surface S, excising the graph's image decomposes the surface into regions.

- Def. the regions of a graph embedded on a surface f:G->S are the connected components of the complement of the image, after compactification (adding the boundary/excision):

regions(f)=components of S-f(G).

- Example: an embedding of K4 on a disk has 4 regions which are triangulated disks.

- Example 4.1.5: a toroidal embedding, with two regions: 1) a polygonal presentation of a cylinder, 2) a 2nd cylinder.

Regions and Their Visualization

- A region is always a surface with boundary, and if the surface has genus zero, it is a polygonal presentation.

- A region mapped on the corresponding surface, i.e. after identifying the various portions of the boundary of the region, is called a visualized region.

- A visualized region does not have to be a surface with boundary, since its boundary may be itself a graph, not just a union of cycles: Fig. 4.7.

- The perimeter of a region is the union of its borders, i.e. the connected components of the boundary.

Surface Presentations

- The collection of regions corresponding to an embedding G->S is a surface presentation, since the surface can be obtained by gluing the pairs of arcs from the perimeter, having the same labels.

- A surface presentation is a generalization of polygonal presentations, in that the components of the presentation are surfaces with boundary cycles, representing the gluing data to construct the surface.

- Examples: Fig. 4.7, 4.8, 4.9.

4.2 Polygonal Embeddings

- Polygonal embeddings are the graphs embeddings whose regions are polygonal presentations. The skeleton of the polygonal presentation is the image of the embedded graph.

- Theorem (Euler-Poincare Equation) If G(p,q) is a graph embedded on the surface Sn then: p-q+r=2-2n. Proof. I.e. the EP-characteristic of the presentation corresponding to the regions of the embedding corresponds to the EP-characteristic of the surface, and therefore with its genus.

Rotation Systems- Def. A rotation system of the graph G is a permutation Rho = Prod c(vi), i=1..p(G), where c(vi) are cycles permuting the arcs emanating from the node Ai of the graph.

- An enumeration Ai, of the nodes of the graph, together with a cyclic order at each node, defines a rotation system. - A graph embedding eps:G->S on an orietable surface S defines such a rotation system.- Example: Fig. 4.3, 4.6.

Rotation Systems and Graph Embeddings- Def. The arc involution of a graph is the permutation of its arcs reversing the orientation: IG(a)=a^(-1).

- Proposition Let eps be a polygonal embedding of G on S. The perimeters of the polygons (regions) of eps are the cycles of the composition Rho(eps) o IG.

- Proposition Any rotation system of a connected graph G is a clockwise rotation system for an orientable polygonal embedding of G on a surface S. - Example 4.2.6

4.3 Embedding a Fixed Graph

- Embeddings of G in a bordered surface ó embeddings in a surface without boundary - Proposition: G embeds in S(n,b) iff n³g(G) - Minimal embedding- A minimal embedding of a connected graph is also polygonal- Amalgamation of two graphs at a vertex - Proposition g(G1+G2)=g(G1)+g(G2) - Proposition g(G)=Sum g(Ci) (Ci: connected components)

4.4 Voltage Graphs

- Def. voltage graph: connected graph G with edges labeled by elements of a group G, s.t. j(a-1)=j(a)-1 for all directed arcs a- Examples (enough to specify j for representative arcs) - Def. covering graph G’ of a voltage graph G: Nodes: G x G , Arcs: if a=(u->v) is in G then ag=((u,g)->(v,j(a)g)) is in G’- Interpretation: G is a discrete space-time, G x G is a “principle bundle” and the voltage graph is a “connection” (fix a base point => a horizontal section: parallel vectors via translations) - Examples

Coverings

- Lifts (covers) of arcs in G: a & g => ag- Lifting a walk in a cover of G- Def. isotropy group at a node G(v)={j(walk at v)}- Def. G is full iff G(v)=G for some v (hence for all v)(“irreducible connection”) - Theorem G connected => number of components of the covering graph G’ is [G,G(v)].

