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Eva Horvat

Publications and source records attributed to Eva Horvat.

18 recordsLinked to original sources
arXiv2026

The fundamental quandle of ribbon concordances

We describe the fundamental quandle of a properly embedded surface $F$ (possibly with boundary) in $\mathbb{R} ^{3}\times I$, and derive its presentation in terms of a motion picture diagram or a CH-diagram of $F$. Our study is based on the topological definition of the fundamental quandle. We prove that a ribbon concordance $C$ from a classical knot $K_1$ to $K_0$ gives rise to an injective quandle homomorphism $Q(K_0)\to Q(C)$ and a surjective quandle homomorphism $Q(K_1)\to Q(C)$.

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arXiv2025

On discrete symmetries of the cube of smoothings

A link diagram with a labeled barycentric subdivision endows the cube of resolutions with an additional combinatorial structure. We study the set of symmetries, preserving this structure. We construct a combinatorial link invariant, based on the symmetries of triangulated smoothings.

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arXiv2025

On 4-dimensional 3-handle attachments

Kirby diagrams for smooth four-dimensional manifolds typically depict only the 1- and 2-handles, omitting the 3-handles. In this work, we investigate 3-handle attachments and provide tools to explicitly include them in handle diagrams. We show a set of moves involving 3-handles to extend the classical Kirby calculus. Under assumptions and the condition that the number of 3-handles equals the rank of the spherical part of the specific boundary's second homology group, we establish a homological criterion that identifies a geometric basis of disjoint embedded spheres in the boundary corresponding to 3-handle attachments, yielding a uniqueness theorem for 3-handle attachments.

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arXiv2024

Knot quandle decomposition along a torus

We study the structure of the augmented fundamental quandle of a knot whose complement contains an incompressible torus. We obtain the relationship between the fundamental quandle of a satellite knot and the fundamental quandles/groups of its companion and pattern knots. General presentations of the fundamental quandles of a link in a solid torus, a link in a lens space and a satellite knot are described. In the last part of the paper, an algebraic approach to the study of affine quandles is presented and some known results about the Alexander module and quandle colorings are obtained.

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arXiv2023

Nonsmooth manifold decompositions

We study the structure induced on a smooth manifold by a continuous selection of smooth functions. In case such selection is suitably generic, it provides a stratification of the manifold, whose strata are algebraically defined smooth submanifolds. When the continuous selection has nondegenerate critical points, stratification descends to the local topological structure. We analyze this structure for the maximum of three smooth functions on a 4-manifold, which provides a new perspective on the theory of trisections.

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arXiv2022

Flattening knotted surfaces

A knotted surface in the 4-sphere may be described by means of a hyperbolic diagram that captures the 0-section of a special Morse function, called a hyperbolic decomposition. We show that every hyperbolic decomposition of a knotted surface K defines a projection of K onto a 2-sphere, whose set of critical values is the hyperbolic diagram of K. We apply such projections, called flattenings, to define three invariants of knotted surfaces: the layering, the trunk and the partition number. The basic properties of flattenings and their derived invariants are obtained. Our construction is used to study flattenings of satellite 2-knots.

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arXiv2020

Algebraic links in lens spaces

The lens space $L_{p,q}$ is the orbit space of a $\mathbb{Z}_{p}$-action on the three sphere. We investigate polynomials of two complex variables that are invariant under this action, and thus define links in $L_{p,q}$. We study properties of these links, and their relationship with the classical algebraic links. We prove that all algebraic links in lens spaces are fibered, and obtain results about their Seifert genus. We find some examples of algebraic knots in lens spaces, whose lift in the $3$-sphere is a torus link.

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arXiv2019

Constructing biquandles

We define biquandle structures on a given quandle, and show that any biquandle is given by some biquandle structure on its underlying quandle. By determining when two biquandle structures yield isomorphic biquandles, we obtain a relationship between the automorphism group of a biquandle and the automorphism group of its underlying quandle. As an application, we determine the automorphism groups of Alexander and dihedral biquandles. We also discuss product biquandles and describe their automorphism groups.

