arXiv subjects

Oğuz Şavk

Publications and source records attributed to Oğuz Şavk.

7 recordsLinked to original sources

arXiv2026

On homology spheres of the trivial local equivalence class

In the homology cobordism group $Θ_\mathbb{Z}^3$, it is not known if there are non-trivial linear dependences between Seifert fibered spheres. Based on involutive Heegaard Floer theory, Hendricks, Manolescu, and Zemke introduced the local equivalence group $\mathfrak{I}$ along with the homomorphism $h:Θ_\mathbb{Z}^3 \rightarrow \mathfrak{I}$. Using the work of Dai and Stoffregen, one can find non-trivial linear dependences between the images of Seifert fibered spheres under $h$. Therefore, it is interesting to ask if such dependences in $\mathfrak{I}$ originate from $Θ_\mathbb{Z}^3$. In this paper, by employing the $r_s$-invariants from the filtered instanton Floer homology developed by Nozaki, Sato, and Taniguchi, we provide certain conditions to guarantee that such relations are not realized even in the rational homology cobordism group. We also discuss the local equivalence class of the $\operatorname{Pin(2)}$-equivariant Seiberg--Witten Floer stable homotopy type.

math.GT↗

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

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

On homology planes and contractible $4$-manifolds

We call a non-trivial homology sphere a Kirby-Ramanujam sphere if it bounds both a homology plane and a Mazur or Poénaru manifold. In 1980, Kirby found the first example by proving that the boundary of the Ramanujam surface bounds a Mazur manifold and it has remained a single example since then. By tracing their initial step, we provide the first additional examples and we present three infinite families of Kirby-Ramanujam spheres. Also, we show that one of our families of Kirby-Ramanujam spheres is diffeomorphic to the splice of two certain families of Brieskorn spheres. Since this family of Kirby-Ramanujam spheres bound contractible $4$-manifolds, they lie in the class of the trivial element in the homology cobordism group; however, both splice components are separately linearly independent in that group.

math.GT↗

arXiv2024

A survey of the homology cobordism group

In this survey, we present most recent highlights from the study of the homology cobordism group, with a particular emphasis on its long-standing and rich history in the context of smooth manifolds. Further, we list various results on its algebraic structure and discuss its crucial role in the development of low-dimensional topology. Also, we share a series of open problems about the behavior of homology $3$-spheres and the structure of $Θ^3_\mathbb{Z}$. Finally, we briefly discuss the knot concordance group $\mathcal{C}$ and the rational homology cobordism group $Θ^3_\mathbb{Q}$, focusing on their algebraic structures, relating them to $Θ^3_\mathbb{Z}$, and highlighting several open problems. The appendix is a compilation of several constructions and presentations of homology $3$-spheres introduced by Brieskorn, Dehn, Gordon, Seifert, Siebenmann, and Waldhausen.

math.GT↗

arXiv2023

Almost simple linear graphs, homology cobordism and connected Heegaard Floer homology

Continuing our previous work, we effectively compute connected Heegaard Floer homologies of two families of Brieskorn spheres realized as the boundaries of almost simple linear graphs. Using Floer theoretic invariants of Dai, Hom, Stoffregen, and Truong, we show that these Brieskorn spheres also generate infinite rank summands in the homology cobordism group. Our computations also have applications to the concordance of classical knots and $0$-concordance of $2$-knots.

math.GT↗