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Rhea Palak Bakshi

Publications and source records attributed to Rhea Palak Bakshi.

13 recordsLinked to original sources
arXiv2026

Two non-semisimple quantum invariants of lens spaces

We calculate the chromatic spherical invariant $\mathscr{K}_{\mathrm{Rep}(H)}(L(p,q))$ of lens spaces $L(p, q)$ for arbitrary finite-dimensional unibalanced, unimodular Hopf algebras $H$. In the case of the index-2 Nichols Hopf algebra $\mathcal{K}_2$ and $q=1$, we find that the invariant of $L(p, 1)$ depends only on the parity of $p$ for $p\geq 1$. We also describe the duals $\mathcal{S_I}(M)^*$ of the admissible skein modules of handlebodies $M=H_g$ and of $M=L(p, q)$ for arbitrary tensor ideals $\mathcal{I}$ of a ribbon tensor category. In particular, $\mathcal{S_I}(L(p, q))^*$ corresponds with cyclic traces on $\mathcal{I}$ satisfying a property reminiscent of the modified trace condition.

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arXiv2026

Kauffman bracket skein module of the connected sum of two solid tori

We determine the structure of the Kauffman bracket skein module of the connected sum of two genus one handlebodies over the ring of Laurent polynomials $\mathbb Z[q^{\pm 1}]$, thereby proving a conjecture posed by the first and third authors. Our results lay the groundwork for computing the Kauffman bracket skein module of arbitrary connected sums over the ring $\mathbb Z[q^{\pm 1}]$.

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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

On the Kauffman bracket skein module of $(S^1 \times S^2) \ \# \ (S^1 \times S^2)$

Determining the structure of the Kauffman bracket skein module of all $3$-manifolds over the ring of Laurent polynomials $\mathbb Z[A^{\pm 1}]$ is a big open problem in skein theory. Very little is known about the skein module of non-prime manifolds over this ring. In this paper, we compute the Kauffman bracket skein module of the $3$-manifold $(S^1 \times S^2) \ \# \ (S^1 \times S^2)$ over the ring $\mathbb Z[A^{\pm 1}]$. We do this by analysing the submodule of handle sliding relations, for which we provide a suitable basis. Along the way we compute the Kauffman bracket skein module of $(S^1 \times S^2) \ \# \ (S^1 \times D^2)$. We also show that the skein module of $(S^1 \times S^2) \ \# \ (S^1 \times S^2)$ does not split into the sum of free and torsion submodules. Furthermore, we illustrate two families of torsion elements in this skein module.

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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

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

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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arXiv2020

Kauffman Bracket Skein Module of the Connected Sum of Handlebodies: A Counterexample

In this paper we disprove a twenty-two year old theorem about the structure of the Kauffman bracket skein module of the connected sum of two handlebodies. We achieve this by analysing handle slidings on compressing discs in a handlebody. We find more relations than previously predicted for the Kauffman bracket skein module of the connected sum of handlebodies, when one of them is not a solid torus. Additionally, we speculate on the structure of the Kauffman bracket skein module of the connected sum of two solid tori.

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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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