Turk's head knots and links: a survey

TL;DR

This survey compiles and discusses various properties and results related to Turk's head knots and links, highlighting their mathematical significance and open problems in knot theory.

Contribution

It provides a comprehensive overview of Turk's head knots and links, including new insights into their properties and a focus on the specific case of Th(3,q).

Findings

Turk's head links are alternating, fibered, hyperbolic, invertible, non-split, periodic, and prime.

Th(p,q) links are positive and negative amphichiral if p is odd.

Several open problems and conjectures are presented for future research.

Abstract

We collect and discuss various results on an important family of knots and links called Turk's head knots and links $Th (p,q)$. In the mathematical literature, they also appear under different names such as rosette knots and links or weaving knots and links. Unless being the unknot or the alternating torus links $T(2,q)$, the Turk's head links $Th (p,q)$ are all known to be alternating, fibered, hyperbolic, invertible, non-split, periodic, and prime. The Turk's head links $Th (p,q)$ are also both positive and negative amphichiral if $p$ is chosen to be odd. Moreover, we highlight and present several more results, focusing on Turk's head knots $Th (3,q)$. We finally list several open problems and conjectures for Turk's head knots and links. We conclude with a short appendix on torus knots and links, which might be of independent interest.