Mirror-curve codes for knots and linksMirror-curve codes for knots and links

Ljiljana Radovic, Slavik JablanLjiljana Radovic, Slavik Jablan

3.-8.9. 2012 , Zlatibor3.-8.9. 2012 , Zlatibor

Snake with interlacing coil, Cylinder seal, Ur, Mesopotamia, 2600-2500 B.C.

9x5

Tchokwe sand drawings

``The Tchokwe people of northeast Angola are wellknown for their beautiful decorative art. When they meet, they illustrate their conversations by drawings on the ground. Most ofthese drawings belong to a long tradition. They refer to proverbs, fables, games, riddles, etc. and play an important role in thetransmission of knowledge from one generation to the other.'‘

(Gerdes, 1990)

Tchokwe sand drawings

"During the harvest month, Tamil women in South India draw designs in front of the thresholds of their houses. In order to prepare their drawings, they set out a rectangular reference frame of equidistant points. Then curves are drawn in such a way that they surround the dots without touching them. The (culturally) ideal design is composed of a single closed line."

P.P.J.Gerdes: On ethnomathematical research and symmetry, Symmetry: Culture and Science 1, 2 (1990), 154-170.

Tamil drawings

Tamil knots (“Pavitram” or “Brahma Mudi”)

Tamil Knots

``Leonardo spent much time in making a regular design of a series ofknots so that the cord may be traced from one end to the other, thewhole filling a round space...''

Bain, G.: Celtic Art - the Methods of Construction, Dover, New York, 1973. 

Plates

Number of curves(components) cin regular squaregrid RG[a,b]

c = GCD(a,b)

10×3

Celtic knots

Celtictangles

Celtic monolinear design withbroken symmetry

Celtic mirror curves

Mirror curvesMirror curves

Reflection in a mirror.

Square grid with a two-sided mirror between cells.

Mirror curve in a square grid 3x3 with two internal mirrors.

“Over-under” alternating.

Mirror-curves can be constructed any polygonal edge-to-edge tilingT of a part of an arbitrary surface. We propose the followingconstruction:

First, construct all different curves in T containing lines that connect different cell-edge midpoints until T is uniformly covered by k components. Then, in order to obtain a single curve, place internal mirrors according tothe following rules:

1) any mirror placed in a crossing point of two distinctcurves connects them in one curve;

2) depending on the position of a mirror, a mirror placedinto a self-crossing point of an (oriented) curve either does notchange the number of curves, or breaks the curve in two closedcurves.

Construction rules

Construction rules

Construction of a single mirror curve from the tiling (a)by connecting edge mid-points (b), tracing components (c) andintroducing a mirror (d).

(a) (b)

(c) (d)

Lomonaco, S. J. and Kauffman,L.H.: Quantum knots and mosaics, Quantum Information Processing 7, 2-3 (2008),85--115 (arXiv:quant-ph/0805.0339v1).

212=2 =

RII

412=4

Consider a rectangular grid RG[p,q] with edges of integer lengths p and q. Every internal edge will carry labels 2, -2, 1, and -1, where 2 denotes a two-sided mirror incident with the edge, -2 two-sided mirror perpendicular to the edge in its middle point, 1 the positive crossing +1, and -1 the negative crossing in the middle point of the edge (Fig. a). Mirror curves will be denoted by a list of lists (i.e., a matrix), containing labels of internal edges corresponding to rows and columns of the RG[p,q]. For example, to the labeled RG[3,2] (Fig. b) corresponds the code Ul={{-2,-1,-1,2},{1,2,-1,1},{2,1,-1},{1,-2,-1},{1,-2,-1}}.

Ul={{-2,-1,-1,2},{1,2,-1,1},{2,1,-1},{1,-2,-1},{1,-2,-1}}

2 = a mirror incident to an internal edge-2 = a mirror perpendicular to an internal edge-1 = negative crossing 1 = positive crossing

Reidemeister moves (knots and links are Reidemeister moves (knots and links are equivalence classes with regard to equivalence classes with regard to ambient isotopies)ambient isotopies)

I II III

I loopsII two standsIII three stands

I II

II III

Every unknot or unlink can be reduced to the code containing only labels 2 and -2.

For example, unknot with three crossings given by the code Ul = {{-2, -1}, {1, 1}} (Fig. a) after RII applied on the upper right crossing gives the code Ul={{-2, -2},{1, -2}} (Fig. b), which after RI applied on the remaining crossing becomes Ul = {{-2, -2}, {2, -2}}. The obtained code contains only labels 2 and -2, so it is unknot.

Minimal diagrams of mirror curves correspond to the codes with the minimal number of labels ±1.

In order to make the reduction (or ambient isotopy), we use themirror-moves.

Mirror moves

Reducing the code Ul={{-2,-1,-1,2},{1,2,-1,1},{2,1,-1},{1,-2,-1},{1,-2,-1}} of the 2-component link shown on Fig. a) to the 2-component unlink (f) is more complicated.

