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

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arXiv · 2310.19979

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

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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 $(\sigma_1 \sigma_2^{-1})^n$. 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.

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Anthony Christiana, Huizheng Guo, Jozef H. Przytycki. 2023-10-30. Using Fibonacci Numbers and Chebyshev Polynomials to Express Fox Coloring Groups and Alexander-Burau-Fox Modules of Diagrams of Wheel Graphs. https://arxiv.org/abs/2310.19979

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