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Boštjan Gabrovšek

Publications and source records attributed to Boštjan Gabrovšek.

18 recordsLinked to original sources
arXiv2026

Classification of (uncolored) bonded knots and links

We present a systematic classification of uncolored bonded knots with singularity number at most seven. Bonded knots provide a topological model for closed protein chains with intramolecular bridges, such as disulfide bonds. Following the tradition of knot tabulation, we describe a procedure based on the generation of planar graphs, their conversion into bonded knot diagrams, and the use of the Yamada polynomial together with brute-force Reidemeister moves to distinguish topological knotted types.

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arXiv2026

A table of knotoids in $S^3$ up to seven crossings

We present a complete classification of spherical knotoids with up to six crossings and conjecture that our classification up to seven crossings is complete. Our work extends the tradition of knot tabulation to the setting of knotoids introduced by Turaev. We describe the methods used to enumerate diagrams, simplify them, and distinguish equivalence classes using a collection of invariants including the Kauffman bracket, the Arrow polynomial, the Affine index polynomial, the Mock Alexander polynomial, and the Yamada polynomial of the closure. We also investigate the chirality and rotational symmetries of these knotoids. Applications to protein entanglement illustrate the importance of such classifications.

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arXiv2025

Bonded braids and the Markov theorem

Bonded knots arise naturally in topological protein modeling, where intramolecular interactions such as disulfide bridges stabilize folded configurations. These structures extend classical knot theory by incorporating embedded graphs, and have been formalized as bonded knots. In this paper, we develop the algebraic theory of bonded braids, introducing the bonded braid monoid in the topological and rigid settings, which encodes both classical braid crossings and (rigid) bonded connections. We prove bonded analogues of the Alexander and Markov theorems, establishing that every bonded knot arises as the closure of a bonded braid and that two bonded knots are equivalent if and only if their braid representatives are related by a finite sequence of algebraic (Markov-like) moves. In addition, we define the bonded Burau and reduced bonded Burau representations of the monoid, extending classical braid group representations to the bonded setting, and analyze their (non-)faithfulness in low dimensions.

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arXiv2025

Classification of algebraic tangles

We study algebraic tangles as fundamental components in knot theory, developing a systematic approach to classify and tabulate prime tangles using a novel canonical representation. The canonical representation enables us to distinguish mutant tangles, which fills the gaps in previous classifications. Moreover, we increase the classification of prime tangles up to 14 crossings and analyze tangle symmetry groups. We provide a database of our results: https://tangleinfo.cent.uw.edu.pl.

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arXiv2025

The bracket polynomial of bonded knots and applications to entangles proteins

We model proteins with intramolecular bonds, such as disulfide bridges, using the concept of bonded knots -- closed loops in three-dimensional space equipped with additional bonds that connect different segments of the knot. We extend the Kauffman bracket polynomial (which is closely related to the Jones polynomial) to bonded knots through the introduction of the bonded version of the Kauffman bracket skein module. We show that this module is infinitely generated and torsion-free for both the rigid and topological case of bonded knots, providing an invariant of such structures.

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arXiv2023

A fresh look to a randomized massively parallel graph coloring algorithm

Petford and Welsh introduced a sequential heuristic algorithm for (approximately) solving the NP-hard graph coloring problem. The algorithm is based on the antivoter model and mimics the behaviour of a physical process based on a multi-particle system of statistical mechanics. It was later shown that the algorithm can be implemented in massively parallel model of computation. The increase of processing power in recent years allow us to perform an extensive analysis of the algorithms on a larger scale, leading to possibility of a more comprehensive understanding of the behaviour of the algorithm including the phase transition phenomena.

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arXiv2023

On nonlocal Dirichlet problems with oscillating term

In this paper, a class of nonlocal fractional Dirichlet problems is studied. By using a variational principle due to Ricceri (whose original version was given in J. Comput. Appl. Math. 113 (2000), 401-410), the existence of infinitely many weak solutions for these problems is established by requiring that the nonlinear term $f$ has a suitable oscillating behaviour either at the origin or at infinity.

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arXiv2022

Invariants of multi-linkoids

In this paper, we extend the definition of a knotoid that was introduced by Turaev, to multi-linkoids that consist of a number of knot and knotoid components. We study invariants of multi-linkoids that lie in a closed orientable surface, namely the Kauffman bracket polynomial, ordered bracket polynomial, the Kauffman skein module, and the $T$-invariant in relation with generalized $Θ$-graphs.

