Searcharxiv⌕ Search

arXiv subjects

Charles Frohman

Publications and source records attributed to Charles Frohman.

At least 19 recordsLinked to original sources
arXiv2025

The Geometric P=W conjecture and Thurston's compactification

In this paper, we use new results together with established facts about Thurston's compactification of Teichm\"uller space to address the geometric P=W conjecture for $\mathrm{SL}(2,\mathbb{C})$, which concerns projective compactifications of character varieties of closed surfaces. In particular, we construct a projective compactification of the $\mathrm{SL}(2,\mathbb{C})$-character variety of any closed surface of genus $g>1$, in which the boundary divisors are toric varieties and the dual intersection complex is a sphere. A main technical step, of independent interest, is the derivation of an explicit formula for a well-known embedding of the set of isotopy classes of multicurves on a closed surface of genus $g$ into $\mathbb{N}^{9g-9}$.

math.GT↗
arXiv2024

The Kauffman Bracket Skein Module at an irreducible representation

In this paper, we study the Kauffman bracket skein module of closed oriented three-manifolds at a non-multiple-of-four roots of unity. Our main result establishes that the localization of this module at a maximal ideal, which corresponds to an irreducible representation of the fundamental group of the manifold, forms a one-dimensional free module over the localized unreduced coordinate ring of the character variety. We apply this by proving that the dimension of the skein module of a homology sphere with finite character variety, when the order of the root of unity is not divisible by $4$, is greater than or equal to the dimension of the unreduced coordinate ring of the character variety. This leads to a computation of the dimension of the skein module with coefficients in rational functions for homology spheres with tame universal skein module.

math.GT↗
arXiv2024

Sliced skein algebras and geometric Kauffman bracket

The sliced skein algebra of a closed surface of genus $g$ with $m$ punctures, $\mathfrak{S}=Σ_{g,m}$, is the quotient of the Kauffman bracket skein algebra $\mathcal{S}_ξ(\mathfrak{S})$ corresponding to fixing the scalar values of its peripheral curves. We show that the sliced skein algebra of a finite type surface is a domain if the ground ring is a domain. When the quantum parameter $ξ$ is a root of unity we calculate the center of the sliced skein algebra and its PI-degree. Among applications we show that any smooth point of a sliced character variety is a fully Azumaya point of the skein algebra $\mathcal{S}_ξ(\mathfrak{S})$. For any $SL_2(\mathbb{C})$--representation $ρ$ of the fundamental group of an oriented connected 3-manifold $M$ and a root of unity $ξ$ with odd $ord(ξ^2)$, we introduce the $ρ$-reduced skein module $\mathcal{S}_{ξ,ρ}(M)$. We show that $\mathcal{S}_{ξ,ρ}(M)$ has dimension 1 when $M$ is closed and $ρ$ is irreducible. We also show that if $ρ$ is irreducible the $ρ$-reduced skein module of a handlebody, as a module over the skein algebra of its boundary, is simple and has the dimension equal to the PI-degree of the skein algebra of its boundary.

math.GT↗
arXiv2023

Skein Algebras of Three-Manifolds at 4th Roots of Unity

This paper introduces an algebra structure on the part of the skein module of an arbitrary $3$-manifold $M$ spanned by links that represent $0$ in $H_1(M;\mathbb{Z}_2)$ when the value of the parameter used in the Kauffman bracket skein relation is equal to $\pm {\bf i}$. It is proved that if $M$ has no $2$-torsion in $H_1(M;\mathbb{Z})$ then those algebras, $K_{\pm {\bf i}}^0(M)$, are naturally isomorphic to the corresponding algebras when the value of the parameter is $\pm 1$. This implies that the algebra $K_{\pm{\bf i}}^0(M)$ is the unreduced coordinate ring of the variety of $PSL_2(\mathbb{C})$-characters of $π_1(M)$ that lift to $SL_2(\mathbb{C})$-representations.

