Knot TheoryKnot Theoryandand

Statistical Mechanics

KNOTS: A 3-dimensional loop projected onto a

2-dimensional surface

Unknot Trefoil Figure 8g

LINK: The entanglement of 2 or more loops

2389

Unknot Unknot

Trefoil Hopf linkTrefoil Hopf link

Unknot Unknot

EQUIVALENT KNOTS: Two knots are equivalent if they

Trefoil Hopf link

EQUIVALENT KNOTS: Two knots are equivalent if theyare the projections of the same 3-dimensional knot

Trefoil Hopf link

An unknot

AAn

An unknot An unknot

A

KNOT INVARIANTS: Algebraic functions (polynomials or numbers) constructed from the knot projectionsA

n

) p jthat are the same for equivalent knots

Reviews of Modern Physics, 64 1099-1131 (1992)

Unknot Unknot

EQUIVALENT KNOTS: Two knots are equivalent if they

Trefoil Hopf link

EQUIVALENT KNOTS: Two knots are equivalent if theyare the projections of the same 3-dimensional knot

Trefoil Hopf link

Equivalent knots can be transformed into eachEquivalent knots can be transformed into each other by Reidemeister moves of lines

There are 3 types of moves: I, II, and III

IA

IB

IIA

IIB

IIIAIIIAu v

w

IIIBIIIB

ENTANGLED CONFIGURATIONSENTANGLED CONFIGURATIONS

Sufficient moves for all equivalent knots

Any one of the IIIA movesAny one of the IIIA moves

Yang-Baxter equationYang Baxter equation

Example of deducing a IIIB move using IIB and IIIA moves

Traditionally, there are two approaches to deduce knot invariants

Algebraic approach:Algebraic approach: Convert knots into braids and use group-theoretic properties of the braid group to deduce invariants

Geometric approach: Use knot graphs to deduce invariants

Algebraic approach

Example of converting a knot into a braid

Braid group:Braid group:

iσi 1+i

111 +++ = iiiiii σσσσσσ 111 +++ iiiiii

,ijji σσσσ = 2≥− ji

iσiσ

σσi 1+i

σσσ

ii σσ 1+

iii σσσ 1+

1+iσ 1+iσi 1+i

1+iiσσ

11 ++ iii σσσ

Jones considers representations of the braid group

Von Neumann algebra the Jones polynomialg p y

Using the Temperley-Lieb algebra akin to the Potts model leads to a one variable polynomial invariantleads to a one-variable polynomial invariant

Hecke algebra Homfly polynomialHecke algebra Homfly polynomial

Using the Hecke algebra leads to a two-variableUsing the Hecke algebra leads to a two variable polynomial invariant

Temperley, 1998

(Harold Neville Vazeille Temperley 1915 2007)

Temperley, 1998

(Harold Neville Vazeille Temperley, 1915 – 2007)

Lieb (2003)Lieb (2003)

Geometrical approach (directed knots)

Consider three knots which are identical except at crossing

+L −L 0LAssociate each knot with a polynomial P(x, y, z) such thatThe three polynomials are related by

+

The skein relation

),,(),,(),,(0

zyxzPzyxyPzyxxP LLL =++

)()()(0

yyyy LLL −+

Starting from1),,( =zyxPunknot

Skein relationSkein relation

),,,(z 1y 1 2 xyxPx l⋅=⋅+⋅z

yxP +=l2 z

xzyxyxPz

zyxyzyxxP HopfHopf

22

,1 ),,( +−=⋅=

+⋅+

xzz

xzyxyxzyzyxxPTrefoil

22

1 ),,( −−⋅=⋅+

xz

2

22 2),,( yxyzzyxPTrefoil−−

=This gives2)(

xyTrefoil

Alexander-Conway polynomial :

tzyt

x 1 ,1 ,1−=−==

)(tΔ

)(1)()(0

ttt

tt LLL Δ⋅⎟⎠

⎞⎜⎝

⎛ −=Δ−Δ−+

Skein relation:

tt

t ⎠⎝

Jones polynomial :)(tV

ttzty

tx 1 , ,1

−=−==

⎞⎛Skein relation: )(1)()(10tV

tttVttV

t LLL ⋅⎟⎠

⎞⎜⎝

⎛ −=⋅−⋅−+

Homfly polynomial :),( ztPty

tx −== ,1

t

Skein relation: ),(),(),(10

ztPzztPtztPt LLL ⋅=⋅−⋅

−+

Statistical Mechanical ApproachStatistical Mechanical Approach

1 Construct a lattice from a given knot1. Construct a lattice from a given knot.

2 Define a statistical mechanical model on the lattice2. Define a statistical mechanical model on the lattice.

3. Assign model parameters such that the partition function of the statistical mechanical model is invariant under Reidemeister moves of lines.

4. The partition function is by definition a knot invariant.

5 Different invariants are obtained by using different models5. Different invariants are obtained by using different models.

