Applications Of Knot Theory In Physics

Philip Walker and Melvin Carter

31 April 2013

Introduction

Knots and links have had a multitude of applications over the years. This

paper details the history of Knot Theory development, their applications in

algebraic topology, the in�uence of the Jones polynomial on the expansion of

the study of knots and physics simultaneously, and the applications found in

Special Relativity (speci�cally Topological Quantum Field Theory).

The Beginnings

Alexander-Theophile Vandermonde �rst developed a mathematical theory of

knots in 1771 by noting the importance of topological features in relation to the

properties in knots relative to their geometry of position. Vandermonde was a

French mathematician who did work in determinant theory and also worked on

symmetric functions and solutions of cyclotomic polynomials. Actual studies of

knots would not begin until the 19th century when Carl Friedrich Gauss de�ned

the linking integral. In 1833 Gauss developed the Gauss linking integral which

is used for computing the linking number of two knots. The linking integral is

de�ned is well de�ned. Given two non-intersecting di�erentiable curves (γ1, γ2:

S1� R3), we de�ne the Gauss map G from the torus to the sphere by:

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Γ(s,t) =g1(s)− g2(t)

|g1(s)− g2(t)|

Pick a point in the unit sphere, w, so that orthogonal projection of the link to the

plane perpendicular to w gives a link diagram. Observe that a point(s, t) that

goes to w under the Gauss map corresponds to a crossing in the link diagram

where γ1is over γ2 . Also, a neighborhood of (s, t) is mapped under the Gauss

map to a neighborhood of w where the orientation is either preserved or reversed

depending on the orientation. Therefore counting the signed number of times

the Gauss map covers w will compute the linking number of the corresponding

diagram. The resulting value will be the precise degree of the Gauss map due to

the fact that w is a regular value. Any other regular value would give the same

number, so the linking number doesn't depend on any particular link diagram

(Linking coe�cient). This formulation of the linking number of γ1and γ2 enables

an explicit formula as a double line integral, the Gauss linking integral:

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4p

ˆg1

ˆg2

r1 − r2|r1 − r2|

(dr1dr2)

This integral computes the total signed area of the image of the Gauss map

and then divides by 4p which is the area of the sphere [1 ].

Classifying Knots

In 1860 English physicist William Thompson was working on problems which

related to the structure of matter. At the time there were two sides with dif-

fering opinions; one side supported Corpuscular theory and those who thought

matter was a superposition of waves dispersed in space time. Corpuscular the-

ory states that matter consists of atoms, where atoms are rigid and occupy a

precise position in space. Both of these theories can explain certain phenomena

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but are incomplete in that they cannot explain other phenomena. Thompson

was able to �nd a way to combine these theories; his theory stated that matter

consists of atoms. The atoms, which he refers to as vortex atoms, are not point

like objects but rather they are knots. This means that an atom is like a wave

that bends back on itself. These knots can be quite complicated, and this would

mean that molecules consist of knots which are interwoven. Interwoven knots

are referred to as links by mathematicians [2 ][3 ].

Thompson began classifying knots in order to develop his theory further.

Once knots were classi�ed it would be possible to classify atoms. One of Thomp-

son's friends, Peter Guthrie Tait, was determined to solve this problem. Tait

stated that since a knot was a closed curve in space it could be represented by

a planar curve projected perpendicularly on the horizontal plane. These pro-

jections would have crossings which take place when two strands of the curve

cross each other.

Figure 1:

In order for Tait to classify knots he had to �rst determine which knots were

in the same class and this meant he had to de�ne the equivalence of knots.

To do this he formulated Tait's conjectures on alternating knots. In 1885 Tait

published a table of classi�cations for knots that were up to ten crossings, this

was known as the Tait Conjectures. One of these conjectures is that in certain

circumstances the crossing number was a knot invariant. Speci�cally, the cross-

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ing number of a reduced and alternating link is a knot invariant. This would

not be proven until 1987 by Morwen Thistlethwaite, Louis Kau�man, and K.

