Twisted torus knots with Horadam parameters

TL;DR

The paper generalizes twisted torus knots by using Horadam sequences to define Horadam twisted torus knots with consecutive parameters $(m,n;1,1)$, then analyzes their knot types via braid representations and Lee's results to determine torus, satellite, or non-torus status. It proves that all $(m,n;1,1)$ Horadam TT knots are primitive/primitive and provides comprehensive lists of primitive/primitive and primitive/Seifert knots, including corrections and extensions to prior work. The study links Horadam TT knots to Kadokami's results and clarifies which parameter families yield Horadam knots within broader primitive/seifert frameworks, while highlighting open questions about overlaps with non-Horadam families. Collectively, this work expands the taxonomy of twisted torus knots and informs potential Seifert-fibered surgery structures in knot complements, with explicit braid and Euclidean-algorithm analyses guiding the classifications.

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.

Figures (11)

• Figure 1: $T(5,3)$ with $D$ intersecting 8 strands
• Figure 2: $\sigma_i$
• Figure 3: Constructing $K(p,q,p+q,n)$
• Figure 4: $K(p,q,p+q, n)$ as a braid
• Figure 5: Braids representing $K(p,q,p-q, +1)$

Theorems & Definitions (35)

• Lemma 1: Lee LeeTTKTypes
• Definition 1
• Lemma 2: Horadam Horadam1
• Lemma 3
• proof
• Lemma 4
• proof
• Remark 1
• Lemma 5
• proof