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On the Qazaqzeh-Chbili-Lowrance conjecture

Shreev Goyal, Joshua Kou, Christine Ruey Shan Lee, Emma Wu, Helen Yang, Aaron Zhou

TL;DR

The work characterizes a five-parameter family L = D(r, s, t, −u, −v) of Turaev genus 2 links in the context of the Qazaqzeh-Chbili-Lowrance conjecture, by computing c(L), g_T(L), and span(V_L(t)) to assess adequacy through the defect δ(L) = c(L) − g_T(L) − span(V_L(t)). Under the key regime r = s, u = v, and t > max{r+1, u+1}, the authors show the leading and trailing Jones coefficients are 2 and derive span(V_L(t)) = r + t + u, yielding δ(L) = s + v − 2 and, for s + v > 4, strict inequality span(V_L(t)) < c(L) − g_T(L). This provides evidence for non-adequacy in this family and supports the conjecture’s characterization of adequacy via the Jones-span relation, while also detailing how Reidemeister moves can affect the Turaev genus and proposing a path toward algorithmic searches for counterexamples to genus additivity. The results combine Kauffman bracket analysis, the Kauffman 2-variable polynomial, and state-graph techniques to simultaneously determine minimal crossing numbers and Turaev-genus properties for a structured, twist-region-rich family of links. Overall, the paper advances understanding of how Turaev genus interacts with Jones polynomial data and adequacy in a concrete, parameterized link family, with implications for identifying potential counterexamples and guiding future explorations of genus additivity.

Abstract

We compute the minimum crossing numbers for a family of Turaev genus 2 links to verify the Qazaqzeh-Chbili-Lowrance conjecture for the family.

On the Qazaqzeh-Chbili-Lowrance conjecture

TL;DR

The work characterizes a five-parameter family L = D(r, s, t, −u, −v) of Turaev genus 2 links in the context of the Qazaqzeh-Chbili-Lowrance conjecture, by computing c(L), g_T(L), and span(V_L(t)) to assess adequacy through the defect δ(L) = c(L) − g_T(L) − span(V_L(t)). Under the key regime r = s, u = v, and t > max{r+1, u+1}, the authors show the leading and trailing Jones coefficients are 2 and derive span(V_L(t)) = r + t + u, yielding δ(L) = s + v − 2 and, for s + v > 4, strict inequality span(V_L(t)) < c(L) − g_T(L). This provides evidence for non-adequacy in this family and supports the conjecture’s characterization of adequacy via the Jones-span relation, while also detailing how Reidemeister moves can affect the Turaev genus and proposing a path toward algorithmic searches for counterexamples to genus additivity. The results combine Kauffman bracket analysis, the Kauffman 2-variable polynomial, and state-graph techniques to simultaneously determine minimal crossing numbers and Turaev-genus properties for a structured, twist-region-rich family of links. Overall, the paper advances understanding of how Turaev genus interacts with Jones polynomial data and adequacy in a concrete, parameterized link family, with implications for identifying potential counterexamples and guiding future explorations of genus additivity.

Abstract

We compute the minimum crossing numbers for a family of Turaev genus 2 links to verify the Qazaqzeh-Chbili-Lowrance conjecture for the family.
Paper Structure (18 sections, 19 theorems, 55 equations, 20 figures, 1 table)

This paper contains 18 sections, 19 theorems, 55 equations, 20 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The Jones polynomial via the Kauffman bracket state sum
  4. The Turaev genus
  5. The degree span of the Jones polynomial
  6. The Kauffman 2-variable polynomial
  7. Extreme coefficients of the Jones polynomial.
  8. Maximum degree
  9. Minimum degree
  10. Case $\sigma_u + \sigma_v = u+v$
  11. Case $\sigma_u + \sigma_v \not= u+v$
  12. The span of the Jones polynomial
  13. A link example $L = D(3, 3, 5, -3, -3)$.
  14. When is $L = D(r, s, t, -u, -v)$ a link and when is it a knot?
  15. Crossing number from the Kauffman polynomial and the Jones polynomial
  16. ...and 3 more sections

Key Result

Theorem 1.2

Let $L = D(r, s, t, -u, -v)$, then

Figures (20)

  • Figure 1: The knot $K(r, s, t, -u, -v) = K(3, 3, 4, -3, -3)$.
  • Figure 4: A Kauffman state at a crossing.
  • Figure 5: $D_+$: a positive crossing. $D_-$: a negative crossing.
  • Figure 6: $A$-and $B$-resolutions and the dashed edges recording the location of the crossing.
  • Figure 7: A bridge of length 2.
  • ...and 15 more figures

Theorems & Definitions (39)

  • Conjecture 1.1
  • Definition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 2.1: Kauffman state
  • Lemma 2.1
  • Definition 2.2
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6: Turaev genus
  • ...and 29 more