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Razvan Gelca

Publications and source records attributed to Razvan Gelca.

At least 19 recordsLinked to original sources
arXiv2024

A recursive relation in the $(2p+1,2)$ torus knot

It is known that the colored Jones polynomials of a knot in the 3-dimensional sphere satisfy recursive relations, it is also known that these recursive relations come from recurrence polynomials which have been related, by the AJ conjecture, to the geometry of the knot complement. In this paper we propose a new line of thought, by extending the concept of colored Jones polynomials to knots in the 3-dimensional manifold such as a knot complement, and then examining the case of one particular knot in the complement of the $(2p+1,2)$ torus knot for which an analogous recursive relation exists, and moreover, this relation has an associated recurrence polynomial. Part of our study consists of the writing in the standard basis of the genus two handlebody of two families of skeins in this handlebody.

math.GT↗
arXiv2023

Algorithms for Skein Manipulation in a Genus-2 Handlebody

We present a series of algorithms for skein manipulation in a genus-2 handlebody, implementing a novel strand sorting method to reduce any skein to a skein in a 2-punctured disk. This reduction guarantees resolution as a linear combination of basis elements of the Kauffman Bracket Skein Module. Manually, these skein manipulations prove to be computationally intensive due to the inherent exponential nature of skein relations (i.e., a skein diagram with $n$ crossings yields $2^n$ new skein diagrams, each in $\mathbb{C}[t,t^{-1}]$, the Laurent polynomials with complex coefficients). Thus, as the number of crossings in a skein diagram increases, manual computations become intractable and automation desirable. We enable the automation of all skein computations in the genus-2 handlebody by first converting the skein diagram into an equivalent array, reducing the task of performing skein computations to that of implementing array operators, and then proving that we can always recover the resulting complex Laurent polynomial.

math.QA↗
arXiv2022

Kauffman bracket versus Jones polynomial skein modules

This paper resolves the problem of comparing the skein modules defined using the skein relations discovered by R. Kirby and P. Melvin that underlie the Reshetikhin-Turaev model for $SU(2)$ Chern-Simons theory to the Kauffman bracket skein modules. Several applications and examples are presented.

math.GT↗
arXiv2021

The action of the Kauffman bracket skein algebra of the torus on the Kauffman bracket skein module of the 3-twist knot complement

We determine the action of the Kauffman bracket skein algebra of the torus on the Kauffman bracket skein module of the complement of the 3-twist knot. The point is to study the relationship between knot complements and their boundary tori, an idea that has proved very fruitful in knot theory. We place this idea in the context of Chern-Simons theory, where such actions arose in connection with the computation of the noncommutative version of the A-polynomial that was defined by Frohman, Gelca and Lofaro, but they can also be interpreted as quantum mechanical systems. Our goal is to exhibit a detailed example in a part of Chern-Simons theory where examples are scarce.

math.GT↗
arXiv2015

From classical theta functions to topological quantum field theory

Abelian Chern-Simons theory relates classical theta functions to the topological quantum field theory of the linking number of knots. In this paper we explain how to derive the constructs of abelian Chern-Simons theory directly from the theory of classical theta functions. It turns out that the theory of classical theta functions, from the representation theoretic point of view of A. Weil, is just an instance of Chern-Simons theory. The group algebra of the finite Heisenberg group is described as an algebra of curves on a surface, and its Schrodinger representation is obtained as an action on curves in a handlebody. A careful analysis of the discrete Fourier transform yields the Reshetikhin-Turaev formula for invariants of 3-dimensional manifolds. In this context, we give an explanation of why the composition of discrete Fourier transforms and the non-additivity of the signature of 4-dimensional manifolds under gluings obey the same formula.

math-ph↗
arXiv2010

Quantum mechanics and non-abelian theta functions for the gauge group $SU(2)$

This paper outlines an approach to the non-abelian theta functions of the $SU(2)$-Chern-Simons theory with the methods used by A. Weil for studying classical theta functions. First we translate in knot theoretic language classical theta functions, the action of the finite Heisenberg group, and the discrete Fourier transform. Then we explain how the non-abelian counterparts of these arise in the framework of the quantum group quantization of the moduli space of flat $SU(2)$-connections on a surface, in the guise of the non-abelian theta functions, the action of a skein algebra, and the Reshetikhin-Turaev representation of the mapping class group. We prove a Stone-von Neumann theorem on the moduli space of flat $SU(2)$-connections on the torus, and using it we deduce the existence and the formula for the Reshetikhin-Turaev representation on the torus from quantum mechanical considerations. We show how one can derive in a quantum mechanical setting the skein that allows handle slides, which is the main ingredient in the construction of quantum $3$-manifold invariants.

