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Helen Wong

Publications and source records attributed to Helen Wong.

At least 19 recordsLinked to original sources
arXiv2025

A relationship between the Kauffman bracket skein algebras and Roger-Yang skein algebras of some small surfaces

We calculate the Roger-Yang skein algebra of the annulus with two interior punctures, $ \mathcal S^{RY}(\Sigma_{0, 2, 2})$, and show there is a surjective homomorphism from this algebra to the Kauffman bracket skein algebra of the closed torus. Using this homomorphism, we characterize the irreducible, finite-dimensional representations of $ \mathcal S^{RY}(\Sigma_{0, 2, 2})$, showing that they can be described by certain complex data and that the correspondence is unique if certain polynomial conditions are satisfied. We also use the relationship with the skein algebra of the torus to compute structural constants for a bracelets basis for $ \mathcal S^{RY}(\Sigma_{0, 2, 2})$, giving evidence for positivity.

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arXiv2025

Cluster algebras and skein algebras for surfaces

We consider two algebras of curves associated to an oriented surface of finite type - the cluster algebra from combinatorial algebra, and the skein algebra from quantum topology. We focus on generalizations of cluster algebras and generalizations of skein algebras that include arcs whose endpoints are marked points on the boundary or in the interior of the surface. We show that the generalizations are closely related by maps that can be explicitly defined, and we explore the structural implications, including (non-)finite generation. We also discuss open questions about the algebraic structure of the algebras.

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arXiv2025

Center of generalized skein algebras

We consider a generalization of the Kauffman bracket skein algebra of a surface that is generated by loops and arcs between marked points on the interior or boundary, up to skein relations defined by Muller and Roger-Yang. We compute the center of this Muller-Roger-Yang skein algebra and show that it is almost Azumaya when the quantum parameter $q$ is a primitive $n$-th root of unity with odd $n$. We also discuss the implications on the representation theory of the Muller-Roger-Yang generalized skein algebra.

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arXiv2024

Fast Algorithm for Multiplication on the Skein Algebra of One-hole Torus

The Kauffman bracket skein algebra of a surface is a generalization of the Jones polynomial invariant for links and plays a principal role in the Witten-Reshetikhin- Turaev topological quantum field theory. However, the multiplicative structure of the skein algebra is not well understood, with a priori exponential complexity. We consider the case of one-hole torus, and provide a polynomial algorithm for computing multiplication of any two skein elements. Some closed form formulas for multiplication of curves with low crossing number are also given.

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arXiv2024

Consequences of the compatibility of skein algebra and cluster algebra on surfaces

We investigate two algebra of curves on a topological surface with punctures - the cluster algebra of surfaces defined by Fomin, Shapiro, and Thurston, and the generalized skein algebra constructed by Roger and Yang. By establishing their compatibility, we resolve Roger-Yang's conjecture on the deformation quantization of the decorated Teichmuller space. We also obtain several structural results on the cluster algebra of surfaces. The cluster algebra of a positive genus surface is not finitely generated, and it differs from its upper cluster algebra.

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arXiv2022

Asymptotics of quantum invariants of surface diffeomorphisms II: The figure-eight knot complement

In earlier work, the authors introduced a conjecture which, for an orientation-preserving diffeomorphism $φ\colon S \to S$ of a surface, connects a certain quantum invariant of $φ$ with the hyperbolic volume of its mapping torus $M_φ$. This article provides a proof of this conjecture in the simplest case where it applies, namely when the surface $S$ is the one-puncture torus and the mapping torus $M_φ$ is the complement of the figure-eight knot.

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arXiv2021

Asymptotics of quantum invariants of surface diffeomorphisms I: conjecture and algebraic computations

The Kashaev-Murakami-Murakami Volume Conjecture connects the hyperbolic volume of a knot complement to the asymptotics of certain evaluations of the colored Jones polynomials of the knot. We introduce a closely related conjecture for diffeomorphisms of surfaces, backed up by numerical evidence. The conjecture involves isomorphisms between certain representations of the Kauffman bracket skein algebra of the surface, and the bulk of the article is devoted to the development of explicit methods to compute these isomorphisms. These combinatorial and algebraic techniques are exploited in two subsequent articles, which prove the conjecture for a large family of diffeomorphisms of the one-puncture torus and are much more analytic.

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arXiv2019

The Roger-Yang skein algebra and the decorated Teichmuller space

Based on hyperbolic geometric considerations, Roger and Yang introduced an extension of the Kauffman bracket skein algebra that includes arcs. In particular, their skein algebra is a deformation quantization of a certain commutative curve algebra, and there is a Poisson algebra homomorphism between the curve algebra and the algebra of smooth functions on decorated Teichmuller space. In this paper, we consider surfaces with punctures which is not the 3-holed sphere and which have an ideal triangulation without self-folded edges or triangles. For those surfaces, we prove that Roger and Yang's Poisson algebra homomorphism is injective, and the skein algebra they defined have no zero divisors. A section about generalized corner coordinates for normal arcs may be of independent interest.

