Quantum traces for representations of surface groups in SL_2

The pillowcase and perturbations of traceless representations of knot groups

We introduce explicit holonomy perturbations of the Chern-Simons functional on a 3-ball containing a pair of unknotted arcs. These perturbations give us a concrete local method for making the moduli spaces of flat singular SO(3) connections relevant to Kronheimer and Mrowka's singular instanton knot homology non-degenerate. The mechanism for this study is a (Lagrangian) intersection diagram which arises, through restriction of representations, from a tangle decomposition of a knot. When one of the tangles is trivial, our perturbations allow us to study isolated intersections of two Lagrangians…

Decision problems, complexity, traces, and representations

In this article, we study connections between representation theory and efficient solutions to the conjugacy problem on finitely generated groups. The main focus is on the conjugacy problem in conjugacy separable groups, where we measure efficiency in terms of the size of the quotients required to distinguish a distinct pair of conjugacy classes.

The trace embedding lemma and spinelessness

We demonstrate new applications of the trace embedding lemma to the study of piecewise-linear surfaces and the detection of exotic phenomena in dimension four. We provide infinitely many pairs of homeomorphic 4-manifolds $W$ and $W'$ homotopy equivalent to $S^2$ which have smooth structures distinguished by several formal properties: $W'$ is diffeomorphic to a knot trace but $W$ is not, $W'$ contains $S^2$ as a smooth spine but $W$ does not even contain $S^2$ as a piecewise-linear spine, $W'$ is geometrically simply connected but $W$ is not, and $W'$ does not admit a Stein structure but $W$ do…

Quantum Teichmüller spaces and quantum trace map

We show how the quantum trace map of Bonahon and Wong can be constructed in a natural way using the skein algebra of Muller, which is an extension of the Kauffman bracket skein algebra of surfaces. We also show that the quantum Teichmüller space of a marked surface, defined by Chekhov-Fock (and Kashaev) in an abstract way, can be realized as a concrete subalgebra of the skew field of the skein algebra.

Trace Coordinates on Fricke spaces of some simple hyperbolic surfaces

The conjugacy class of a generic unimodular 2 by 2 complex matrix is determined by its trace, which may be an arbitrary complex number. In the nineteenth century, it was known that a generic pair (X,Y) of such pairs is determined up to conjugacy by the triple of traces (tr(X),tr(Y),tr(XY), which may be an arbitary element of C^3. This paper gives an elementary and detailed proof of this fact, which was published by Vogt in 1889. The folk theorem describing the extension of a representation to a representation of the index-two supergroup which is a free product of three groups of order two, is …

迹收缩长度与汇常数

Trace Systoles and Sink Constant

该研究定义了曲面基本群到SL(2,ℂ)表示的迹收缩长度，借助Markoff映射对应关系，显式算出三类曲面相对特征簇上的最优界，并得到双曲流形收缩不等式与非Fuchsian表示的相关推论。

The quantum trace as a quantum non-abelianization map

We prove that the balanced Chekhov-Fock algebra of a punctured triangulated surface is isomorphic to a skein algebra which is a deformation of the algebra of regular functions of some abelian character variety. We first deduce from this observation a classification of the irreducible representations of the balanced Chekhov-Fock algebra at odd roots of unity, which generalizes to open surfaces the classification of Bonahon, Liu and Wong. We re-interpret Bonahon and Wong's quantum trace map as a non-commutative deformation of some regular morphism between this abelian character variety and the S…

曲面上的秩-$N$ 二聚体模型

Rank-$N$ Dimer Models on Surfaces

本文将Douglas等人的网迹定理推广至高亏格曲面，展开自旋结构对应的扭曲Kasteleyn矩阵，并利用嵌入环的自旋几何与赛车道构造给出了平面情形的几何推导。

Hecke algebra trace algorithm and some conjectures on weaving knots

Computing polynomial invariants for knots and links using braid representations relies heavily on finding the trace of Hecke algebra elements. There is no easy method known for computing the trace and hence it becomes difficult to compute the known polynomial invariants of knots using their braid representations. In this paper, we provide an algorithm to compute the trace of the Hecke algebra representation of any braid. We simplify this algorithm and write a Mathematica program to compute the invariants such as Alexander polynomial, Jones polynomial, HOMFLY-PT polynomial and Khovanov homology…

曲面 ${\rm SL}_3$ 量子迹映射的自然性

Naturality of ${\rm SL}_3$ quantum trace maps for surfaces

本文开发了不同三角剖分的立方根量子环面代数间的量子突变映射，证明了 ${\rm SL}_3$ 量子迹映射在这些映射下的自然性，并由此得出量子 ${\rm SL}_3$-${\rm PGL}_3$ 对偶映射独立于理想三角剖分的选择。

Real Places and Surface Bundles

Our results complement D. Calegari's result that there are no hyperbolic once-punctured torus bundles over $S^1$ with trace field having real place. We exhibit several infinite families of pairs $(-χ, p)$ such that there exist hyperbolic surface bundles with over $S^1$ with fiber having $p$ punctures and Euler characteristic $χ$ with trace field of having a real place. This supports our conjecture that there exist such examples for each pair $(-χ, p)$ with finitely many known exceptions.