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Alessio Di Prisa

Publications and source records attributed to Alessio Di Prisa.

8 recordsLinked to original sources
arXiv2026

Turk's head knots and links: a survey

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.

math.GT↗
arXiv2025

On the detection of knotted spheres by their traces in high dimensions

For every $n \geq 4$, we demonstrate the existence of non-isotopic smooth $(n-2)$-knots in $S^n$ with diffeomorphic traces by generalising the RBG link construction to all dimensions. Conversely, we prove that for every $n \geq 4$, the unknot in $S^n$ is detected by the diffeomorphism type of its surgery and hence by its trace.

math.GT↗
arXiv2025

Every negative amphichiral knot is rationally slice

In 2009, Kawauchi proved that every strongly negative amphichiral knot is rationally slice. However, as shown by Hartley in 1980, there are examples of negative amphichiral knots that are not strongly negative amphichiral. In this paper, we prove that every negative amphichiral link whose amphichiral map preserves each component is rationally slice. Our proof relies on a systematic analysis of the action induced by the negative amphichiral map on the JSJ decomposition of the link exterior. Moreover, we provide sufficient conditions on such an action to deduce when a negative amphichiral knot is either isotopic to, or concordant to, a strongly negative amphichiral knot. In particular, we prove that every fibered negative amphichiral knot is strongly negative amphichiral, answering a question asked by Kim and Wu in 2016 on Miyazaki knots.

math.GT↗
arXiv2024

Equivariant Q-sliceness of strongly invertible knots

We introduce and study the notion of equivariant $\mathbb{Q}$-sliceness for strongly invertible knots. On the constructive side, we prove that every Klein amphichiral knot, which is a strongly invertible knot admitting a compatible negative amphichiral involution, is equivariant $\mathbb{Q}$-slice in a single $\mathbb{Q}$-homology $4$-ball, by refining Kawauchi's construction and generalizing Levine's uniqueness result. On the obstructive side, we show that the equivariant version of the classical Fox-Milnor condition, proved recently by the first author, also obstructs equivariant $\mathbb{Q}$-sliceness. We then introduce the equivariant $\mathbb{Q}$-concordance group and study the natural maps between concordance groups as an application. We also list some open problems for future study.

math.GT↗
arXiv2024

Equivariant algebraic concordance of strongly invertible knots

By considering a particular type of invariant Seifert surfaces we define a homomorphism $Φ$ from the (topological) equivariant concordance group of directed strongly invertible knots $\widetilde{\mathcal{C}}$ to a new equivariant algebraic concordance group $\widetilde{\mathcal{G}}^{\mathbb{Z}}$. We prove that $Φ$ lifts both Miller and Powell's equivariant algebraic concordance homomorphism, and Alfieri and Boyle's equivariant signature. Moreover, we provide a partial result on the isomorphism type of $\widetilde{\mathcal{G}}^{\mathbb{Z}}$, and we obtain a new obstruction to equivariant sliceness, which can be viewed as an equivariant Fox-Milnor condition. We define new equivariant signatures and using these we obtain novel lower bounds on the equivariant slice genus. Finally, we show that $Φ$ can obstruct equivariant sliceness for knots with Alexander polynomial one.

math.GT↗
arXiv2023

Solvability of concordance groups and Milnor invariants

Using Milnor invariants, we prove that the concordance group $\mathcal{C}(2)$ of $2$-string links is not solvable. As a consequence we prove that the equivariant concordance group of strongly invertible knots is also not solvable, and we answer a conjecture posed by Kuzbary.

math.GT↗
arXiv2023

A new invariant of equivariant concordance and results on 2-bridge knots

We study the equivariant concordance classes of two-bridge knots, providing an easy formula to compute their butterfly polynomial, and we give two different proofs that no two-bridge knot is equivariantly slice. Finally, we introduce a new invariant of equivariant concordance for strongly invertible knots. Using this invariant as an obstruction we strengthen the result on two-bridge knots, proving that their equivariant concordance order is always infinite.

math.GT↗