math.GT↗
arXiv2023

On compactifications of the SL(2,C) character varieties of punctured surfaces

This paper addresses some conjectures and questions regarding the absolute and relative compactifications of the $\SL(2,\C)$-character variety of an $n$-punctured Riemann surface without boundary. We study a class of projective compactifications determined by ideal triangulations of the surface and prove explicit results concerning the boundary divisors of these compactifications. Notably, we establish that the boundary divisors are toric varieties and confirm a well-known conjecture asserting that the (dual) boundary complex of any (positive dimensional) relative character variety is a sphere. In a different vein, we enhance and streamline Komyo's compactification method, which utilizes a projective compactification of $\SL(2,\C)$ to compactify the (relative) character varieties. Specifically, we construct a uniform relative compactification over the base space of $\C^n$ and determine its monodromy, addressing a question posed by Simpson.

math.AG↗
arXiv2021

SU(3)-skein algebras and webs on surfaces

The $SU_3$-skein algebra of a surface $F$ is spanned by isotopy classes of certain framed graphs in $F\times I$ called $3$-webs subject to the skein relations encapsulating relations between $U_q(sl(3))$-representations. These skein algebras are quantizations of the $SL(3)$-character varieties of surfaces. It is expected that their theory parallels that of the Kauffman bracket skein algebras. We make the first step towards developing that theory by proving that the reduced $SU_3$-skein algebra of any surface of finite type is finitely generated. We achieve that result by developing a theory of canonical forms of webs in surfaces. Specifically, we show that for any ideal triangulation of $F$ every reduced $3$-web can be uniquely decomposed into unions of pyramid formations of hexagons and disjoint arcs in the faces of the triangulation with possible additional "crossbars" connecting their edges along the ideal triangulation. We show that such canonical position is unique up to "crossbar moves". That leads us to an associated system of coordinates for webs in triangulated surfaces (counting intersections of the web with the edges of the triangulation and their rotation numbers inside of the faces of the triangulation) which determine a reduced web uniquely. Finally, we relate our skein algebras to $\cal A$-varieties of Fock-Goncharov and to $\text{Loc}_{SL(3)}$-varieties of Goncharov-Shen. We believe that our coordinate system for webs is a manifestation of a (quantum) mirror symmetry conjectured by Goncharov-Shen.

math.GT↗
arXiv2020

A Projective Representation of the Modular Group

Quantum Teichmuller theory assigns invariants to three-manifolds via projective representations of mapping class groups derived from the representation of a noncommutative torus. Here, we focus on a representation of the simplest non-commutative torus which remains fixed by all elements of the mapping class group of the torus, $SL_2(\mathbb{Z})$. Also known as the modular group. We use this representation to associate a matrix to each element of $SL_2(\mathbb{Z})$; we then compute the trace and determinant of the associated matrix.

math.GT↗
arXiv2019

Unicity for Representations of the Kauffman bracket Skein Algebra

This paper resolves the unicity conjecture of Bonahon and Wong for the Kauffman bracket skein algebras of all oriented finite type surfaces at all roots of unity. The proof is a consequence of a general unicity theorem that says that the irreducible representations of a prime affine $k$-algebra over an algebraically closed field $k$, that is finitely generated as a module over its center, are generically classified by their central characters. The center of the Kauffman bracket skein algebra of any orientable surface at any root of unity is characterized, and it is proved that the skein algebra is finitely generated as a module over its center. It is shown that for any orientable surface the center of the skein algebra at any root of unity is the coordinate ring of an affine algebraic variety.

math.GT↗
arXiv2019

Dimension and Trace of the Kauffman Bracket Skein Algebra

Let $F$ be a finite type surface and $ζ$ a complex root of unity. The Kauffman bracket skein algebra $K_ζ(F)$ is an important object in both classical and quantum topology as it has relations to the character variety, the Teichmüller space, the Jones polynomial, and the Witten-Reshetikhin-Turaev Topological Quantum Field Theories. We compute the rank and trace of $K_ζ(F)$ over its center, and we extend a theorem of Frohman and Kania-Bartoszynska which says the skein algebra has a splitting coming from two pants decompositions of $F$.

math.GT↗
arXiv2018

Locality and the Uniqueness of Quantum Invariants

We introduce the notion of a "state function" for framed tangles in a disk. After choosing a finite set of states for each marked disk, a state function is a projection from the vector space spanned by all tangles to the vector space spanned by the states, that is local, and topologically invariant. Given the states for the Kauffman bracket, and the quantum $SU(3)$-invariant we classify all state functions, and then compare our results to the literature.