There are three main different types of lattice models:

1. Vertex models

2. Interaction-round-face (IRF) models

3. Edge-interaction spin models

Vertex modelsbd bd

The statistical model is a vertex model with vertex weights

a cThe statistical model is a vertex model with vertex weights.Each lattice edge can be in q different states.Specify the states of the 4 edges at a vertex by variables a b c d = 1 2 q and denote the vertexvariables a, b, c, d = 1, 2, …, q and denote the vertex weight by

),|,( dcba±ω

∑ ∏ )|( dbZ

The partition function of the lattice is given by

∑ ∏ ±=states vertices

),|,( dcbaZ ω

For un-oriented knots the Reidemeister moves areFor un oriented knots, the Reidemeister moves are

The Reidemeister moves require the following conditions on the weights:

For oriented knots, it is convenient to introduce a parameterinto the model. Then the Reidemeister moves require

λinto the model. Then the Reidemeister moves require

(IIIA)±=3,2,1

The equation (IIIA) is the Yang-Baxter equation. It i t d t i d ti ith ti6qIt is a set over-determined equation with equationsand unknown vertex weights.

q43q

The Yang-Baxter equation

eq ations nkno ns6 43qequations, unknowns 6q 3q

The Yang-Baxter equation

eq ations nkno ns6 43qequations, unknowns 6q 3q

A solution of the Y-B equation exists for each exactly solved modelsq y

Condition I is a unitarity condition which is usually satisfied.

Condition IIA can usually be satisfied by choosing model parameters.

The Yang-Baxter equation IIIA is satisfied for exactly solved models.

This implies that from each exactly solved model,one can construct a knot invariant !

Interaction-around-face (IRF) models

functionPartition

∑∏= ),,,( functionPartition

dcbaW

),,,(ghtVertex wei

dcbaW= ),,,( dcbaW

Yang-Baxter equation for IRF modelsYang Baxter equation for IRF models

W'W

WW

W'W

'W''W ''W

WW W

∑c

bbbcWcbcaWacbaW )','',,('' )',',,''(' )'',,,(

∑=c

c

abbcWcbbaWacaaW )',','',( ),'',,(' )',,,''('' c

Simple spin models with pair interactions

Shade alternate faces and put spins in shaded faces

There are 2 kinds of pair interactions along dotted linesThere are 2 kinds of pair interactions along dotted lines.a

a bab

b

∑∏ ±± = ),( 1)( 2/ baWq

WZ Nq

T I R id i tType I Reidemeister moves

∑ =± baW mα),(1 ∑bq

1)( mW 1),( mα=± aaW

Type II Reidemeister moves

∑ =abWbaW IIA)()(1 δ∑ =+−b

caabWbaWq

IIA ).(),( ,δ

IIB1)()( bWbW IIB 1),(),( =−+ baWbaW

Type III Reidemeister moves

),(),(),(),(),(),(1 acWcbWbaWdcWdbWdaWq d

±±±±∑ = mm

Example: The Potts model – a soluble (unphysical) case

TakebaKeAbaW ,),( δ±

±± =

− −=−= −+ eet KKwith

4/1±± = tA

4/3

12−

−

=

++=

tttq

α −= tαThen this gives the Jones polynomial

)()()( 4/32/1 −− WZttV nn )()()( 4/32/1±

+−−= WZtqtV nn

whereknotin thecrossings ofnumber the ±=±n g±

Jones polynomialKnot Jones polynomialKnot

The Jones polynomial can also be derived from theThe Jones polynomial can also be derived from thebracket polynomial of Kauffman (1987)

The Jones polynomial can also be derived from theThe Jones polynomial can also be derived from thebracket polynomial of Kauffman (1987)

The bracket polynomial is identical to a q-state non-intersecting string model of Perk and Wu (1986)

Kauffman approach: Un-oriented knot

Reidemeister moves:

Theorem: Partition function Z(w) is a regularIsotopy of knot invariants (Kauffman, 1987)

The non-intersecting string model of Perk and Wu g g(1986)

= polynomial in q, A, B

33 33Bq

223 ABq

BqA23q

32 Aq

32223 33 AqBqAABqBqZ +++=

Reidemeister moves II:

1−= AB)( 22 −+−= AAq

Reidemeister moves I:

)()()(

4/3

4/13

ZVtAA

nn −

−=−=α

)()()( 4/3 tZttV nn +− −−−= Is the Jones polynomial

Alexander-Conway polynomial

Knot Alexander (1928) Conway (1970)

ttz 1−=

Knot invariants from the exact solution of a 19-vertex modelKnot invariants from the exact solution of a 19-vertex model(Pant and Wu, 1995)

Jones polynomialV F R J B ll A M h S 12 (198 ) 103 112V. F. R. Jones, Bull. Am. Math. Soc. 12 (1985) 103-112.

Bracket polynomialL. H. Kauffman, Topology 26 (1987) 395-407.

Nonintersecting string modelNonintersecting string modelJ. H. H. Perk and F. Y. Wu, J. Stat. Phys.42 (1986) 727-742.

Knot invariants and statistical mechanicsKnot invariants and statistical mechanicsV. F. R. Jones, Pacific J. Math. 17 (1989) 311-354.F. Y. Wu, Rev. Mod. Phys. 64 (1992) 1099-1131.

Knot invariants from the chiral Potts modelF. Y. Wu, P. Pant, C. King, Phys. Rev. Lett. 72 (1994) 3937-3940.

The End