Bi-partite vertex model and multi-colored link invariants

TL;DR

This paper extends the algebraic construction of link invariants from single-spin models to bi-partite vertex models, enabling the study of multi-colored link invariants with different spins on component knots.

Contribution

It introduces a new method to derive braid group representations from bi-partite vertex models for multi-colored link invariants.

Findings

Derived a formula for multi-colored link invariants using bi-partite vertex models.

Generalized previous single-spin invariants to multi-spin, multi-colored cases.

Provided algebraic tools for studying complex knot and link structures.

Abstract

Construction of representations of braid group generators from $N$-state vertex models provide an elegant route to study knot and link invariants. Using such a braid group representation, an algebraic formula for the link invariants was put forth when the same spin $(N-1)/2$ are placed on all the component knots. In this paper, we generalise the procedure to deduce representations of braiding generators from bi-partite vertex models. Such a representation allows the study of multi-colored link invariants where the component knots carry different spins. We propose a multi-colored link invariant formula in terms of braiding generators derived from $R$ matrices of bi-partite vertex models.