Recursion Formulas for HOMFLY and Kauffman Invariants

TL;DR

This paper establishes new recursion relations between HOMFLY and Kauffman polynomials for framed links, linking them to embeddings of quantized universal enveloping algebras, including novel relations for specific Lie algebra embeddings.

Contribution

It introduces new recursion formulas connecting HOMFLY and Kauffman invariants based on embeddings of certain Lie algebras, expanding the algebraic understanding of these polynomials.

Findings

Derived recursion relations for HOMFLY and Kauffman polynomials

Connected link invariants to embeddings of quantized universal enveloping algebras

Presented new relations for specific Lie algebra embeddings such as $so_{2n+1}$, $so_{2n}$, and $sp_{2n}$

Abstract

In this note we describe the recursion relations between two parameter HOMLFY and Kauffman polynomials of framed links These relation correspond to embeddings of quantized universal enveloping algebras. The relation corresponding to embeddings $g_{n}\supset g_{k}\times sl_{n-k}$ where $g_{n}$ is either $so_{2n+1}$, $so_{2n}$ or $sp_{2n}$ is new.