arXiv subjects

Gabriel Montoya-Vega

Publications and source records attributed to Gabriel Montoya-Vega.

15 recordsLinked to original sources

arXiv2025

A Study of Gram Determinants in Knot Theory

Historically originated as a sub-field of topology, knot theory is an active area of mathematical investigation that has strong connections with a diverse set of scientific fields such as algebra, biology, and statistical mechanics. A popular and important concept in linear algebra, Gram determinants enjoy a connection with the mathematical theory of knots. In this article, we expose this concept and present several types of Gram determinants in what can be considered as a survey of the current Gram determinants of interest to knot theorists; examples are included to illustrate the definitions. In particular, we pay special attention to a recently defined determinant from a Möbius band and we further study its structure. At the end, some speculation is presented regarding the closed formula for the Gram determinant of type $(Mb)_1$, a problem that arouses serious interest among knot theorists.

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arXiv2025

A Glimpse of the Khovanov Homology of T(2,n) Via Long Exact Sequence

Khovanov homology is a powerful link invariant: a categorification of the Jones polynomial that enjoys a rich and beautiful algebraic structure. This homology theory has been extensively studied and it has become an ubiquitous topic in contemporary knot theory research. In the same spirit, the Kauffman skein relation, which allows to define the Kauffman bracket polynomial up to normalization of the unknot, can be categorified by means of a long exact sequence. In an expository style, in this article we present how to build Khovanov homology from the Kauffman bracket polynomial and construct its long exact sequence. Furthermore, we present a deviceful and practical way in which this long exact sequence can be used for the computation of the Khovanov homology of torus links of the type $T(2,n)$. This article serves as a partial translation of a Spanish paper to be published on occasion of the Encuentro Internacional de Matemáticas (International Meeting of Mathematics) celebrated at the Universidad del Atlántico in Barranquilla, Colombia in November 2023. This paper offers a first look into the world of Khovanov homology by constructing it from the Kauffman bracket polynomial, as it was first done by Oleg Viro. Moreover, it gives the reader references for further studies from leading experts such as D. Bar-Natan, M. Khovanov, S. Mukherjee, J. Przytycki, and A. Shumakovitch, among others. In particular, one of the main objectives in publishing this article (and this partial translation) is to popularize research in knot theory, more specifically on Khovanov homology in Colombia, and Latin-America in general, acting as a language bridge given that most of the literature is in English.

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arXiv2025

A robot that unknots knots

Consider a robot that remembers only the starting position and walks along a knot once on a knot diagram, switching every undercrossing it meets until it returns to the starting position. We observe that the robot produces an ascending diagram, and we provide a new combinatorial proof that every ascending or descending knot diagram can be transformed into the zero-crossing unknot diagram. Using the machinery developed from the combinatorial proof, we show that the minimal number of Reidemeister moves required for such a transformation is bounded above by (7C+1)C if the diagram has C crossings. Moreover, we provide a new alternative proof that there exist sequences of Reidemeister moves that do not increase the number of crossings and transform ascending or descending knot diagrams into zero-crossing unknot diagrams.

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arXiv2025

Fundamentals of cubic skein modules

Over the past thirty-seven years, the study of linear and quadratic skein modules has produced a rich and far-reaching skein theory, intricately connected to diverse areas of mathematics and physics, including algebraic geometry, hyperbolic geometry, topological quantum field theories, and statistical mechanics. However, despite these advances, skein modules of higher degree-those depending on more parameters than the linear and quadratic cases-have received comparatively little attention, with only a few isolated explorations appearing in the literature. In this article, we undertake a systematic study of the cubic skein module, the first representative of this broader class. We begin by investigating its structure and properties in the $3$-sphere, and then extend the analysis to arbitrary $3$-manifolds. The results presented here aim to establish a foundational framework for the study of higher skein modules, thereby extending the scope of skein theory beyond its classical domains. Furthermore, studying the structure of cubic skein modules may lead to new polynomial invariants of knots.

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arXiv2025

Temperley-Lieb Categories on Non-Orientable Surfaces

In this paper we present the construction of a skeletal diagram category, which we call the square with bands category. This category extends the Temperley-Lieb (TL) category, where morphisms now include diagrams of embedded curves on (possibly) non-orientable bounded surfaces, and involves three parameters associated to simple closed curves. Such diagrams utilise handle decompositions for surfaces and are considered up to a handle slide equivalence. We define a tensor product on this category, extending the well-known tensor product on the TL category, and a full set of monoidal generators is given, which includes the TL generators, a family of orientable genus one diagrams, and a family of non-orientable diagrams. This document constitutes an initial draft of ongoing research with preliminary reporting of some results in the last section; a subsequent version including a detailed introduction and full proofs will follow.

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arXiv2025

Chebyshev polynomials and Gram determinants from the Möbius band

This article explores the connection between Chebyshev polynomials and knot theory, specifically in relation to Gram determinants. We reveal intriguing formulae involving the Chebyshev polynomial of the first and second kind. In particular we show that for Mersenne numbers, $M_k=2^k-1$ where $k\geq 2$, the $M_k$-th Chebyshev polynomial of the second kind is the product of Chebyshev polynomials of the first kind. We then discuss the Gram determinant of type $(Mb)_1$, restate the conjecture of its closed formula in terms of mostly products of Chebyshev polynomials of the second kind, and prove a factor of the determinant that supports the conjecture. We also showcase an algorithm for calculating the Gram determinant's corresponding matrix. Furthermore, we restate Qi Chen's conjectured closed formula for the Gram determinant of type Mb and discuss future directions.

