Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2610.16600

The symmetry group of Turk's head knots

Werner Stanggassinger
View original source ↗Read source document ↗Export metadata ↓

Abstract

For p, q >= 3 with gcd(p,q) = 1 we show that the symmetry group of the Turk's head knot Th(p,q), the closure of the braid (sigma_1 sigma_2^{-1} sigma_3 sigma_4^{-1} ... sigma_{p-1}^{+-1})^q, is the dihedral group of order 4q. This proves Conjecture R of Di Prisa and Şavk for q >= 3; the case p = 3 is due to Sakuma and Weeks. The argument is diagrammatic: by the flyping theorem of Menasco and Thistlethwaite, the symmetries of a knot with a prime, reduced alternating diagram without non-trivial flypes are the symmetries of the diagram (we write out a proof for the diagrams at hand), and we show that the standard diagram of Th(p,q) has no non-trivial flypes and exactly 4q symmetries. The chirality of Th(p,q) for even p, which also follows from a theorem of Murasugi, is visible in the group. We compare the argument with the geometric one of Sakuma and Weeks for p = 3.

Explore related subjects

math.GT

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Werner Stanggassinger. 2026-10-12. The symmetry group of Turk's head knots. https://arxiv.org/abs/2610.16600

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Maps between circle bundles: Fiber-preserving, Finiteness and Realization of mapping degree sets

Let $E_i$ be an oriented circle bundle over a closed oriented $n$-manifold $M_i$ with $i=1,2$ and assume that $M_2$ is aspherical. We prove the following geometric realization result, which extends the classical result for maps between circle bundles over surfaces to all dimensions $\geq 2$. Let $f\colon E_1\to E_2$ be a map. If the induced homomorphism $f_*$ on $π_1$ sends the fiber subgroup of $π_1(E_1)$ to the fiber subgroup of $π_1(E_2)$, then there is a fiber-preserving map in the homotopy class of $f$. Using the above result and a computation of the mapping degree set of fiber-preserving maps between $E_1$ and $E_2$, we obtain the following results on the finiteness and the realization problems of mapping degree sets. ($\mathcal F$) The mapping degree set $D(E_1, E_2)$ is finite if $M_2$ is hyperbolic and the Euler class of $E_2$ is not torsion. ($\mathcal R$) For any finite set $A$ of integers containing $0$ and any $n>2$, $A$ is the mapping degree set $D(M,N)$ for some closed oriented $n$-manifolds $M$ and $N$. Item ($\mathcal F$) extends the previously known $3$-dimensional case to all dimensions $\geq 3$. Item ($\mathcal R$) gives a complete answer to the realization problem on finite sets of integers (containing 0) in all dimensions, including in particular the previously unknown cases in dimensions $ 4$ and $5$.

math.GT↗
arXiv2026

Triple-cup product forms of 3-manifolds and Heegaard diagrams

The triple-cup product form $μ$ is a classical invariant of $3$-manifolds, determining the cohomology ring up to torsion. Given a closed, connected, oriented $3$-manifold $M$, we describe an explicit formula for computing $μ$ from a Heegaard diagram of $M$. Then, we show that the triple-cup product form $μ$ can be recovered as a reduction of Turaev's homotopy intersection form $η$ of the Heegaard surface.

math.GT↗
arXiv2026

A topological interpretation of numbers

We survey recent advancements on the realisation problem for mapping degree sets. In particular, we explain that any finite set containing zero is the mapping degree set between some $n$-manifolds, for each $n\geq3$. Our building factors for the construction of the realising manifolds will be aspherical manifolds. Thus, along the way, we review a couple of open questions related to maps of non-zero degree for aspherical manifolds.

math.GT↗