First we apply RI to the right lower crossing, and three moves RII, in order to obtain the code (Fig. b){{-2, -2, -1, 2}, {-2, 2, 2, -2}, {2, 1, -2}, {-2, -2, -1}, {-2, -2, -2}}, then the mirror-move to the first mirror in the upper row and obtain the code (Fig. c) {{-2, -2, -1, 2}, {-2, 2, -1,-2}, {2, 1,-1}, {-2, -2, -2}, {-2, -2, -2}}, RI and obtain the code (Fig. d) {{-2, -2, -1, 2}, {-2, 2, 2, -2}, {2, 1, -1}, {-2, -2, -2}, {-2, -2, -2}}, RI and obtain the code (Fig. e) {{-2, -2, 2, 2},{-2, 2, 2, -2}, {2, 1, -1}, {-2, -2, -2}, {-2, -2, -2}}, and finally, by RII we eliminate the remaining two crossings and obtain the code (Fig. f) {{-2,-2,2,2},{-2,2,2,-2},{2,2,2},{-2,-2,-2},{-2,-2,-2}} of the two-component unlink.

• Same as about mosaics and quantum knots and links, we can pose many similar open questions about mirror-curves, e.g., the question about mosaic number.

• For knot mosaics this will be the minimal squarefrom which certain knot or link can be obtained. For mirror curves this will be p+q.

• In RG[p,q] every alternating knot or link is given by a code which contains either 1 or -1, but not the both of them.

Which knots and links can be obtained from a RG[p,q]?

From RG[1,1] we can derive only unknot. From RG[2,1] we can also derive two-component unlink. It is clear that from every RG[p,1] we can obtain p-component unlink.

In order to simplify, for every RG[p,q] we will be interested only for new knots or links,i.e., knots or links which cannot be derived from some smaller rectangular grid.

4=412 2=21

2 2=212 2=21

2 31

RG[2,2]

RG[3,2]

(a) 3 1 3 = 74 {{1,1,1},{1,1},{1,1}}(b) 4 2 = 61 {{1,1,-1},{1, 1},{-1,-1}} (c) 3 1 2 = 62 {{1,1,1},{1,1},{-2,1}} (d) 6 = 61

2 {{1,2,1},{1,1},{1,1}}(e) 5 = 51 {{1,2,1},{-2,1},{1,1}} (f) 3 2 = 52 {{1,1,1},{1,1},{-2,-2}} (g) 2 1 2 = 51

2 {{1,1,1},{-2,1},{1,-2}}(h) 2 2 = 41 {{-2,1,1},{1,1},{-2,-2}} (i) 3#3 {{1,-2,1},{1,1},{1,1}}(j) 3#2 {{1,-2,1},{-2,1},{1,1}}(k) 2#2 {{1,-2,1},{1,-2},{-2,1}}.

(a) 3 1 2 1 3 = L10a101 (b) 5 1 3 = 95 (c) 3 1 2 1 2 = 920

(d) 4 1 1 3 = 952

(e) 3 1 3 2 = 982

(f) 3 1 1 1 3 = 992

(g) 5 1 2 = 82

(h) 4 1 3 = 84

(i) 3 1 1 1 2 = 813

(j) 8 = 812

(k) 4 2 2 = 832

(l) 3 2 3 = 842

(m) 3 1 2 2 = 852(n) 2 4 2 = 86

2

(o) 2 12 1 2 = 872

(p) 7 = 71

(q) 5 2 = 72

(r) 2 2 1 2 = 76

(s) 2 1 1 1 2 = 77

(t) 4 1 2 = 712

(u) 3 1 1 2 = 722

(v) 2 3 2 = 732

(x) 2 1 1 2 = 63

(y) 3 3 = 622

(z) 2 2 2 = 632

RG[4,2], prime KLs

Theorem: All rational knots and links can be derived as mirror-curves fromrectangular grids RG[p,2] (p>1).

The main problem is that some KLs created as mirror-curves will be placed in non-minimal rectangle grids. This problem can be solved by grid reduction, where by "all-over move" froma KL placed in a RG[p,q] we obtain the same KL placed in the grid RG[p-1,q].

Reduction of the knot 3 -1 3 placed in the RG[3,2] tothe trefoil placed in its minimal RG[2,2].

The next question is: how to construct a mirror-curve correspondingto some KL given in Conway notation.

Construction of a mirror-curve diagram of figure-eight knot2 2 from its Conway symbol.

Remark: Probably the simplest method of construction is to use the programs KnotAtlas or gridlink to construct grid diagram (arc presentation) of a given link, then transform it into a mirror-curve and reduce the obtained mirror-curve.