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arXiv2019

The label bracket for knotted trivalent graphs

We generalize the construction of Akimova and Manturov, define the label bracket for knotted trivalent graphs in $\mathbb{R}^3$ and show it defines an isotopy invariant of such graphs.

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arXiv2019

Knot quandle decompositions

We show that the fundamental quandle defines a functor from the oriented tangle category to a suitably defined quandle category. Given a tangle decomposition of a link $L$, the fundamental quandle of $L$ may be obtained from the fundamental quandles of tangles. We apply this result to derive a presentation of the fundamental quandle of periodic links, composite knots and satellite knots.

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arXiv2019

The topological biquandle of a link

To every oriented link $L$, we associate a topologically defined biquandle $\widehat{\mathcal{B}}_{L}$, which we call the topological biquandle of $L$. The construction of $\widehat{\mathcal{B}}_{L}$ is similar to the topological description of the fundamental quandle given by Matveev. We find a presentation of the topological biquandle and explain how it is related to the fundamental biquandle of the link.

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arXiv2019

From biquandle structures to Hom-biquandles

We investigate the relationship between the quandle and biquandle coloring invariant and obtain an enhancement of the quandle and biquandle coloring invariants using biquandle structures. We also continue the study of biquandle homomorphisms into a medial biquandle begun by the second author et al., finding biquandle analogs of results about Hom-quandles. We describe the biquandle structure of the Hom-biquandle, and consider the relationship between the Hom-quandle and Hom-biquandle.

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arXiv2019

On the Alexander polynomial of links in lens spaces

We show how the Alexander polynomial of links in lens spaces is related to the classical Alexander polynomial of a link in the 3-sphere, obtained by cutting out the exceptional lens space fibre. It follows from these relationship that a certain normalization of the Alexander polynomial satisfies a skein relation in lens spaces.

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arXiv2018

Knot invariants in lens spaces

In this survey we summarize results regarding the Kauffman bracket, HOMFLYPT, Kauffman 2-variable and Dubrovnik skein modules, and the Alexander polynomial of links in lens spaces, which we represent as mixed link diagrams. These invariants generalize the corresponding knot polynomials in the classical case. We compare the invariants by means of the ability to distinguish between some difficult cases of knots with certain symmetries.

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arXiv2017

Rack invariants of links in $L(p,1)$

We describe a presentation for the augmented fundamental rack of a link in the lens space $L(p,1)$. Using this presentation, the (enhanced) counting rack invariants that have been defined for the classical links are applied to the links in $L(p,1)$. In this case, the counting rack invariants also include the information about the action of $π_{1}(L(p,1))$ on the augmented fundamental rack of a link.

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arXiv2017

Knot symmetries and the fundamental quandle

We establish a relationship between the knot symmetries and the automorphisms of the knot quandle. We identify the homeomorphisms of the pair $(S^{3},K)$ that induce the (anti)automorphisms of the fundamental quandle $Q(K)$. We show that every quandle (anti)automorphism of $Q(K)$ is induced by a homeomorphism of the pair $(S^{3},K)$. As an application of those results, we are able to explore some symmetry properties of a knot based on the presentation of its fundamental quandle, which is easily derived from a knot diagram.

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arXiv2014

Double plumbings of disk bundles over spheres

We consider double plumbings of two disk bundles over spheres. We calculate the Heegaard-Floer homology with its absolute grading of the boundary of such a plumbing. Given a closed smooth 4-manifold $X$ and a suitable pair of classes in $H_{2}(X)$, we investigate when this pair of classes may be represented by a configuration of surfaces in $X$ whose regular neighbourhood is a double plumbing of disk bundles over spheres. Using similar methods we study single plumbings of two disk bundles over spheres inside $X$.

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