# 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 ${\displaystyle G=SU(N)}$ 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 ${\displaystyle 3}$-manifold ${\displaystyle M^{3}}$ with given surgery presentation. This includes all ${\displaystyle 3}$-manifolds, due to a result by Lickorish and Wallace.
6. HOMFLY-PT polynomials in arbitrary ${\displaystyle 3}$-manifolds