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B-type coefficient polynomial

Noboru Ito, Mayuko Kon

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.

B-type coefficient polynomial

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.

Paper Structure

This paper contains 16 sections, 13 theorems, 85 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Preliminaries and notation
  3. Link diagrams and crossings
  4. Local splices
  5. Component-change quantities
  6. Algebraic conventions
  7. Coefficient polynomials
  8. Base points and warping degree
  9. Disjoint union and connected sum
  10. Main Result
  11. Construction and Proof of Theorem \ref{['maintheorem1']}
  12. Uniqueness
  13. Uniqueness under regular isotopy
  14. Uniqueness under ambient isotopy
  15. A direct uniqueness proof for the coefficient polynomials $\alpha_n$
  16. ...and 1 more sections

Key Result

Theorem 3.1

For every unoriented link diagram $D$, there exists a sequence of coefficient polynomials such that the following properties hold.

Figures (7)

  • Figure 1: A crossing $D_p$ and its two smoothings $D_\infty$ and $D_0$.
  • Figure 2: A connected change of the sequence of base points. The base point $a_i$ moves along the component across the crossing $p$ to $a'_i$.
  • Figure 3: Correspondence of the sequence of base points under a Reidemeister move of type I. The four possible orientation cases are shown.
  • Figure 4: A local move relating diagrams $D$ (left) and $D'$ (right). The crossings $p$ and $q$ correspond under this move.
  • Figure 5: Correspondence of the six diagrams appearing in equation \ref{['eq:doubleRII']}, obtained by applying the skein relation twice. The diagram $D_{-\epsilon(p)}$ is further expanded at the crossing $q$. The opposite orientation case, corresponding to a non-braid type bigon, is omitted.
  • ...and 2 more figures

Theorems & Definitions (40)

  • Definition 2.1: link diagrams
  • Definition 2.3: writhe
  • Remark 1
  • Definition 2.5: Component indicator
  • Definition 2.6: Signed component change
  • Remark 2
  • Definition 2.8
  • Definition 2.9: Generating series of the coefficient polynomials
  • Definition 2.10: Normalization
  • Definition 2.11: Sequences of base points and induced directions
  • ...and 30 more