math-ph↗
arXiv2008

El polinomio de Jones y la mecanica cuantica

In this paper we discuss progress made in the study of the Jones polynomial from the point of view of quantum mechanics. This study reduces to the understanding of the quantization of the moduli space of flat SU(2)-connections on a surface with the Chern-Simons lagrangian. We outline some background material, then present the particular example of the torus, in which case it is known that the quantization in question is the Weyl quantization. The paper concludes with a possible application of this theory to the study of the fractional quantum Hall effect, an idea originating in the works of Moore and Read.

math.QA↗
arXiv2008

On the holomorphic point of view in the theory of quantum knot invariants

In this paper we describe progress made toward the construction of the Witten-Reshetikhin-Turaev theory of knot invariants from the geometric point of view. This is done in the perspective of a joint result of the author with A. Uribe which relates the quantum group and the Weyl quantizations of the moduli space of flat SU(2)-connections on the tours. Two results are emphasized: the reconstruction from Weyl quantization of the restriction to the torus of the modular functor, and a description of a basis of the space of quantum observables on the torus in terms of colored curves, which answers a question related to quantum computing.

math.QA↗
arXiv2002

The Weyl quantization and the quantum group quantization of the moduli space of flat SU(2)-connections on the torus are the same

We prove that, for the moduli space of flat SU(2)-connections on the torus, the Weyl quantization and the quantization using the quantum group of SL(2,C) are unitarily equivalent. This is done by comparing the matrices of the operators associated by the two quantization to cosine functions. We also discuss the *-product of the Weyl quantization and show that it satisfies the product-to-sum formula for noncommutative cosines on the noncommutative torus.

math-ph↗
arXiv2002

A product-to-sum formula for the quantum group of SL(2,C)

The paper exhibits a product-to-sum formula for the observables of a certain quantization of the moduli space of flat SU(2)-connections on the torus. This quantization was defined using the topological quantum field theory that was developed by Reshetikhin and Turaev from the quantum group of SL(2,C) at roots of unity. As a corollary it is shown that the algebra of quantum observables is a subalgebra of the noncommutative torus with rational rotation angle. The proof uses topological quantum field theory with corners, and is based on the description of the matrices of the observables in a canonical basis of the Hilbert space of the quantization.

math.QA↗
arXiv2002

The noncommutative A-ideal of a (2,2p+1)-torus knot determines its Jones polynomial

The noncommutative A-ideal of a knot is a generalization of the A-polynomial, defined using Kauffman bracket skein modules. In this paper we show that any knot that has the same noncommutative A-ideal as the (2,2p+1)-torus knot has the same colored Jones polynomials. This is a consequence of the orthogonality relation, which yields a recursive relation for computing all colored Jones polynomials of the knot.

math.GT↗
arXiv2000

Noncommutative trigonometry and the A-polynomial of the trefoil knot

The paper shows the computation of the noncommutative generalization of the A-polynomial of the trefoil knot. The classical A-polynomial was introduced by Cooper, Culler, Gillet, Long and Shalen, and was generalized to the context of Kauffman bracket skein modules by the author in joint work with Frohman and Lofaro. A major step in determining the noncommutative version of the A-polynomial of the trefoil is the description of the action of the Kauffman bracket skein algebra of the torus on the skein module of the knot complement. As such, the computation reduces to operations with noncommutative trigonometric functions.

math.GT↗
arXiv2000

On the relation between the A-polynomial and the Jones polynomial

In previous joint work with Frohman and Lofaro a noncommutative generalization of the A-polynomial of a knot was introduced, consisting of a finitely generated ideal of polynomials (the noncommutative A-ideal) in the quantum plane. The present paper shows that the noncommutative A-ideal of a knot, together with finitely many of its colored Jones polynomials determines all other colored Jones polynomials. Also, for particular knots, such as the unknot and the trefoil, the noncommutative A-ideal completely determines all colored Jones polynomials.

math.QA↗
arXiv1999

On the groupoid of transformations of rigid structures on surfaces

We prove that the groupoid of transformations of rigid structures on surfaces has a finite presentation as a 2-groupoid establishing a result first conjectured by G.Moore and N.Seiberg. An alternative proof was given by B.Bakalov and A.Kirillov Jr. We present some applications to TQFTs. This is also related to recent work on the Grothendieck-Teichmuller groupoid by P.Lochak, A.Hatcher and L.Schneps.

math.GT↗
arXiv1998

The A-polynomial from the noncommutative viewpoint

We develop an invariant of knots that depends on a complex parameter t, describing a left ideal in the noncommutative torus. When the parameter is set equal to -1 we recover the A-polynomial of the knot. We relate the invariant to the colored Jones polynomials of the knot.

math.QA↗