Topological String Theory Methods of Computer-aided Drug Design/Knots, HOMFLY-PT Polynomial, Chern-Simons Theory and Surgery

This chapter covers knot theory and its invariants, including, especially, HOMFLY-PT polynomials. We explain Witten's viewpoint: these polynomials can be interpreted as the vacuum expectation value of Wilson loop operators in Chern-Simons Theory with gauge group $$G = SU(N)$$ and the fundamental representation. This easily leads to the generalization of HOMFLY-PT polynomials to arbitrary gauge groups and representations. We then introduce Guadagnini's amalgamation of Chern-Simons theory and Dehn surgery: this allows the computation of HOMFLY-PT polynomials in an arbitrary $$3$$-manifold $$M^3$$ with given surgery presentation. This includes all $$3$$-manifolds, due to a result by Lickorish and Wallace.

Contents

 * 1) Knot theory
 * 2) Link invariants
 * 3) HOMFLY-PT polynomials
 * 4) Chern-Simons Theory
 * 5) Surgery
 * 6) HOMFLY-PT polynomials in arbitrary $$3$$-manifolds