几何拓扑

We generalize the horofunction compactification to maps that are not distance functions. As an application we define a horofunction counterpart to the Teichm\"uller distance, and discuss its properties.

We show that if $X$ is a finite-dimensional Polish metric space, then the natural bijection $\mathrm{VR}(X;r)\to \mathrm{VR^m}(X;r)$ from the (open) Vietoris-Rips complex to the Vietoris-Rips metric thickening is a homotopy equivalence.…

The $\Delta$-unknotting number for a knot is defined as the minimum number of $\Delta$-moves needed to deform the knot into the trivial knot. We determine the $\Delta$-unknotting numbers for two-bridge knots of type $C(2\beta_1, 2\beta_2,…

We use a decomposition of the tensor of the fundamental representation of the quantum group $U_q(\mathfrak{sl}_N)$ and the Rosso-Jones formula to establish a peculiar ``panhandle'' shape of the HOMFLY-PT polynomial of the reverse parallel…

This paper studies locally linear involutions on S^4. Our main theorem shows that any such involution with a 1-dimensional fixed-point set is necessarily linear, provided the fixed-point set admits an equivariant tubular neighborhood. The…

For links $L \subset \Sigma \times [0,1]$, where $\Sigma$ is a closed orientable surface, we define a $U_q(\mathfrak{gl}(1|1))$ Reshetikhin-Turaev invariant with coefficients in $\mathbb{Z}[H_1(\Sigma)]$. This invariant turns out to be…

We study the existence of branched coverings between closed $3$-manifolds, with emphasis on universal knots and links. We prove that the only closed $3$-manifolds that admit a universal link are spherical. Furthermore, we distinguish…

In this paper, we primarily investigate the following symmetric presentation of the surface group $\pi_1(\Sigma_g)=\left\langle c_1,\dots, c_{2g}\mid c_1\cdots c_{2g}c_1^{-1}\cdots c_{2g}^{-1}\right\rangle$. For every nontrivial element…

Kirby diagrams for smooth four-dimensional manifolds typically depict only the 1- and 2-handles, omitting the 3-handles. In this work, we undertake a study of 3-handle attachments and provide tools to explicitly include them in handle…

A virtual link is said to be almost classical (AC) if it has a homologically trivial representative in some thickened surface $\Sigma \times [0,1]$, where $\Sigma$ is a closed orientable surface. AC links provide a useful window for…

There are 352.2 million prime knots in the 3-sphere with at most 19 crossings. We study which of these knots are slice, in both the smooth and topological categories. While no algorithm is known for deciding whether a given knot is slice in…

Let $S_g$ be a closed, connected, and oriented smooth surface of genus $g\geq 2$. Let the mapping class group of $S_g$ be denoted by $\mathrm{Mod}(S_g)$ and the Teichm\"{u}ller space of $S_g$ by $\mathrm{Teich}(S_g)$. It is known that…

In this paper, we present a complete coarse classification of non-loose Legendrian and transverse torus knots in any contact structure on $S^1\times S^2$.

We present new families of examples of non-simple prime Legendrian and transversal knots in tight Lens spaces, which demonstrate that the botany of Legendrians in Lens space is rich. In fact, there are more non-isotopic Legendrians that are…

In the present paper, we discuss a way of generalising Vassiliev knot invariants and weight systems to framed chord diagrams having framing 0 and 1.

Let $g$ be a non-negative integer, $\Sigma _g$ a closed orientable surface of genus $g$, and $\mathcal{M}_g$ its mapping class group. We classify all the group homomorphisms $\pi _1(\Sigma _g)\to G$ up to the action of $\mathcal{M}_g$ on…

We study the Teichm\"uller space $\mathcal{T}(S,\underline{p})$ of hyperbolic cone-surfaces of fixed topological type with marked cone singularities. Fix a combinatorial triangulation $G$, and let $\mathcal{T}(G)\subset…

We show that for every genus $g \geq 0$, there exist quasipositive knots $K_0^g$ and $K_1^g$ such that there is a cobordism of genus $g=|g_4(K_1^g)-g_4(K_0^g)|$ between $K_0^g$ and $K_1^g$, but there is no ribbon cobordism of genus $g$ in…

We prove that every 4-dimensional oriented handlebody without 3- and 4-handles can be modified to admit infinitely many exotic smooth structures, and moreover prove that their genus functions are pairwise equivalent. We furthermore show…

We give an invariant construction of reduced HOMFLY homology for arbitrary links reduced at components of arbitrary color and prove some structural properties relating this invariant to unreduced HOMFLY homology. Combined with previous…