B-type coefficient polynomial

TL;DR

This paper extends the concept of coefficient polynomials from A-type to B-type skein relations, providing a new inductive scheme that recovers the Kauffman polynomial and establishes invariance under Reidemeister moves.

Contribution

It introduces B-type coefficient polynomials with a novel inductive approach, linking them to the Kauffman polynomial and proving their invariance properties.

Findings

B-type coefficient polynomials recover the Kauffman polynomial

The new inductive scheme handles the four-term skein relation

Generated series are invariant under Reidemeister moves

Abstract

An A-type coefficient polynomial introduced by Kawauchi recovers the HOMFLY-PT polynomial as a formal power series within skein theory. A notable feature of this construction is that each coefficient defines a link invariant, yielding an infinite sequence of invariants, while the low-degree coefficients are relatively easy to compute. In this paper, we extend this viewpoint to the B-type setting. Unlike the A-type case, the B-type setting requires a genuinely new inductive scheme due to the four-term skein relation. More precisely, we introduce coefficient polynomials associated with the B-type skein relation and show that their generating series recovers the Kauffman polynomial. We further prove that these coefficient polynomials are well-defined and that the resulting generating series is invariant under the corresponding Reidemeister moves.