Embedded Graphs

- G embedded on S with rotation system r=r(u)r(v)…- Let n=[G,G(v)]- If r(u)=(abc…) is a typical cycle of r at u then r’(ug)=…(? … g must be in G!) … is a rotation system for G’- Recall: a rotation system determines a graph embedding on a surface- Examples - Relation between regions of G and G’=G xj G - Some results on coverings …

Branched Covers

- Def. rounding of a polygon, k-cover of the unit disk, polygonal k-branching f:P->P’, - Def branched covering: f:S->S’ s.t. there are polygonal presentation P, P’ of S and S’ for which the restriction of f to any polygon P of P is a k-branching onto P’ a polygon of P’. Branched Cover of a Voltage Graph- Voltage graph (G,j,G), rotation system r and associated embedding onto the surface S- Let S be the orientable surface determined by the lifted rotation system r’

Covering Maps

- Theorem There is a branched covering f:S’->S which maps G’ onto G, and for every lift R’ of a region R the restriction f:R’->R is a k-branching where k is the order of j(R) - Def. covering map f:S->S’ which is a k-branching on each region with k=1 Examples1) Two-loop bouquet on the torus2) Three-loop bouquet

Ch. 5 Knots and Links

Graphs -> Surfaces -> Space

Knots and Links: Graphs -> Space

Introduction

- Lord Kelvin (1824-1907) thought of modeling atoms as vortices or knots in ether (substance filling the space)- P. Tait, T. P. Kirkman, C. N. Little – worked on classifying knots (hoping to find a correspondence with atoms)- Knot Theory later found other applications in biology, chemistry, physics (Feynman diagrams), mathematics (classification of 3d-manifolds) etc.

5.1 Preliminaries

- Def. knot, link (its components are knots), projections into a plane, crossings (under/over), knot/link diagram- Def. unknot = knot isotopic to a plane loop; unlink- Def. equivalent knots: modulo ambient isotopy (in R3)- Goals - to find criteria 1) To distinguish non-equivalent knots2) To decide link ~ unlink; W. Haken 1961 (not practical; haven’t been implemented on computers)- Equivalent diagrams => equivalent links- Examples: trifoil (not equivalent with its mirror), 8-knot (equivalent to its mirror)

Rademeister Moves

- Plane isotopy: does not alter the crossings- Theorem (Reidemeister 1926) Two links are equivalent ó their diagrams are related by isotopies or a Reidemeister move:I) Remove a twist, II) Remove overlappingIII) Vertical pass over a crossing- Examples- Invariants: functions compatible with ~

5.2 Labelings

- To distinguish inequivalent links associate a system of modular linear equations: one equation to each crossing- Def. strand of a link (component) : any loop (no crossing) or any arc starting and ending at a crossing- Examples- Remark: not the strands of the associated braiding- Overstrands separates the understrands - Def. p-labeling of a knot (p odd prime) s.t. 1) It is not constant;2) At crossings the labeling equation holds: 2x-y-z=0 mod p (x labels the overstrand)

An Isotopy Inviariant

- The existence of a p-labeling is an isotopy invariant- Theorem if L1~L2 & L1 has a p-labeling => L2 has a p-labeling pf. check Reidemeister moves preserve p-colorability - Corollary trefoil knots are not unknots. - Finding a p-labeling ó solving a system of equation in the field Zp; example: the trefoil (p-labeling => p=3)- p-labeling does not distinguish knots in general (there exist inequivalent knots with no distinct p-labelings)

Labeling Links

- Def. split link diagram (can be separated into two disjoint diagrams); splittable link- Splittable digrams have “locally constant” p-labelings - Proposition If a link with at least 2 components has no p-labeling for a fixed p => it is not splittable - Example: the Borromean rings 1) Have no 3-labeling (solve system => constant solution);2) => Not splittable - Project: study the generalization of p-labelings (not locally constant: labelings using k+1 labels, if k=no. of components)

Solving Systems in Zp

Solving systems in Zp- Add a multiple of an equation to eliminate variables, not having to invert modulo p. The resulting equation: Ax-Ay=0. Any odd prime divisor of A provides a p-labeling: x=0, y=1.