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arXiv2025

On the notion of Khovanov A-adequacy

The concept of adequate links, introduced by Lickorish and Thistlethwaite as a generalization of alternating links, has recently gained interest among knot theorists in the context of Khovanov homology. Przytycki and Silvero introduced the more general concept of Khovanov adequacy: a diagram is Khovanov-adequate if its associated Khovanov chain complexes at both potential maximal and minimal quantum gradings have non-trivial homology. This article explores Khovanov adequacy within the framework of independence complexes and the calculation of the homotopy type of extreme Khovanov spectra.

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arXiv2025

The Montesinos-Nakanishi 3-move conjecture for links up to 20 crossings

Yasutaka Nakanishi formulated the following conjecture in 1981: every link is 3-move equivalent to a trivial link. While the conjecture was proved for several specific cases, it remained an open question for over twenty years. In 2002, Mieczysław D{\c a}bkowski and the last author showed that it does not hold, in general. In this article, we prove the Montesinos-Nakanishi $3$-move conjecture for links with up to 19 crossings and, with the exception of six pairwise non-isotopic links including the Chen link and its mirror image, for links with 20 crossings. Our work completely classifies links up 20 crossings modulo $3$-moves. This work includes computational methods, including new code in Regina that generalises pre-existing knot functions to work with links.

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arXiv2024

Alexander-Conway and Bracket Polynomials of Pretzel Links $\boldsymbol{P(1,1,n)}$

Polynomial invariants constitute a dynamic and essential area of study in the mathematical theory of knots. From the pioneer Alexander polynomial, the revolutionary Jones polynomial, to the collectively discovered HOMFLYPT polynomial, just to mention a few, these algebraic expressions have been central to the understanding of knots and links. The introduction of Khovanov homology has sparked significant interest in the categorification of these polynomials, offering deeper insights into their topological and algebraic properties. In this work, we revisit two prominent polynomial invariants, the Alexander-Conway and the Kauffman bracket polynomials, and focus specifically on the polynomials associated with the family of three strand pretzel links $P(1,1,n)$.

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arXiv2024

Independence complexes of circle graphs

Independence complexes of circle graphs are purely combinatorial objects. However, when constructed from some diagram of a link $L$, they reveal topological properties of $L$, more specifically, of its Khovanov homology. We analyze the homotopy type of independence complexes of circle graphs, with a focus on those arising when the graph is bipartite. Moreover, we compute (real) extreme Khovanov homology of a $4$-strand pretzel knot using chord diagrams and independence complexes.

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arXiv2023

Exploring unimodality of the plucking polynomial with delay function

The plucking polynomial is an invariant of rooted trees with connections to knot theory. The polynomial was constructed in 2014 as a tool to analyze lattice crossings after taking the quotient by the Kauffman bracket skein relations. In this paper we study the plucking polynomial and the plucking polynomial with delay function. We present a formula for the plucking polynomial of hedgehog rooted trees and explore the unimodality of this polynomial. In particular, we consider an anti-unimodal delay function and a delay function with a specific image set. Furthermore, we present a number of interesting examples and make some speculations on the unimodality of plucking polynomials with delay functions of hedgehog rooted trees.

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arXiv2023

Una Mirada Inicial a la Teoría de Nudos y a la Homología de Khovanov

The mathematical theory of knots studies the embeddings of circles into the space $\mathbb{R}^3$, being the classification one of the fundamental problems. The introduction of homology theories results in complex mathematical structures that generate new research opportunities. On occasion of the Encuentro Internacional de Matemáticas (EIMAT) (International Meeting of Mathematics) to be celebrated at the Universidad del Atlántico in Barranquilla, Colombia in November 2023, in this article, in an expository way, we offer a first look into Khovanov homology and the long exact sequence of Khovanov homology. Moreover, we present a summary of the historical origins of the theory which can take us as early as the year 2600 BCE, passing through Italy of the XV century, Scotland of the XIX century, and we give references for further and more detailed discussions. Additionally to showing the construction of Khovanov homology from the Kauffman bracket polynomial, the main objective in publishing this article is to attack the language barrier and popularize knot theory and Khovanov homology in Colombia and Latin-America in general.

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arXiv2023

On a new Gram determinant from the Möbius band

Gram determinants earned traction among knot theorists after E. Witten's presumption about the existence of a 3-manifold invariant connected to the Jones polynomial. Triggered by the creation of such an invariant by N. Reshetikhin and V. Turaev, several mathematicians have explored this line of research ever since. Gram determinants came into play by W. B. Raymond Lickorish's skein theoretic approach to the invariant. The construction of different bilinear forms is possible through changes in the ambient surface of the Kauffman bracket skein module. Hence, different types of Gram determinants have arisen in knot theory throughout the years; some of these determinants are discussed here. In this article, we introduce a new version of such a determinant from the Möbius band and prove some important results about its structure. In particular, we explore its connection to the annulus case and factors of its closed formula.

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arXiv2023

On framings of links in 3-manifolds

We show that the only way of changing the framing of a link by ambient isotopy in an oriented $3$-manifold is when the manifold has a properly embedded non-separating $S^2$. This change of framing is given by the Dirac trick, also known as the light bulb trick. The main tool we use is based on McCullough's work on the mapping class groups of $3$-manifolds. We also relate our results to the theory of skein modules.

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arXiv2023

The Generalized Kauffman-Harary Conjecture is True

For a reduced alternating diagram of a knot with a prime determinant $p,$ the Kauffman-Harary conjecture states that every non-trivial Fox $p$-coloring of the knot assigns different colors to its arcs. In this paper, we prove a generalization of the conjecture stated nineteen years ago by Asaeda, Przytycki, and Sikora: for every pair of distinct arcs in the reduced alternating diagram of a prime link with determinant $δ,$ there exists a Fox $δ$-coloring that distinguishes them.

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