math.GT↗
arXiv2017

The Yang-Mills Measure in the $SU(3)$ Skein Module

Let $A\neq 0$ be a complex number with $ |A|\neq 1$. Let $M$ be a compact smooth oriented $3$-manifold, the $SU(3)$-skein space of $M$, $S_A(M)$, is the vector space over $\mathbb{C}$ generated by framed oriented links (including framed oriented trivalent graphs in $M$) quotient by the $SU(3)$-skein relations due to Kuperberg. For a closed, orientable surface $F$, we construct a local diffeomorphism invariant trace on $S_A(F\times I)$.

math.GT↗
arXiv2017

Spider Evaluation and Representations of Web Groups

The topology of $SU(3)$-representation varieties of the fundamental groups of planar webs so that the meridians are sent to matrices with trace equal to $-1$ are explored, and compared to data coming from spider evaluation of the webs. Corresponding to an evaluation of a web as a spider is a rooted tree. We associate to each geodesic $γ$ from the root of the tree to the tip of a leaf an irreducible component $C_γ$ of the representation variety of the web, and a graded subalgebra $A_γ$ of $H^*(C_γ;\mathbb{Q})$. The spider evaluation of geodesic $γ$ is the symmetrized Poincare polynomial of $A_γ$. The spider evaluation of the web is the sum of the symmetrized Poincare polynomials of the graded subalgebras associated to all maximal geodesics from the root of the tree to the leaves

math.GT↗
arXiv2016

The Structure of the Kauffman Bracket Skein Algebra at Roots of Unity

This paper is focused on the structure of the Kauffman bracket skein algebra of a punctured surface at roots of unity. A criterion that determines when a collection of skeins forms a basis of the skein algebra as an extension over the $SL(2,{\mathbb C})$ characters of the fundamental group of the surface, with appropriate localization is given. This is used to prove that when the algebra is localized so that every nonzero element of the center has a multiplicative inverse, that it is a division algebra. Finally, it is proved that the localized skein algebra can be split over its center as a tensor product of two commutative subalgebras.

math.GT↗
arXiv2016

Tangle Functors from Semicyclic Representations

Let $q$ be a $2N$th root of unity where $N$ is odd. Let $U_q(sl_2)$ denote the quantum group with large center corresponding to the lie algebra $sl_2$ with generators $E,F,K$, and $K^{-1}$. A semicyclic representation of $U_q(sl_2)$ is an $N$-dimensional irreducible representation $ρ:U_q(sl_2)\rightarrow M_N(\mathbb{C})$, so that $ρ(E^N)=aId$ with $a\neq 0$, $ρ(F^N)=0$ and $ρ(K^N)=Id$. We construct a tangle functor for framed homogeneous tangles colored with semicyclic representations, and prove that for $(1,1)$-tangles coming from knots, the invariant defined by the tangle functor coincides with Kashaev's invariant.

math.GT↗
arXiv2015

The Localized Skein Algebra is Frobenius

When $A$ in the Kauffman bracket skein relation is a primitive $2N$th root of unity, where $N\geq 3$ is odd, the Kauffman bracket skein algebra $K_N(F)$ of a finite type surface $F$ is a ring extension of the $SL_2\mathbb{C}$-characters $χ(F)$ of the fundamental group of $F$. We localize by inverting the nonzero characters to get an algebra $S^{-1}K_N(F)$ over the function field of the character variety. We prove that if $F$ is noncompact, the algebra $S^{-1}K_N(F)$ is a symmetric Frobenius algebra. Along the way we prove $K(F)$ is finitely generated, $K_N(F)$ is a finite rank module over $χ(F)$, and the simple closed curves that make up any simple diagram on $F$ generate a finite field extension of $S^{-1}χ(F)$ inside $S^{-1}K_N(F)$.

math.GT↗
arXiv2011

Dubois' Torsion, A-polynomial and Quantum Invariants

It is shown that for knots with a sufficiently regular character variety the Dubois' torsion detects the A-polynomial of the knot. A global formula for the integral of the Dubois torsion is given. The formula looks like the heat kernel regularization of the formula for the Witten-Reshetikhin-Turaev invariant of the double of the knot complement. The Dubois' torsion is recognized as the pushforward of a measure on the character variety of the double of the knot complement coming from the square root of Reidemeister torsion. This is used to motivate a conjecture about quantum invariants detecting the A-polynomial.

math.GT↗