Vertex Models and Knot invariants

Knots are closed non-self-intersecting curves in three dimensional space. In order to classify knots, we require quantitative description which are called ‘knot invariants.’ Well-known knot invariants are Alexander polynomial, Jones’ polynomial, Kauffman and HOMFLYPT polynomials. We will briefly recapitulate the salient features of knots. Then we will present the computation of Jones’ polynomial for a knot. As these knots can be obtained from braids, we can reproduce knot invariants using braid groups and their representations. Six vertex model corresponds to placing spin half states on the edges of a square lattice. Using the Boltzmann weights corresponding to the six vertex model, we review a braid group representation and the construction of Jones’ polynomials. This procedure can be applied to higher spins placed on the edges of the square lattice which we will briefly summarize.