5.3 From Graphs to Links and onto Surfaces- Arcs of graphs form knots and links- Orientation of a knot, oriented link (2k orientations)- Equivalent oriented links, reversed oriented link- Left/right crossing (overstrand -> understrand & orientation)- Def. linking number of two components of a link: ½ sum of types of crossings (right=1, left=-1)- Theorem K1 & K2 disjoint knots => lk(K1,K2) is an invariant of the link K1 È K2 - The linking number is an integer (despite the ½ )

Placements of Graphs

- Switching a crossing (under -> overstrand)- Placement of K6- Moebus band: center line and border are linked- Application to surfaces: the non-orientable Sn can’t be embedded in R3

5.4 The Jones Polynomial

- Distinguishing isotopy classes of links is a measurement process- James W. Alexander 1928 – polynomial: link invariant- Vaugham Jones 1986 – polynomial invariant with deep implications to manifold invariants- Louis H. Kauffman – simplified the definition of the Jones polynomial - Jones polynomial V(L)Îk[A] & V(L)=V(Lr)- Skein relation satisfied by V(L): A4V(L+)-A-4 V(L-)+(A2-A-2) V(L0)=0 where the links differ at only one crossing and: L+ is a right crossing, L- left crossing, L0 no crossing

Examples and Properties

- Examples - The Jones polynomial is the product of two other polynomials (neither is a link invariant):V(D)=(-A)-w(D)<D>, where w(D) is the writhe of the link (numerical weight) and <D> is the bracket polynomial. Interpretation: 1) w(D) is used to shift the degree of V(D) (its coefficients are shifted to the left)2) The formula is reminiscent of a quantum amplitude Y=R eiS …

The Bracket Polynomial

- If D is a link then [D] is determined by the crossings- Labeling crossings “angles”: A-angle: counterclockwise from overstrand->understrand B-angle: clockwise from overstrand->understrand - Important Remark: “Borrowing terminology from electronic circuits, each crossing is a switch with two possible settings or connections” (indeed: quantum computing and superposition of gates/U-matrices, information flow etc.) - The links corresponding to A-connections (join A-A regions) and B-connections (join the B-B regions)

States and the Bracket Polynomial

- Def. a state S of the diagram D (basis element – info flow chart): each crossing is replaced by an A or a B-connection- The link is thought of as a superposition of basis elements determined by the crossings- Each state S is a diagram of some unlink (ex. fig.5.50); the number of its components: |S| (number of loops), - Number of A-connections of s:D->S is a(s) and B-connections is b(s)- Def. the bracket polynomial of D is: [D]=ås Aa(s) Bb(s) d|S|-1- Example

Representations of Links

- Def. the representation of a link as a superposition of states:y(D)= (coproduct or qubit 1|A>+0|B> for each crossing)

Properties of the Bracket Polynomial

1. [0]=12. [D]=A[DA]+B[DB], where DA & DB are obtained by changing just one crossing into an A/B-connection.3. [DÈ0]=[D]d - If rescaling [D] to [D]’=[D]d, replacing |S|-1 by |S|, then [0]’=d, (2) holds and [D]’ is a homomorphism. (Euler-Poincare mapping: sA <-> A->D->>S extension, and D@sAÅsB …?)

Compatibility with Rademeister Moves

- Is derived from the above properties (axioms of []?)- Note: d provides a grading, [D]Îk[q][d], q=a|A>+b|B>- Lemma [D] is invariant to type II R-moves iff (*) B=A-1 & d=-(A2+B2) Pf.- Nore: try complex numbers Z=A+iB, but for a different set of axioms, and A,B noncommutative: [D u 0]=[D], [D]=1, homomorphism …(use [| |] ó i [=] Wick rotation)- Lemma [D] is invariant to type III R-moves iff (*) pf. - Lemma the twist (over->under) ó -A-3 - For ribbons, the twist is essential and corresponds to multiplication by A+Bd=Z if d=i !! Then we need |Z|=1 (quantum state for the qubit!)