Murasugi using the Jones polynomial [4 ]. Another conjecture was that a re-

duced alternating link with zero writhe is not equivalent to its mirror image.

The writhe number is the total number of positive crossings minus the total

number of negative crossings on an oriented link. Tait believed that by making

these tables of knots he was constructing a table of chemical elements de�ned

through his theory.

Figure 2:

Knot Polynomials

James Waddell Alexander II was born on September 19, 1888. He is credited

with the �rst knot polynomial which is known as the �Alexander Polynomial�.

It was introduced in 1923, while other knot polynomials were not found until

over 50 years later. The Alexander Polynomial is a knot invariant that assigns

a polynomial with integer coe�cients to the di�erent knot types. In 1969 a

version of the Alexander polynomial which could be computed using a skein

relationship was shown by John Conway. A skein relation gives a linear relation

between the values of a knot polynomial on a collection of three links which

di�er from each other in a small region. This relation can be used to calculate

the polynomial recursively.

Alexander proved that the Alexander ideal is nonzero and always principal.

Thus an Alexander polynomial always exists, and is clearly a knot invariant.

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Rules for calculating Alexander Polynomial

Rule 1: 1(0) = 1, when 0 is any projection of the unknot

Rule 2: 1(L+)-1(L−)+(t12 − t− 1

2 )1(L0)= 0.

L+, L−, L0 are three oriented link diagrams.

In 1984 Vaughan Jones discovered the Jones Polynomial. The Jones poly-

nomial is an invariant of an oriented knot which assigns to each link a Laurent

polynomial. A Laurent polynomial is a variable over a �eld which is a linear

combination of positive and negative powers of the variable with coe�cients.

The Jones polynomial takes the value 1 on any diagram of the unknot and

satis�es the skein relation .

How to calculate Jones polynomial:

Step 1: Calculate Bracket polynomial

Rule 1: < 0 >= 1, where 0 is any projection of the unknot

Rule 2a: < L+ >= A < Linf > +A−1 < L0 >

Rule 2b: < L− >= A < L0 > +A−1 < Linf >

Rule 3:< LU0 >= (−A2 −A−2) < L >

Step 2: Use Bracket Polynomial to calculate X polynomial

Where: X < L > = (A3)−w(L) < L > , w(L) is the writhe of L

Step 3: Replace each A with t−14 in X polynomial

The need for physical measures in S4

There is a link between the particles and energies in the quantum �eld to the

topological tools used to describe properties of these �elds using Topological

Quantum Field Theory [TQFT]. Quantum concepts have two types of de�ni-

tions the ambiguous and disambiguous use: ambiguous being the measure of

properties and disambiguous being the measurements of the physical interac-

tion {x |x ∈ R}. The Lorentz Equation and other measures of the disambiguous

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are not addressed in this paper. For the ambiguous, Knot Theory, especially in

the form of the Bracket Polynomial, is used to account for the all the potential

topological variations quanta may take (Figure 4).

The Jones polynomial does much to de�ne the topological properties quanta

in a quantum �eld. Yet, we recall that the real-time behaviors of quanta are

not dictated by topology but by Special Relativity. The interchangeability of

mass and energy is expressed in Einstein's famous equation E = mc2. Special

Relativity assumes the speed of light is constant (Em = c2). Newton's model of

the physical world was no longer adequate. The four dimensions (4-manifold)

of the Khovanov homology are adequate and su�cient. For the properties of

the quantum �elds, in the ambiguous interpretation, the intent is to lead the

reader from the Jones polynomial to the Khovanov homology (1998).

Observed quanta phenomenon has given us ideas of the nature of quantum

�elds:

Figure 3:

Figure 3 depicts the electrons of Hydrogen in the 3d orbital (preceded by 1s,

2s, 2p, 3s, 3p, and 4s orbitals). Note that Heisenberg's uncertainty principle is

captured in the graphics as a probability density. The space between the density

'clouds' is the area of interest if we are to make use of the Jones polynomial and

hence Khovanov's homology as properties of the Topological Quantum Field

Theory [TQFT].

