Turk's head knots and links: a survey

TL;DR

This survey consolidates the rich theory of Turk's head knots and links $Th(p,q)$ (aka rosette or weaving knots), highlighting their alternating, fibered, hyperbolic, invertible, non-split, and prime nature, with amphichirality when $p$ is odd. It develops a unified picture via a Murasugi-sum decomposition, exact formulas for invariants (Seifert genera, Alexander polynomials, determinants, and homology theories), and comprehensive treatment of the special case $Th(3,q)$. The work also connects these knots to hyperbolic geometry (volumes, right-angled polyhedra), Floer/Khovanov homologies, and concordance, while outlining extensive open problems across diagrammatic, geometric, and algebraic aspects. By organizing results from classical origins (Fox, Mur65, NY00) to modern developments (CKP16, GJS24, AC23), it clarifies where Turk's head knots sit at the intersection of low-dimensional topology and knot invariants, guiding future research directions.

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.

Figures (10)

• Figure 1: The braids $\sigma_i$ and $\sigma^{-1}_i$.
• Figure 2: The link diagram of $T(p,q)$.
• Figure 3: The link diagram of $Th (p,q)$.
• Figure 4: The infinite weave $\mathcal{W}$.
• Figure 5: The Murasugi sum decomposition: $Th(3,5) = T(2,5) \ \#_{10} \ \overline{T(2,5)}$. The red and blue edges together constitute the gray-shaded $10$-gon inside the Seifert surface. The two $10$-gons are glued by identifying blue edges with blue edges and red edges with red edges.

Theorems & Definitions (20)

• Theorem A
• Theorem B
• Theorem C
• Theorem D
• Theorem E
• Theorem F
• Theorem G
• Theorem H
• Theorem I
• Theorem J