Knots and Physics. Lifang Xia. Dec 12, 2012

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1 Knots and Physics Lifang Xia Dec 12, 2012

2 Knot A knot is an embedding of the circle (S 1 ) into three-dimensional Euclidean space (R 3 ).

3 Reidemeister Moves Equivalent relation of knots with an ambient isotopy (continuous deformation in R 3 without cutting, self-crossing) between them. Type I: Twist and untwist in either direction. Type II: Move one strand completely over another. (regular isotopy) Type III: Move a strand completely over or under a crossing. (regular isotopy)

4 Knot Invariants Distinguish knots Important invariants: knot polynomials, knot groups, and hyperbolic invariants. Bracket polynomial is a polynomial invariant under regular isotopy (type II, III Reidemeister moves). Jones polynomial (normalized bracket polynomial), is currently among the most useful invariants for distinguishing knots from one another, and connecting with quantum invariants.

5 Bracket Polynomial K = K σ ( A,B,d) = K σ d σ Partition function in statistic mechanics

6 Normalized Bracket Polynomial & Jones Polynomial Writhe of K: the number of positive crossings (K + ) minus the number of negative crossings (K - ), invariant of regular isotopy ω(k) = p ε(p) Normalized bracket: invariant of ambient isotopy L K (A) = ( A 3 ) ω(k ) K Jones Polynomial: L K (t 1/ 4 ) = V K (t)

7 Vassiliev Invariance Vassiliev invariants Lie algebras V (K * ) = V (K + ) V (K ) Lemma: If a graph G has exactly k nodes, the value of a Vassiliev invariant of type k on G, v k (G) is independent of the embedding of G. (Proof omitted) Knot invariants derived from Lie algebras are all built from Vassiliev invariants of finite type, which are directly related to Witten s functional integral.

8 Witten s Functional Integration Knot invariants partition function K = σ K σ d σ Link invariants functional integral Z K = DAe (ik / 4 π )L M T K = DAe(ik / 4π )L M A tr Pe K L M : Chern-Simons Lagrangian K M 3 : compact oriented three-manifold A: gauge fields (Lie algebra valued) tr K : product of traces of path-ordered exponential of the integral of the field around the closed circuit K Wilson Loop: W K (A) = tr(pe A K ) Use gauge field SU(2), Pauli matrices generators, commutation relations Small loops of K: tr K measures curvature of the gauge field A L M : curvature tensor arises as the variation of the CS Lagrangian wrt the gauge field

9 Vassiliev Invariance Behavior of Witten integral under a small deformation of the loop K δz(k) = (4πi /k) dae (ik / 4 π )S dx r dx s dx t ε rst T a T a W K (A) 1. Regular isotopy (type II, III) without creating a local volume: invariant 2. Switching a crossing (type I) Z(K + ) Z(K ) = (4πi /k) DAe (ik / 4π )S T a T a < K ** A >= (4πi /k)z(t a T a K ** )

10 Summary Knot Bracket Invariant Partition function of statistical mechanics Link Invariant Witten s Functional Integral Knot theory quantum field theory, loop quantum gravity, quantum computation

11 References L. H. Kauffmann, Knots and Physics, World Scientific Publishers (1991) L. H. Kauffmann, Knot Theory and Physics,

Knots and Physics. Louis H. Kauffman

Knots and Physics. Louis H. Kauffman Knots and Physics Louis H. Kauffman http://front.math.ucdavis.edu/author/l.kauffman Figure 1 - A knot diagram. I II III Figure 2 - The Reidemeister Moves. From Feynman s Nobel Lecture The character of