Twisted torus knots with Horadam parameters

TL;DR

This paper explores twisted torus knots with parameters defined by the Horadam sequence, generalizing previous Fibonacci-based classifications and identifying new families of such knots.

Contribution

It introduces a new class of twisted torus knots with parameters following the Horadam sequence, extending Lee's Fibonacci-based results.

Findings

Families of twisted torus knots with Horadam parameters are characterized.

Lists of primitive/primitive and primitive/Seifert twisted torus knots are provided.

Connections between these lists and Horadam twisted torus knots are established.

Abstract

Sangyop Lee has done much work to determine the knot types of twisted torus knots, including classifying the twisted torus knots which are the unknot. Among the unknotted twisted torus knots are those of the form $K(F_{n+2}, F_n, F_{n+1}, -1)$, where $F_i$ is the $i$th Fibonacci number. Here, we consider twisted torus knots with parameters that are defined recursively, similarly to the Fibonacci sequence. We call these Horadam parameters, after the generalization of the Fibonacci sequence introduced by A.F. Horadam. Here, we provide families of twisted torus knots that generalize Lee's work with Horadam parameters. Additionally, we provide lists of primitive/primitive and primitive/Seifert twisted torus knots and connect these lists to the Horadam twisted torus knots.