arXiv research

Study genus-three Torelli maps and their fixed point sets in representation varieties.

Study on mapping class groups for low genus surfaces.

The study explores SU(2) representations in 3-manifolds and knots with specific Heegaard genus constraints.

Researchers create projective representations of Hecke groups using TQFT.

Researchers compute cohomology of mapping class groups with Prym representations, showing instability for large genus.

Classifies finite orbits of mapping class group actions on surface group representations.

New lower bound for knot genus using Links-Gould invariant.

Nonorientable surfaces have a faithful linear representation of a specific dimension.

A realization of a virtual link diagram is obtained by choosing over/under markings for each virtual crossing. Any realization can also be obtained from some representation of the virtual link. (A representation of a virtual link is a link diagram on an oriented 2-dimensional surface.) We prove that if a minimal genus …

Study of low dimensional representations of mapping class groups of surfaces, focusing on genus ≥ 7.

The paper studies representations of surface groups into PSL(2,R) and their geometric realizations.

We construct integral bases for the SO(3)-TQFT-modules of surfaces in genus one and two at roots of unity of prime order and show that the corresponding mapping class group representations preserve a unimodular Hermitian form over a ring of algebraic integers. For higher genus surfaces the Hermitian form sometimes must…

We prove that the sequence of projective representations of the mapping class group obtained from the projective flat connection in the SU(n)-Verlinde bundles over Teichmuller space is asymptotically faithful, that is the intersection over all levels of the kernels of these representations is trivial, whenever the genu…

The Frey--Mazur conjecture states that an elliptic curve over $\mathbb{Q}$ is determined up to isogeny by its $p$-torsion Galois representation for $p\geq 17$. We study a geometric analog of this conjecture, and show that the map from isogeny classes of "fake elliptic curves"---abelian surfaces with quaternionic multip…

New polynomial connects knot genus to 3-manifold geometry.

New knots with specific properties have larger 4-genus than previously known bounds.

Researchers create coordinates for hyperbolic surfaces, proving a magic formula.

Paper uses harmonic surfaces to model surfaces of any complexity.

Improved lower bounds for faithful linear representations of mapping class groups.

Study calculates Alexander polynomials for a specific knot type.

Quantum representations of mapping class groups are locally rigid at prime levels.

Classifies representations up to dimension 3g-3 for surface mapping class groups.

Paper constructs a minimal surface with specific ends and curvature.

In this paper we construct a faithful representation of the mapping class group of the genus two surface into a group of matrices over the complex numbers. Our starting point is the Lawrence-Krammer representation of the braid group B_n, which was shown to be faithful by Bigelow and Krammer. We obtain a faithful repres…

Chart descriptions are a graphic method to describe monodromy representations of various topological objects. Here we introduce a chart description for genus-two Lefschetz fibrations, and show that any genus-two Lefschetz fibration can be stabilized by fiber-sum with certain basic Lefschetz fibrations.

We decompose into irreducible factors the ${\rm SU}(2)$ Witten-Reshetikhin-Turaev representations of the mapping class group of a genus $2$ surface when the level is $p=4r$ and $p=2r^2$ with $r$ an odd prime and when $p=2r_1r_2$ with $r_1$, $r_2$ two distinct odd primes. Some partial generalizations in higher genus are…

Characterizes area-minimizing maps for surfaces of genus ≥ 2.

We show the triviality of representations of the mapping class group of a genus $g$ surface in $GL(n,C), Diff(S^2)$ and $Homeo(T^2)$ when appropriate restrictions on the genus $g$ and the size of $n$ hold. For example, if $S_g$ is a surface of finite type and $φ: MCG(S_g) \to GL(n,C)$ is a homomorphism, then $φ$ is tri…

The study examines modular fusion categories with trivial Torelli group actions.

A well-known conjecture asserts that the mapping class group of a surface (possibly with punctures/boundary) does not virtually surject onto $\Z$ if the genus of the surface is large. We prove that if this conjecture holds for some genus, then it also holds for all larger genera. We also prove that if there is a counte…

Researchers study Prym representations for handlebody groups, focusing on cyclic cases.

Bounds on knot polynomials for Lie superalgebras of type I.

The study examines dynamics on SU(2)-representation varieties for surfaces and non-orientable surfaces.

Formula for colored Links-Gould polynomial with genus bounds.

Proof that $SU(2)$ character variety of genus 2 surface is ${\mathbb C} P^3$.

Morifuji computed the twisted Alexander polynomial of twist knots for nonabelian representations. In this paper we compute the twisted Alexander polynomial and the Reidemeister torsion of genus one two-bridge knots, a class of knots which includes twist knots. As an application, we give a formula for the Reidemeister t…

Let $e$ denote the Euler class on the space $Hom(Γ_g, PSL(2,\mathbb R))$ of representations of the fundamental group $Γ_g$ of the closed surface $Σ_g$ of genus $g$. Goldman showed that the connected components of $Hom(Γ_g, PSL(2,\mathbb R))$ are precisely the inverse images $e^{-1}(k)$, for $2-2g\leq k\leq 2g-2$, and t…

The SU(2) TQFT representation of the mapping class group of a closed surface of genus g, at a root of unity of prime order, is shown to be irreducible. Some examples of reducible representations are also given.

We prove that the image of the mapping class group by the representations arising in the SU(2)-TQFT is infinite, provided that the genus is bigger than 2 and the level r of the theory is different from 2,3,4,6. In particular the quotient of the mapping class group by the normaizer of the r-th power of a Dehn twist is i…

We give new explicit formulas for the representations of the mapping class group of a genus one surface with one boundary component which arise from Integral TQFT. Our formulas allow one to compute the h-adic expansion of the TQFT-matrix associated to a mapping class in a straightforward way. Truncating the h-adic expa…

In the genus expansion of the HOMFLY polynomials their representation dependence is naturally captured by symmetric group characters. This immediately implies that the Ooguri-Vafa partition function (OVPF) is a Hurwitz tau-function. In the planar limit involving factorizable special polynomials, it is actually a trivia…

Study on kernels of SO(3) WRT representations for surfaces of genus g≥3.

Braid groups help create complex surfaces in algebraic geometry.

Study knot invariants to answer questions about slice genus and clasp numbers.

For large genus, precise monodromy groups are calculated for surface covers.

We use geometric techniques to explicitly find the topological structure of the space of SO(3)-representations of the fundamental group of a closed surface of genus 2 quotient by the conjugation action by SO(3). There are two components of the space. We will describe the topology of both components and describe the cor…

In the planar limit of the 't Hooft expansion, the Wilson-loop average in 3d Chern-Simons theory (i.e. the HOMFLY polynomial) depends in a very simple way on representation (the Young diagram), so that the (knot-dependent) Ooguri-Vafa partition function becomes a trivial KP tau-function. We study higher genus correctio…

Investigates minimal genus of second homology classes in RAAGs, finding bounds and specific cases.