Twisted torus links that are unlinks

TL;DR

This work completely classifies twisted torus links $T(p,q,r,s)$ that are unlinks. By combining linking-number computations with Lee’s classification of unknotted twisted torus knots and a component-wise analysis of how the twisting interacts with the torus components, the authors reduce the problem to a finite set of parameter families. The main result identifies three explicit families for the two-component case and two additional families for links with at least three components, depending on the gcd $d=\gcd(p,q)$. The classification provides a precise answer within the broader framework of generalized T-links and extends Lee’s knot-level results to the link setting, illustrating the constraints that unlinking imposes on the twisting data.

Abstract

A twisted torus link $T(p,q,r,s)$ is obtained by performing $s$ full twists on $r$ adjacent strands of the $(p,q)$-torus link. In this paper, we classify twisted torus links that are unlinks. We give a complete characterization of all parameter families $(p,q,r,s)$ for which the associated twisted torus link is an unlink.

Figures (4)

• Figure 1: Left hand side: The red $(5,2)$-torus knot (with clockwise orientation). By convention in this paper, crossings in this diagram of $T(5,2)$ are considered to be positive, and the braid contained in this diagram of $T(5,2)$ is presented as a product of positive braid group genenrators $\sigma_i$. The blue circle indicates the disk along which the full twist is performed. Right hand side: The twisted torus knot $(5,2,3,-1)$. This is constructed from the $(5,2)$-torus knot by adding $-1$ full twist to the first $3$ strands.
• Figure 2: The twisted torus knots $(5,2,6,s)$ are constructed from the $(5,2)$-torus knot by performing full twists along the disk enclosed by the blue circle in this figure. Exactly $6$ red strands pass through this disk.
• Figure 3: The braid $(\sigma_{2n}^{-1}\sigma_{2n-1}^{-1} \dots\sigma_{1}^{-1})\sigma_{2n+1}\sigma_{2n}\dots\sigma_{2}$. Its closure is the unlink with two components, given by the red strands and the blue strands respectively. The red component is placed above the blue component, and each component is an unknot.
• Figure 4: A projection of the embedded annulus, with two boundary components colored red and blue. For any $n$, the link $(4n,n,2n,-1)$ is supported on this annulus.

Theorems & Definitions (34)

• Theorem 1
• Lemma 2
• proof
• Lemma 3
• proof
• Theorem 4: Theorem 1.1 in Lee
• Proposition 5
• proof
• Proposition 6
• proof