
Boštjan GabrovšekUniversity of Ljubljana · Faculty of Mechanical Engineering
Boštjan Gabrovšek
PhD
About
34
Publications
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276
Citations
Introduction
Boštjan Gabrovšek currently works at the University of Ljubljana. Boštjan does research in Knot theory and low dimensional topology. Their most recent publication is 'A Markov theorem for generalized plat decomposition.'
Skills and Expertise
Publications
Publications (34)
We compute the HOMFLYPT skein module of the lens spaces Lp,1Lp,1 and present a free basis of this module for each p.
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...
We explore properties of the Alexander polynomial and twisted Alexander polynomial of links in the lens spaces. In particular, we calculate the Alexander polynomial of some families of links and show how the Alexander polynomial is connected with the classical Alexander polynomial of the link in $S^3$, obtained by cutting out the exceptional lens s...
Petford and Welsh introduced a sequential heuristic algorithm to provide an approximate solution to the NP-hard graph coloring problem. The algorithm is based on the antivoter model and mimics the behavior of a physical process based on a multi-particle system of statistical mechanics. It was later shown that the algorithm can be implemented in a m...
Entanglement in proteins is a fascinating structural motif that is neither easy to detect via traditional methods nor fully understood. Recent advancements in AI-driven models have predicted that millions of proteins could potentially have a nontrivial topology. Herein, we have shown that long short-term memory (LSTM)-based neural networks (NN) arc...
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 oscilla...
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 oscil...
In this paper, we extend the definition of a knotoid to multi-linkoids that consist of a finite number of knot and knotoid components. We study invariants of multi-linkoids, such as the Kauffman bracket polynomial, ordered bracket polynomial, the Kauffman skein module, and the T-invariant in relation with generalized Θ\documentclass[12pt]{minimal}...
By suitably adjusting the tropical algebra technique we compute the rainbow independent domination numbers of several infinite families of graphs including Cartesian products Cn□Pm\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{u...
In this paper, we study knotoids with extra graphical structure (bonded knotoids) in the settings of rigid vertex and topological vertex graphs. We construct bonded knotoid invariants by applying tangle insertion and unplugging at bonding sites of a bonded knotoid. We demonstrate that our invariants can be used for the analysis of the topological s...
In this paper, we study bonded knotoids in the settings of rigid vertex and topological vertex graphs. We construct bonded knotoid invariants by applying tangle insertion and unplugging at bonding sites of a bonded knotoid.
An essential step towards gaining a deeper insight into intricate mechanisms underlying the formation and functioning of complex networks is extracting and understanding their building blocks encoded in the clustering structure. At its core, the problem of partitioning vertices into clusters may be regarded as a dual problem to vertex colouring and...
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-invaria...
We equip a knot K with a set of colored bonds, that is, colored intervals properly embedded into . Such a construction can be viewed as a structure that topologically models a closed protein chain including any type of bridges connecting the backbone residues. We introduce an invariant of such colored bonded knots that respects the HOMFLYPT relatio...
The k-assignment problem (or, the k-matching problem) on k-partite graphs is an NP-hard problem for k≥3. In this paper we introduce five new heuristics. Two algorithms, Bm and Cm, arise as natural improvements of Algorithm Am from (He et al., in: Graph Algorithms And Applications 2, World Scientific, 2004). The other three algorithms, Dm, Em, and F...
We obtain new results on independent 2- and 3-rainbow domination numbers of generalized Petersen graphs P ( n , k ) for certain values of n , k ∈ N . By suitably adjusting and applying a well established technique of tropical algebra (path algebra) we obtain exact 2-independent rainbow domination numbers of generalized Petersen graphs P ( n , 2 ) a...
We can equip a knot $K$ with a set of colored bonds, that is, colored intervals properly embedded into $\mathbb{R}^3 \setminus K$. Such a construction can be viewed as a structure that topologically models a closed protein chain including any type of bridges connecting the backbone. We show that the HOMFLYPT skein module of (rigid-vertex) colored b...
We present a reduced Burau-like representation for the mixed braid group on one strand representing links in lens spaces and show how to calculate the Alexander polynomial of a link directly from the mixed braid.
In this paper, we consider the following nonlinear Kirchhoff type problem: \[ \left\{\begin{array}{lcl}-\left(a+b\displaystyle\int_{\mathbb{R}^3}|\nabla u|^2\right)\Delta u+V(x)u=f(u), & \textrm{in}\,\,\mathbb{R}^3,\\ u\in H^1(\mathbb{R}^3), \end{array}\right. \] where $a,b>0$ are constants, the nonlinearity $f$ is superlinear at infinity with subc...
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 by mixed link diagrams. These invariants generalize the corresponding knot polynomials in the classical case. We compare the invariants by means of th...
We present a reduced Burau-like representation for the mixed braid group on one strand representing links in lens spaces and show how to calculate the Alexander polynomial of a link directly from the mixed braid.
In this paper, we consider the following nonlinear Kirchhoff type problem: [Formula presented]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 combinatio...
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 th...
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 diagr...
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 $\Sigma_g$ for $M$ we represent each link in $M$ as the plat closure of a braid in the surface braid group $B_{g,2n}=\pi_1(C_{2n}(\Sigma_g))$ and analyze how to trans...
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 diagr...
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...
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.
Khovanov homology, an invariant of links in R3, 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 et al. ['Categorification of the Kauffman bracket skein module of I-bundles over surfaces', Algebr. Geom. Topol. 4 (2004), 1177-1210] gen...
We classify non-affine, prime knots in the solid torus up to 6 crossings. We establish which of these are amphicheiral: almost all knots with symmetric Jones polynomial are amphicheiral, but in a few cases we use stronger invariants, such as HOMFLYPT and Kauffman skein modules, to show that some such knots are not amphicheiral. Examples of knots wi...
We present the algorithm for passing from classical diagrams of links in the solid torus to arrow diagrams of such links and vice versa. We also show how to pass from a canonical basis of the KBSM (Kauffman bracket skein module) of the solid torus to the basis of the KBSM of RP 3 .
Questions
Question (1)
What are some references with details how to draw planar knot diagrams by computer (preferably using Bezier curves)?