Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Second-to-Top Term of \hat{HFK} of Closed 3-Braids

Zhaojun Chen

TL;DR

The paper computes the second-to-top term of $\widehat{HFK}$ for closed 3-braids by leveraging the Ozsváth–Szabó skein exact triangle together with Xu's classification into specific word forms. Through a detailed, case-by-case analysis (Cases 1–5) and a set of skein-based lemmas, it provides explicit Maslov gradings and rank decompositions for all possibilities, culminating in a complete determination of $\widehat{HFK}(L,u(L)-1)$ for $L=cl(w)$. It also proves a graded version of Sivek's rank inequality and discusses equality conditions when $L$ is a knot, extending prior results on the top term to the second-to-top term for closed 3-braids. The work thus yields a comprehensive description of the second-to-top term across Xu's classes and confirms the expected rank constraints in this setting.

Abstract

In this paper, we use the skein exact sequence and other techniques to compute the second-to-top term of HFK of closed 3-braids. We do it case-by-case according to Xu's classification. We also verify the rank inequality conjectured by Sivek.

Second-to-Top Term of \hat{HFK} of Closed 3-Braids

TL;DR

The paper computes the second-to-top term of for closed 3-braids by leveraging the Ozsváth–Szabó skein exact triangle together with Xu's classification into specific word forms. Through a detailed, case-by-case analysis (Cases 1–5) and a set of skein-based lemmas, it provides explicit Maslov gradings and rank decompositions for all possibilities, culminating in a complete determination of for . It also proves a graded version of Sivek's rank inequality and discusses equality conditions when is a knot, extending prior results on the top term to the second-to-top term for closed 3-braids. The work thus yields a comprehensive description of the second-to-top term across Xu's classes and confirms the expected rank constraints in this setting.

Abstract

In this paper, we use the skein exact sequence and other techniques to compute the second-to-top term of HFK of closed 3-braids. We do it case-by-case according to Xu's classification. We also verify the rank inequality conjectured by Sivek.
Paper Structure (12 sections, 22 theorems, 42 equations, 2 figures, 2 tables)

This paper contains 12 sections, 22 theorems, 42 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction and Terminologies
  2. Preliminaries
  3. Computing Maslov Grading of the Top Term
  4. Computing Maslov Grading of the Second-to-top Term
  5. Preliminary Lemmas
  6. Case 1: Suppose $w=\alpha^d P$ is a word in Xu's form, $d>0,$ $L=cl(w).$
  7. Case 2: Suppose $w=NP,$ $l(N)>1,$ $l(P)>1,$ $L=cl(w).$
  8. Case 3: Suppose $w=NP$, $1=l(N)\leq l(P),$ $L=cl(w)$.
  9. Case 4: Suppose $w=P$ is in Xu's form, $w$ starts with $a_1$ and ends with $a_3$, $L=cl(w).$
  10. Case 5: $w=a_1^{n}$, $n\geq 0$. $L=cl(w).$
  11. Proof of Main Theorem and the Rank Conjecture
  12. Acknowledgment

Key Result

Theorem 1.1

For $w\in B_3,$ let $L$ be $cl(w).$ Let $\zeta(w)$ be the absolute value of the coefficient of the second-to-top term of $(t^{\frac{-1}{2}}-t^{\frac{1}{2}})^{|L|-1}\Delta_{L}(t)$. $\widehat{HFK}(L,u(L)-1)$ is as follows:

Figures (2)

  • Figure 1: Generators of $B_3$
  • Figure 2: The skein relation, with $L_+$$L_-$ and $L_0$ from left to right.

Theorems & Definitions (42)

  • Theorem 1.1
  • Remark 1.2
  • Corollary 1.3
  • Proposition 2.1
  • Proposition 2.2
  • Lemma 3.1
  • proof
  • Proposition 3.2
  • proof
  • Proposition 3.3
  • ...and 32 more