From RG[3,3] with its corresponding alternating 3-component link 8*2:2:2:2 (a) with n=12 we derive a lot of new KLs, among them the smallest basic polyhedron- Borromean rings 6* = 62

3 {{-1,-1,-2},{-2,-1,-2},{-2,2,-2},{-1,-1,-1}} and the first non-alternating 3-component link 2,2,-2 =63

3 {{-1,-1,1},{-1,-1,1},{-2,2,-2},{-2,2,-2}}.

(a) RG[3,3] with 3-component link 8*2:2:2:2 (b) non-alternating 3-component link 2,2,-2 =63

3.

(a) Figure-eight knot and (b) Borromean rings mosaic transformed into mirror-curves.

The approach based on mirror-curves is equivalent to the approach based on link mosaics: every link mosaic can be easily transformed into a mirror-curve and vice versa.

Even more illustrative are knot mosaics from T. Kyria’s paper: On a Lomonaco-Kauffman Conjecture,arXiv:math.GT/0811.0710v3 (2008), page 15: take any of them, rotate it by 45o, cut the emptyparts and add the two-sided mirrors in appropriate places.

4151 52

62 74

Since Takahito Kuriya proved Lomonaco-Kauffman Conjecture, showing that the knot mosaic theory and the tame knot theory are equivalent, the same holds for the mirror-curve theory and the tame knot theory. According to the Proposition 8.4 from the same paper, there is also a correspondence between knot mosaics and grid diagrams, enabling the connection between grid diagrams and mirror-curves.

Definition: The mosaic number m(L) of a link L is the smallest integer n for which L is representable as a knot n-mosaic.

For every link L, the mosaic number m(L) equals p+q, where p and q are dimensions of the minimal RG[p,q] in which L can be placed, and the dimension of the grid (arc) representation matrix equals m(L)+1=p+q+1.

Using this we can prove Kyria’s Conjecture 10.4, claiming that the mosaic number of the knot 2 1 1 2 = 63 is 6, since its minimal rectangular grid is RG[3,3].

63

Conjecture: For direct product of two links and their mosaic numbers holds the relationship:m(L1#L2)= m(L1)+m(L2)-3.

Product of mirror-curves

In order to define a product of mirror-curves derived from the same RG[p,q] we can substitute in their codes 2, -2, 1, and -1 by the elements of some semigroup of order 4. For example, if S is the semigroup of order 4, with elements A={a,aba}, B={b,bab}, C={ab}, and D={ba}, given by the Cayley table

A B C D A A C C A B D B B D C A C C A D D B B D

after substitutions 2 → a, -2 → b, 1→ ab, -1→ ba, the product of mirror-curvesM1={{-2,-2,1,1},{1,2},{-1,1},{-1,-2}} andM2={{-2,-2,1,1},{-1,-2},{1,-1},{2,-1}} will beM1*M2={{-2,-2,1,1},{2,1},{-2,2},{-1,-1}}.

(a) 2*2 → 2; (b) 2*-2 → 1; (c) 2*1 → 1; (d) 2*-1 → 2; (e) -2*2 → -1; (f) -2*-2 → -2; (g) -2*1 → -2; (h) -2*-1 → -1; (i) 1*2 → 2; (j) 1*-2 → 1; (k) 1*1 → 1; (l) 1*-1 → 2; (m) -1*2 → -1; (n) -1*-2 → -2; (o) -1*1 → -2; (p) -1*-1 → -1.

Since the elements a, b, ab and ba are idempotents, for every mirror-curve M holds the relationship M*M=M2=M. If M[p,q] is the set of all mirror-curves derived from RG[p,q],

the basis (minimal set of mirror-curves from which M[p,q] can be obtained by the operation of product) is the subset of all mirror-curves with codes consisting only from 2 and -2, i.e. the set of all unlinks belonging to RG[p,q]. The product of two different mirror-curves belonging to the basis is a mirror-curve not belonging to the basis, i.e., a mirror-curve with at least onecrossing. In particular, as the product of mirror-curves containing only vertical and horizontal mirrors we obtain alternating knot or link corresponding to RG[p,q].

Alternating link 3 1 2 1 3 = L10a101 corresponding to RG[4,2] obtained as the product M1*M2={{1,1,1,1},{1,1},{1,1},{1,1}} of mirror-curves M1={{2,2,2,2},{2,2},{2,2},{2,2}} andM2={{-2,-2,-2,-2},{-2,-2},{-2,-2},{-2,-2}}.

Minimal representation of mirror-curves

If a link L is given by a mirror-curve M=M1*M2, its mirror-image is M'=M2*M1.

Moreover, every mirror curve M can be decomposed a unique way in a product of two mirror-curves representing two Kauffman states, meaning that for every mirror-curve we can obtain exactly one pair of mirror-curves (M1,M2) such that M=M1*M2, and M1 and M2 represent Kauffman states (i.e., mirror-curves containing only 2 and -2 in their codes).