Dr. A. Zee stated:

�Quantum �eld theory is needed when we confront simultaneously

the two great physics innovations of the last century of the previ-

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ous millennium: special relativity and quantum mechanics... It is in

the peculiar con�uence of special relativity and quantum mechanics

that a new set of phenomena arises: Particles can be born and par-

ticles can die. It is this matter of birth, life, and death that requires

the development of a new subject in physics, that of quantum �eld

theory. Let me give a heuristic discussion. In quantum mechan-

ics the uncertainty principle tells us that the energy can �uctuate

wildly over a small interval of time. According to special relativity,

energy can be converted into mass and vice versa. With quantum

mechanics and special relativity, the wildly �uctuating energy can

metamorphose into mass, that is, into new particles not previously

present.� [6 ].

Figure 3 does not depict an electron but the statistically probable location of

electrons. The same depiction on the right veri�es the Heisenberg uncertainty

principle by showing the possible locations as a probability function.

Further developments in the de�ning of knot prop-

erties

The topological de�nitions of the Jones polynomial are for the valuation of

knots in S3. There are limitations inherent in the use of the Jones polynomial

though: �rst that it does not detect the unknot and second it is limited to the

S3 or speci�cally the 2-D view of the x, y, and z dimensions. This meant that

determining all outcomes of a quantum �eld is not possible with only the Jones

polynomial. Perception and detection was needed for the full quantum �eld

(4 −manifold or S4). Mikhail Khovanov introduced a method to evaluate S4

. The ability to analyze Statistical Mechanics, Quantum Group, and knots and

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links in S3, using the Jones polynomial (J. Birman, 1990), were enhanced by

the Khovanov Homology. This development has the means to provide de�nition

to all the topological �elds generated by quanta. TQFT is the analysis of

quantum spacetime in the 4-manifold or four dimensions. In this case we choose

the Khovanov Homology.

To detect the unknot we need an accessible invariant that will distinguish L

from an unknot. Recall that when L ⊂ S3 (Jones polynomial) we fail to detect

the unknot or distinguish it from other knots. The Jones polynomial uses the

bracket polynomial < 0 >= 1 as an identity function. The unknot, then, has no

value per se but preserves the value of the sum operation of the given link. An

unlink or unknot is the identity element with respect to the knot sumoperation.

Figure 4:

In order to detect the unlink or unknot the Floer homology (Heegaard Floer)

is used by Khovanov. The Khovanov homology makes use of the Knot Floer

homology (Osvath, Szabo, and Rasmussen). It is a generalization of the Jones

polynomial but is crucial in its ability to detect the unknot. The Knot Floer ho-

mology is in the framework L ⊂ S3 and, using the notation HFK (L) [Heegaard

Floer Knot], is a generator of an abelian group that is bi-graded. A bi-graded

module is a module of links Lm,a that is indexed by the pair of integers m and a

(with each module over a �xed abelian group). In this case HFK (L)=⊕ where

m, a ∈ |L|−12 + Z or HFKm (L, a) and where |L| is the number of components

in L (m is the Maslov grading and a is the Alexander grading)[7 ]. Dr. Yi Ni

de�ned the Khovanov homology as a bi-grade of L ⊂ S3 → Khi,j(L) such that

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∑i,j

(−1)iqj , or rank of the Khi,j(L) =

(q + q−1

)VL (q). This is true where i is

the homological grading and j is quantum grading [8 ]. Dr. Yi Ni states that

though the bi-grading work for Kh, �...[captures] the geometric and topological

meaning of the bi-gradings.... in Kh�. This would mean that the full expression

of the Khovanov module is for TQFT. Given that the bi-grading is by the

Maslov and Alexander grading there is a simultaneous association with physics

and knots.

