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Using Fibonacci Numbers and Chebyshev Polynomials to Express Fox Coloring Groups and Alexander-Burau-Fox Modules of Diagrams of Wheel Graphs

Anthony Christiana, Huizheng Guo, Jozef H. Przytycki

TL;DR

The paper derives closed formulas for the reduced Fox coloring groups of wheel-graph diagrams, showing Col^{red}(D_n) is determined by Fibonacci numbers and Lucas numbers via wheel-graph combinatorics. It then generalizes to the Alexander-Burau-Fox module over $\mathbb{Z}[t^{\pm1}]$ using Chebyshev polynomials of the second kind, providing an explicit module decomposition into cyclic summands and computing the determinant of a reduced ABF matrix. The work illuminates a deep connection between colorings, Chebyshev polynomials, and Burau representation, linking knot-theoretic invariants to branched-covering topology through product-to-sum identities and explicit matrix reductions. Overall, the results yield compact, computable formulas for both the reduced coloring groups and ABF modules for the class of closures of $(\sigma_1\sigma_2^{-1})^n$, with potential implications for branched-covering homology and orderability questions in 3-manifold topology.

Abstract

In this paper we compute the Reduced Fox Coloring Group of the diagrams of Wheel Graphs which can also be represented at the closure of the braids $(σ_1 σ_2^{-1})^n$. In doing so, we utilize Fibonacci numbers and their properties. Following this, we generalize our result to compute the Alexander-Burau-Fox Module over the ring $\mathbb{Z}[t^{\pm 1}]$ for the same class of links. In our computation, Chebyshev polynomials function as a generalization of Fibonacci Numbers.

Using Fibonacci Numbers and Chebyshev Polynomials to Express Fox Coloring Groups and Alexander-Burau-Fox Modules of Diagrams of Wheel Graphs

TL;DR

The paper derives closed formulas for the reduced Fox coloring groups of wheel-graph diagrams, showing Col^{red}(D_n) is determined by Fibonacci numbers and Lucas numbers via wheel-graph combinatorics. It then generalizes to the Alexander-Burau-Fox module over using Chebyshev polynomials of the second kind, providing an explicit module decomposition into cyclic summands and computing the determinant of a reduced ABF matrix. The work illuminates a deep connection between colorings, Chebyshev polynomials, and Burau representation, linking knot-theoretic invariants to branched-covering topology through product-to-sum identities and explicit matrix reductions. Overall, the results yield compact, computable formulas for both the reduced coloring groups and ABF modules for the class of closures of , with potential implications for branched-covering homology and orderability questions in 3-manifold topology.

Abstract

In this paper we compute the Reduced Fox Coloring Group of the diagrams of Wheel Graphs which can also be represented at the closure of the braids . In doing so, we utilize Fibonacci numbers and their properties. Following this, we generalize our result to compute the Alexander-Burau-Fox Module over the ring for the same class of links. In our computation, Chebyshev polynomials function as a generalization of Fibonacci Numbers.
Paper Structure (10 sections, 24 theorems, 49 equations, 7 figures)

This paper contains 10 sections, 24 theorems, 49 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Fibonacci numbers and the Reduced Coloring Group of Wheel Graphs, $Col^{red}(D)$
  3. Alexander-Burau-Fox matrix of a link diagram
  4. Chebyshev Polynomials
  5. Product-to-Sum Formulas
  6. Alexander-Burau-Fox Module
  7. Computing determinant of the matrix $A'_n$
  8. Relation to Burau representation
  9. Speculations and Future Directions
  10. Acknowledgments

Key Result

Theorem 1.6

Let $F_k$ be the Fibonacci sequence defined by Let $D_n$ be the closure of the braid $(\sigma_1\sigma_2^{-1})^n,$ that is, $D_n=D(W_n)$ as in Figure W7. Then In particular, for $n=2, 3, 4, 5, 6, 7,$ we have $\mathbb{Z}_5$, $\mathbb{Z}_4 \oplus \mathbb{Z}_4$, $\mathbb{Z}_{15} \oplus \mathbb{Z}_3$, $\mathbb Z_{11} \oplus \mathbb Z_{11}$, $\mathbb Z_{40} \oplus \mathbb Z_{8}$, and $\mathbb Z_{29} \

Figures (7)

  • Figure 1: Crossing from an edge
  • Figure 2: Coloring of a crossing
  • Figure 3: Wheel graph $W_7$ and its Tait diagram, $D_7=D(W_7)$ representing the closure of the braid $(\sigma_1 \sigma_2^{-1})^7.$
  • Figure 4: Labeling part of $(\sigma_1\sigma_2^{-1})^n$
  • Figure 5: ABF Module Relations
  • ...and 2 more figures

Theorems & Definitions (52)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Definition 1.4
  • Definition 1.5
  • Theorem 1.6
  • Proposition 2.1
  • proof
  • Lemma 2.2
  • Lemma 2.3
  • ...and 42 more