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arXiv2019

Infinitely many sign-changing solutions for Kirchhoff type problems in $\mathbb{R}^3$

In this paper, we consider the following nonlinear Kirchhoff type problem: \[ \left\{\begin{array}{lcl}-\left(a+b\displaystyle\int_{\mathbb{R}^3}|\nabla u|^2\right)Δu+V(x)u=f(u), & \textrm{in}\,\,\mathbb{R}^3,\\ u\in H^1(\mathbb{R}^3), \end{array}\right. \] where $a,b>0$ are constants, the nonlinearity $f$ is superlinear at infinity with subcritical growth and $V$ is continuous and coercive. For the case when $f$ is odd in $u$ we obtain infinitely many sign-changing solutions for the above problem by using a combination of invariant sets method and the Ljusternik-Schnirelman type minimax method. To the best of our knowledge, there are only few existence results for this problem. It is worth mentioning that the nonlinear term may not be 4-superlinear at infinity, in particular, it includes the power-type nonlinearity $|u|^{p-2}u$ with $p\in(2,4]$.

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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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arXiv2019

A Markov theorem for generalized plat decomposition

We prove a Markov theorem for tame links in a connected closed orientable 3-manifold $M$ with respect to a plat-like representation. More precisely, given a genus $g$ Heegaard surface $Σ_g$ for $M$ we represent each link in $M$ as the plat closure of a braid in the surface braid group $B_{g,2n}=π_1(C_{2n}(Σ_g))$ and analyze how to translate the equivalence of links in $M$ under ambient isotopy into an algebraic equivalence in $B_{g,2n}$. First, we study the equivalence problem in $Σ_g\times [0,1]$, and then, to obtain the equivalence in $M$, we investigate how isotopies corresponding to "sliding" along meridian discs change the braid representative. At the end we provide explicit constructions for Heegaard genus 1 manifolds, i.e. lens spaces and $S^2\times S^1$.

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arXiv2018

The categorification of the Kauffman bracket skein module of $\mathbb{R}P^3$

Khovanov homology, an invariant of links in $\mathbb{R}^3$, is a graded homology theory that categorifies the Jones polynomial in the sense that the graded Euler characteristic of the homology is the Jones polynomial. Asaeda, Przytycki and Sikora generalized this construction by defining a double graded homology theory that categorifies the Kauffman bracket skein module of links in $I$-bundles over surfaces, except for the surface $\mathbb{R}P^2$, where the construction fails due to strange behaviour of links when projected to the non-orientable surface $\mathbb{R}P^2$. This paper categorifies the missing case of the twisted $I$-bundle over $\mathbb{R}P^2$, $\mathbb{R}P^2$ \widetilde{\times} I \approx \rpt \setminus \{\ast\}$, by redefining the differential in the Khovanov chain complex in a suitable manner.

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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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arXiv2018

Link diagrams in Seifert manifolds and applications to skein modules

In this survey paper we present results about link diagrams in Seifert manifolds using arrow diagrams, starting with link diagrams in $F\times S^1$ and $N\hat{\times}S^1$, where $F$ is an orientable and $N$ an unorientable surface. Reidemeister moves for such arrow diagrams make the study of link invariants possible. Transitions between arrow diagrams and alternative diagrams are presented. We recall results about %the knot group presentation for lens spaces and the Kauffman bracket and HOMFLYPT skein modules of some Seifert manifolds using arrow diagrams, namely lens spaces, a product of a disk with two holes times $S^1$, $\mathbb{R}P^3 \# \mathbb{R}P^3$, and prism manifolds. We also present new bases of the Kauffman bracket and HOMFLYPT skein modules of the solid torus and lens spaces.

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arXiv2016

Tabulation of prime knots in lens spaces

Using computational techniques we tabulate prime knots up to five crossings in the solid torus and the infinite family of lens spaces $L(p,q)$. For these knots we calculate the second and third skein module and establish which prime knots in the solid torus are amphichiral. Most knots are distinguished by the skein modules. For the handful of cases where the skein modules fail to detect inequivalent knots, we calculate and compare the hyperbolic structures of the knot complements. We were unable to resolve a handful of 5-crossing cases for $p\geq 13$.

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arXiv2015

On the KBSM of links in lens spaces

In this paper the properties of the Kauffman bracket skein module of $L(p,q)$ are investigated. Links in lens spaces are represented both through band and disk diagrams. The possibility to transform between the diagrams enables us to compute the Kauffman bracket skein module on an interesting class of examples consisting of inequivalent links with equivalent lifts in the $3$-sphere. The computation show that the Kauffman bracket skein module is an essential invariant, that is, it may take different values on links with equivalent lifts. We also show how the invariant is related to the Kauffman bracket of the lift in the $3$-sphere.

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arXiv2015

On the Seifert fibered space link group

We introduce generalized arrow diagrams and generalized Reidemeister moves for diagrams of links in Seifert fibered spaces. We give a presentation of the fundamental group of the link complement. As a corollary we are able to compute the first homology group of the complement and the twisted Alexander polynomials of the link.

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