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

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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 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 -manifold with given surgery presentation. This includes all -manifolds, due to a result by Lickorish and Wallace.

Contents[edit | edit source]

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