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arXiv2016

Representations of the Kauffman bracket skein algebra II: punctured surfaces

In earlier work, we constructed invariants of irreducible representations of the Kauffman skein algebra of a surface. We introduce here an inverse construction, which to a set of possible invariants associates an irreducible representation that realizes these invariants. The current article is restricted to surfaces with at least one puncture, a condition that will be lifted in subsequent work of the authors that relies on this one. A step in the proof is of independent interest, and describes the algebraic structure of the Thurston intersection form on the space of integer weight systems for a train track.

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arXiv2016

Representations of the Kauffman bracket skein algebra III: closed surfaces and naturality

This is the third article in the series begun with [BonWon3, BonWon4], devoted to finite-dimensional representations of the Kauffman bracket skein algebra of an oriented surface $S$. In [BonWon3] we associated a classical shadow to an irreducible representation $ρ$ of the skein algebra, which is a character $r_ρ\in \mathcal R_{\mathrm{SL}_2(\mathbb C)}(S)$ represented by a group homomorphism $π_1(S) \to \mathrm{SL}_2(\mathbb C)$. The main result of the current article is that, when the surface $S$ is closed, every character $r\in \mathcal R_{\mathrm{SL}_2(\mathbb C)}(S)$ occurs as the classical shadow of an irreducible representation of the Kauffman bracket skein algebra. We also prove that the construction used in our proof is natural, and associates to each group homomorphism $r\colon π_1(S) \to \mathrm{SL}_2(\mathbb C)$ a representation of the skein algebra $\mathcal S^A(S)$ that is uniquely determined up to isomorphism.

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arXiv2015

Representations of the Kauffman bracket skein algebra I: invariants and miraculous cancellations

We study finite-dimensional representations of the Kauffman skein algebra of a surface S. In particular, we construct invariants of such irreducible representations when the underlying parameter q is a root of unity. The main one of these invariants is a point in the character variety consisting of group homomorphisms from the fundamental group of S to SL_2(C), or in a twisted version of this character variety. The proof relies on certain miraculous cancellations that occur for the quantum trace homomorphism constructed by the authors. These miraculous cancellations also play a fundamental role in subsequent work of the authors, where novel examples of representations of the skein algebra are constructed.

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arXiv2015

The Witten-Reshetikhin-Turaev representation of the Kauffman skein algebra

For A a primitive 2N-root of unity with N odd, the Witten-Reshetikhin-Turaev topological quantum field theory provides a representation of the Kauffman skein algebra of a closed surface. We show that this representation is irreducible and we compute its classical shadow, in the sense of earlier work of the authors (arXiv:1206.1638).

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arXiv2013

Knotted and linked products of recombination on $T(2,n)\#T(2,m)$ substrates

We develop a topological model of site-specific recombination that applies to substrates which are the connected sum of two torus links of the form $T(2,n)\#T(2,m)$. Then we use our model to prove that all knots and links that can be produced by site-specific recombination on such substrates are contained in one of two families, which we illustrate.

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arXiv2011

Quantum invariants of random 3-manifolds

We consider the SO(3) Witten-Reshetikhin-Turaev quantum invariants of random 3-manifolds. When the level r is prime, we show that the asymptotic distribution of the absolute value of these invariants is given by the standard Rayleigh distribution and independent of the choice of level. Hence the probability that the quantum invariant certifies the Heegaard genus of a random 3-manifold of a fixed Heegaard genus g is positive but very small, less than 10^-7 except when g < 4. We also examine random surface bundles over the circle and find the same distribution for quantum invariants there.

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arXiv2010

Quantum traces for representations of surface groups in SL_2

We consider two different quantizations of the character variety consisting of all representations of surface groups in SL_2. One is the skein algebra considered by Przytycki-Sikora and Turaev. The other is the quantum Teichmuller space introduced by Chekhov-Fock and Kashaev. We construct a homomorphism from the skein algebra to the quantum Teichmuller space which, when restricted the classical case, corresponds to the equivalence between these two algebras through trace functions.

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arXiv2010

Kauffman brackets, character varieties, and triangulations of surfaces

A Kauffman bracket on a surface is an invariant for framed links in the thickened surface, satisfying the Kauffman skein relation and multiplicative under superposition. This includes representations of the skein algebra of the surface. We show how an irreducible representation of the skein algebra usually specifies a point of the character variety of homomorphisms from the fundamental group of the surface to PSL_2(C), as well as certain weights associated to the punctures of the surface. Conversely, we sketch a proof of the fact that each point of the character variety, endowed with appropriate puncture weights, uniquely determines a Kauffman bracket. Details will appear elsewhere.

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