The Kauffman Bracket

The Kauffman bracket (Laurent) polynomial <D>- It is [D] when A & B satisfy: B=A-1 & d=-(A2+B2) The Writhe of a Link- Def. D oriented link, w(D)=å w(C) (+1/-1)- Examples- Properties: 1) w(D)=w(Dr), so w is defined for unoriented knots (and links)2) It is invariant to type II & III R-moves3) w(D with twist)=-1 + w(D)

The Jones Polynomial

The Jones Polynomial- Def. V(L)=(-A)-3w(D)<D>, where D is a diagram of the oriented link L- Examples: unknot, 3b - Theorem V(L) is invariant under R-moves, and therefore it is an isotopy invariant of oriented links - Jones polynomial is an invariant of unoriented links

Properties of the Jones Polynomial

- Proposition 1) V(L)=V(Lr) where L is an oriented link 2) K unoriented knot & oK and oKr its two oriented knots, then V(oK)=V(oKr) Skein relation of V(L)Oriented links: L+, L-, L0 satisfy the skein relationA4V(L+)-A-4 V(L-)+(A2-A-2) V(L0)=0 - Note: it is similar to a finite difference equation for V: B-alg->Poly (V’(L)=-V(L)/V(s+s*)- Notation: R=A2+A-2

Split Union of Links and Diagrams

- Def. L1 È L2 is called the split union (similar for D1 È D2)- Proposition V(L1 È L2)=- R V(L1) V(L2)- Power in R is the number of unlinked components (tensor algebra degree); it corresponds to: [D]=d [D1][D2] and <D>=-R <D1><D2> Sum of Links (“merger”)- Def. sum of links L1+L2 for disjoint links (eliminate an “annihilation-creation” pair or a pair of “max-min”); - Def. (in particular) connected sum of two knots - Proposition V(L1+L2)=V(L1)V(L2)

Mirror Link and Prime Knots

- Def. mirror link (obverse) Lm: apply the transformation (x,y,z)->(x,y,-z); compare with the 2-dim analog for diagrams- Remark: D and Dm corresponding to L and Lm, have all the crossings reversed (right into a left crossing)- Interpretation: changing spin up into spin down for L - Proposition1) [Dm](A,B,d)=[D](B,A,d) 2) <Dm>(A)=<D>(A-1)3) w(Dm)=-w(D) 4) V(Lm)=V(L)(A-1), - Def. prime knots: K ¹ K1 + K2 (its oriented versions cannot be expressed as connected sums)- Examples

Crossing Number

- Def. the crossing number of a Knot: the minimum number of crossings in any of its diagrams Some Graduate Research Projects- Jones polynomial & B-algebras of knots (Lie theory & DE)- Jones polynomial & OPEs- Links and information flow

5.5 The Jones Polynomial and Alternating Diagrams- Def. alternating diagram of knots (following a component of the link, R & Left crossings alternate)- Def. a crossing is an isthmus iff two of its angles belong to the same region; example: the 8 unknot- Def. a diagram is reduced iff it has no isthmus - Theorem The number of crossings in reduced alternating diagrams of a knot is an isotopy invariant of the knot

5.6 Knots and Surfaces

- Knot theory applied to topology; example: non-orientable surfaces cannot be embedded in R3- Conversely: theory of surfaces can be used to knot theory- Example: an unknot can be deformed into the plane, and bounds a disc- Every knot is the boundary of an orientable surface, called a spanning surface- Constructing spanning surfaces of a diagram 1) Shade the outside regions of a diagram (orientable) 2) Shading the interior surfaces (non-orientable)- Proposition Every knot has a spanning surface Pf. knot projections are graphs with degree 4 nodes. They define Eulerian maps => 2-colorable.

