Three-partite vertex model and knot invariants

TL;DR

This paper develops a new algebraic approach using three-partite vertex models to construct braid group representations, enabling the study of knot invariants for multi-colored links with different spins, connecting to Jones and HOMFLY-PT polynomials.

Contribution

It introduces a novel algebraic formula for knot invariants derived from three-partite vertex models, extending the analysis to multi-colored links with varying spins.

Findings

Derived a braid generator representation from three-partite vertex models.

Formulated a knot invariant formula for multi-colored links with different spins.

Connected the resulting invariant to Jones and HOMFLY-PT polynomials.

Abstract

This work is dedicated to the consideration of the construction of a representation of braid group generators from vertex models with $N$-states, which provides a great way to study the knot invariant. An algebraic formula is proposed for the knot invariant when different spins $(N-1)/2$ are located on all components of the knot. The work summarizes procedure outputting braid generator representations from three-partite vertex model. This representation made it possible to study the invariant of a knot with multi-colored links, where the components of the knot have different spins. The formula for the invariant of knot with a multi-colored link is studied from the point of view of the braid generators obtained from the $R$-matrices of three-partite vertex models. The resulting knot invariant $5_2$ corresponds to the Jones polynomial and HOMFLY-PT.