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Sujoy Mukherjee

Publications and source records attributed to Sujoy Mukherjee.

At least 19 recordsLinked to original sources
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

Algebraic invariants of multi-virtual links

Multi-virtual knot theory was introduced in $2024$ by the first author. In this paper, we initiate the study of algebraic invariants of multi-virtual links. After determining a generating set of (oriented) multi-virtual Reidemeister moves, we discuss the equivalence of multi-virtual link diagrams, particularly those that have the same virtual projections. We introduce operator quandles (that is, quandles with a list of pairwise commuting automorphisms) and construct an infinite family of connected operator quandles in which at least one third of right translations are distinct and pairwise commute. Using our set of generating moves, we establish the operator quandle coloring invariant and the operator quandle $2$-cocycle invariant for multi-virtual links, generalizing the well-known invariants for classical links. With these invariants at hand, we then classify certain small multi-virtual knots based on the existing tables of small virtual knots due to Bar-Natan and Green. Finally, to emphasize a key difference between virtual and multi-virtual knots, we construct an infinite family of pairwise nonequivalent multi-virtual knots, each with a single classical crossing. Many open problems are presented throughout the paper.

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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

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, Mieczys{\l}aw 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

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

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 $\delta,$ there exists a Fox $\delta$-coloring that distinguishes them.

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arXiv2022

Algebraic concordance order of almost classical knots

Torsion in the concordance group $\mathscr{C}$ of knots in $S^3$ can be studied with the algebraic concordance group $\mathscr{G}^{\mathbb{F}}$. Here $\mathbb{F}$ is a field of characteristic $χ(\mathbb{F}) \ne 2$. The group $\mathscr{G}^{\mathbb{F}}$ was defined by J. Levine, who also obtained an algebraic classification when $\mathbb{F}=\mathbb{Q}$. While the concordance group $\mathscr{C}$ is abelian, it embeds into the non-abelian virtual knot concordance group $\mathscr{VC}$. It is unknown if $\mathscr{VC}$ admits non-classical finite torsion. Here we define the virtual algebraic concordance group $\mathscr{VG}^{\mathbb{F}}$ for almost classical knots . This is an analogue of $\mathscr{G}^{\mathbb{F}}$ for homologically trivial knots in thickened surfaces $Σ\times [0,1]$, where $Σ$ is closed and oriented. The main result is an algebraic classification of $\mathscr{VG}^{\mathbb{F}}$. A consequence of the classification is that $\mathscr{G}^{\mathbb{Q}}$ embeds into $\mathscr{VG}^{\mathbb{Q}}$ and $\mathscr{VG}^{\mathbb{Q}}$ contains many nontrivial finite-order elements that are not algebraically concordant to any classical Seifert matrix. For $\mathbb{F}=\mathbb{Z}/2\mathbb{Z}$, we give a generalization of the Arf invariant.

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arXiv2022

Hyperbolic Knots and Torsion in Khovanov Homology

In this note, we show that if there is a knot in $S^3$ having $\mathbb{Z}_m$ torsion in its Khovanov homology, then there are infinitely many hyperbolic knots and infinitely many prime satellite knots having $\mathbb{Z}_m$ torsion in their Khovanov homology. As an application, we give the first known examples of hyperbolic knots and links with odd (and other non-$\mathbb{Z}_2$ torsion) in their Khovanov homology.

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arXiv2019

On odd torsion in even Khovanov homology

This short note resolves the most important part of the PS braid conjecture while introducing the first known examples of knots and links with odd torsion of order 9, 27, 81, and 25 in their even Khovanov homology.

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arXiv2019

A Generalization of the Gram determinant of type A

The Gram determinant of type $A$ was introduced by Lickorish in his work on invariants of 3 - manifolds. We generalize the theory of the Gram determinant of type $A$ by evaluating, in the annulus, a bilinear form of non-intersecting connections in the disc. The main result provides a closed formula for this Gram determinant. We conclude the paper by discussing Chen's conjecture about the Gram determinant of type $Mb$ evaluated in the Klein bottle.

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arXiv2018

Schur Multipliers and Second Quandle Homology

We define a map from second quandle homology to the Schur multiplier and examine its properties. Furthermore, we express the second homology of Alexander quandles in terms of exterior algebras. Additionally, we present a self-contained proof of its structure and provide some computational examples.

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arXiv2018

On multiplying curves in the Kauffman bracket skein algebra of the thickened four-holed sphere

Based on the presentation of the Kauffman bracket skein module of the torus given by the third author in previous work, Charles D. Frohman and Răzvan Gelca established a complete description of the multiplicative operation leading to a famous product-to-sum formula. In this paper, we study the multiplicative structure of the Kauffman bracket skein algebra of the thickened four-holed sphere. We present an algorithm to compute the product of any two elements of the algebra, and give an explicit formula for some families of curves. We surmise that the algorithm has quasi-polynomial growth with respect to the number of crossings of a pair of curves. Further, we conjecture the existence of a positive basis for the algebra.

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arXiv2017

On the rack homology of graphic quandles

This paper has partially a novel and partially a survey character. We start with a short review of rack (two term) homology of self distributive algebraic structures (shelves) and their connections to knot theory. We concentrate on a sub-family of quandles satisfying the graphic axiom. For a large family of graphic quandles (including infinite ones), we compute the second rack homology groups. Finally, we propose conjectures based on our computational data.

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arXiv2017

Rooted trees with the same plucking polynomial

In this paper we give a sufficient and necessary condition for two rooted trees with the same plucking polynomial. Furthermore, we give a criteria for a sequence of non-negative integers to be realized as a rooted tree.

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arXiv2017

Search for torsion in Khovanov homology

In the Khovanov homology of links, presence of $\mathbb{Z}_2$-torsion is a very common phenomenon. Finite number of examples of knots with $\mathbb{Z}_n$-torsion for $n>2$ were also known, none for $n>8$. In this paper, we prove that there are infinite families of links whose Khovanov homology contains $\mathbb{Z}_n$-torsion for $2 < n < 9$ and $\mathbb{Z}_{2^s}$-torsion for $s < 24$. We also introduce $4$-braid links with $\mathbb{Z}_3$-torsion which are counterexamples to the PS braid conjecture. We also provide an infinite family of knots with $\mathbb{Z}_5$-torsion in reduced Khovanov homology and $\mathbb{Z}_3$-torsion in odd Khovanov homology.

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arXiv2016

A Homology Theory For A Special Family Of Semi-groups

In this paper, we construct a new homology theory for semi-groups satisfying the self distributivity axiom or the idempotency axiom. Next, we consider the geometric realization corresponding to the homology theory. We continue with the comparison of this homology theory with one term and two term (rack) homology theories of self-distributive algebraic structures. Finally, we propose connections between the homology theory and knot theory via Temperley-Lieb algebras.

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arXiv2016

Homology of Associative Shelves

Homology theories for associative algebraic structures are well established and have been studied for a long time. More recently, homology theories for self-distributive algebraic structures motivated by knot theory, such as quandles and their relatives, have been developed and investigated. In this paper, we study associative self-distributive algebraic structures and their one-term and two-term (rack) homology groups.

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