The Khovanov Homology was de�ned by Hedden and Ni as a moduleKh (L;F)

that make use of the Floer homology, F , to detect the unlink over the ring

F [X0, ..., Xn−1] \[X20, ..., X

2n−1]. Hedden and Ni's theorem2declaresthattheunlinkisfoundwhen

Kh (L;F) ∼= F [X0, ..., Xn−1] \[X20, ..., X

2n−1][7 ]. Dr. Dror Bar-Natan extends the

usefulness of the Khovanov homology to all four dimensions by adding the Wil-

son operator W and the height shift [s] to de�ne the full spectrum of options

available in a 4-manifold or S4:

�Khovanov's �categori�cation� idea is to replace polynomials by graded

vector spaces2 of the appropriate �graded dimension", so as to turn

the Jones polynomial into a homological object. With the diagram

[see �gure 3] as the starting point the process is straight forward and

essentially unique. Let us start with a brief on some necessary gen-

eralities: De�nition 3.1 Let W = ⊕mWm be a graded vector space

(where ⊕m is the vector by Maslov's grading) with homogeneous

components{Wm}[the Wilson operator with respect to the Maslov

portion of the bi-grading]. The graded dimension of W is the power

series∑

m qmdimW , De�nition 3.2 Let · {l} be the �degree shift� op-

eration on graded vector spaces. That is, ifW = ⊕mWm is a graded

vector space, we setW {l}m := Wm−l, so thatW {l}m = q1 q dimW ,

De�nition 3.3 Likewise, let [s] be the �height shift� operation on

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chain complexes. That is, if C is a chain complex ... C r dr→C r+1

... of (possibly graded) vector spaces (we call r the �height� of a

piece C r of that complex), and if C = C[s], then C r = C r−s

(with all di�erentials shifted accordingly). Likewise, let [s] be the

height shift" operation on chain complexes. That is, if C is a chain

complex:::! Cr dr→!C

r+1

:::of (possibly graded) vector spaces (we call

r the ”height” of a piece Crof that complex), and if C = C [s], then

Cr= C

r=s (with all di�erentials shifted accordingly).� [9 ]. The

4-manifold is de�ned by the (m, a) bi-grading of HFKm (L, a), the

Wilson operator �shift�, and [s] be the �height shift� (see �gure 7).

Edward Witten clari�es the Wilson operator's use in Special Rela-

tivity and links it to the Jones Polynomial, �the quantum formula

for the Jones polynomial is just JK =< WK >�[10 ].

The depiction of this shift is given by comparing a normal depiction

(�gure 5) of the bracket polynomial for the trefoil and the �shifts�

of the Wilson operator and [s] in �gure 6 and �gure 7. An example

has been made of bracket 100 on both �gures.

Figure 5:

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Figure 6:

Figure 7:

Conclusion

Topological Quantum Field Theory sees its maximum fruition in the form of bor-

disms. Once we have the 4-manifold we examine topological generators within

S4. The value of the Khovanov homology is in de�ning knots and links in

terms of vector spaces. The summation of all these spaces is a bordism. For

instance, if a function M is onto N it forms a bordism. If M is also onto N

then M → N →M and forms the bordism:

Figure 8:

At last there is a full representation of the value of the 4-manifold vector

space given by Mikhail Khovanov's homology.

Bibliography

[1] Linking coe�cient. A.V. Chernavskii (originator), Encyclopedia of Mathe-

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[2] Alexander, J.W.. "Topological Invariants of Knots and Links." . Ameri-

can Mathematical Society, n.d. Web. 05 Apr 2013. <http://homepages.math.uic.edu/~kau�man/Alex.pdf>.

[3] Kau�man, Louis. "Knot Theory and Physics." American Mathematical

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[7] Ni, Yi & Hedden, Matthew. �Khovanov module and the detection of un-

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[8] Ni, Yi. �Khovanov module and the detection of unlinks�. 2012. Lecture

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27:02 to 28:08. < http://www.youtube.com/watch?v=UTA6ZUVIev0>.

[9] Bar-Natan, Dror. �On Khovanov's categorication of the Jones polyno-

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ISSN 1472-2739 (on-line) 1472-2747 (printed).

[10] Witten, Edward. �Knots and Quantum Theory�. Spring 2011. The

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