Seifert Surfaces

- Def. orientable spanning surfaces are called Seifert surfaces- L. S. Pontrjagin and F. Frankl 1930: all knots have a Seifert surface Constructing Seifert surfaces (Herbert Seifert 1934)- Orient the diagram D of a knot- At each crossing of D follow the “wrong” pair of strings, modifying the crossing (flip the switch to remove the crossing)- D => disjoint disks (Seifert circles)- The interior of a Seifert circle is a Seifert disk- The Seikert disks are stacked (the circles may be nested)- Restore the crossing: at each crossing paste a small arc having opposite orientations

Remarks

- In other words “localize the twists” and build the surface from stacked discs and twisted passages (½ a turn bridges); it is a RS with boundary (orientable), NOT ts-parametrized! - Alternative: at each crossing, swipe a surface as if time flows and space (a segment) fills them in between; It’s a ts-parametriation of a RS with boundary, but not orientable!(see fig. 5.68)

Applications to Physics

- Green’s Theorem (Stokes”s Th. on a surface with boundary, embedded in space)- Corollary: In R3 curl(F)=0 => F conservative ()

- Application to sums of knots- Def. The orientable genus: min genus(Seifert surface)- Theorem: The orientable genus of knots is additive- Corollary: K non-trivial => K+K’ non-trivial- Theorem: Knot K with alternating diagram D => genus(K)=orientable genus(D)

Ch. 6 The Differential Geometry of Surfaces

Introduction

- History- Work of Gauss => directions of geometry for 2 centuries

6.1 Surfaces, Normal, Tangent Plane

- Parametrized surfaces, normal xu x xv- Examples- Tangent vectors to curves, including parametric curves- Normal, unit normal

6.2 Gaussian Curvature

- Def. Gauss map: P|->P*=g(P)=normal(P)- Lemma If g(S) is a surface => TP(S)=TP*(g(P)) Curvature- Curve: rate of change of unit tangent w.r.t. arc length- Surface: curvature in a direction, total curvature=product o curvatures in orthogonal principal directions- Total curvature of a surface S0: K(S0)=+/- area(g(S0)) (depending weather g(P) is sense-reversing or not)- Def. Gaussian curvature K(P)=lim area(g(S0))/area(S0) (S0->P)

Gauss' Theorem

- Theorem(Gaussian curvature) K(P)=(LN-M2)/(EG-F2) (determinants)- Examples- Example: the graph of a function K=det(Hessian)/(1+||f’(P)||2)2- Corollary: K(S0)=òòD K(P)dA

6.2 The First Fundamental Form

- A surface contains curves forming angles: E, F, G allow to evaluate angles; - Length of a curve: ò(||X’(t)||)dt=- Def. the first fundamental form ds2=Edu2+2Fdudv+G2dv2- It’s a generalization of the concept of distance, by generalizing Pythagora’s Theorem; it’s also called a metric of the surface- Distinguish: topologic surface (topology) and metric surface (geometry)- Examples

Angles and Geodesics

- A surface contains curves forming angles: E, F, G allow to evaluate angles; - Length of a curve: ò(||X’(t)||)dt=- Def. the first fundamental form ds2=Edu2+2Fdudv+G2dv2- It’s a generalization of the concept of distance, by generalizing Pythagora’s Theorem; it’s also called a metric of the surface- Distinguish: topologic surface (topology) and metric surface (geometry)- Examples

Equations of Geodesics

- Proposition (Geodesics on a surface of revolution) If X(u,v)=(x(u), y(u) cos(v), y(u) sin(v)) Then the meridians (v=const) are geodesics Pf. 1) fundamental form; 2) q=0 satisfies Gauss’s GE. - Corollary: The geodesics of the spheres are arcs of great circles - Proposition (Equation of geodesics) g(t) geodesics iff q’(t)Ödet=…- Def. positive definite quadratic forms- f(x,y) positive definite ó A,C>0 and AC-B2>0- Proposition the fundamental form is a positive definite quadratic form.

6.4 Normal Curvature

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