3-辫子的共形不变量与计数函数

Invariants of closed braids via counting surfaces

A Gauss diagram is a simple, combinatorial way to present a link. It is known that any Vassiliev invariant may be obtained from a Gauss diagram formula that involves counting subdiagrams of certain combinatorial types. In this paper we present simple formulas for an infinite family of invariants in terms of counting surfaces of a certain genus and number of boundary components in a Gauss diagram associated with a closed braid. We then identify the resulting invariants with partial derivatives of the HOMFLY-PT polynomial.

Braids, Conformal Module and Entropy

The conformal module of conjugacy classes of braids implicitly appeared in a paper of Lin and Gorin in connection with their interest in the 13. Hilbert Problem. This invariant is the supremum of conformal modules (in the sense of Ahlfors) of certain annuli related to the conjugacy class. This note states that the conformal module is inverse proportional to a popular dynamical braid invariant, the entropy. The entropy appeared in connection with Thurston's theory of surface homeomorphisms. An application of the concept of conformal module to algebraic geometry is given.

Counting Dirac braid relators and hyperelliptic Lefschetz fibrations

We define a new invariant $w$ for hyperelliptic Lefschetz fibrations over closed oriented surfaces, which counts the number of Dirac braids included intrinsically in the monodromy, by using chart description introduced by the second author. As an application, we prove that two hyperelliptic Lefschetz fibrations of genus $g$ over a given base space are stably isomorphic if and only if they have the same numbers of singular fibers of each type and they have the same value of $w$ if $g$ is odd. We also give examples of pair of hyperelliptic Lefschetz fibrations with the same numbers of singular f…

辫子、共形模与熵

Braids, conformal module and entropy

该研究证明辫子共轭类的共形模与熵成反比，可实现两类不变量研究方法的互用，给出希尔伯特第13问题相关定理的短证明，并将共形模用于研究Gromov的Oka原理存在障碍的情形。

Counting genus one fibered knots in lens spaces

The braid axis of a closed 3-braid lifts to a genus one fibered knot in the double cover of S^3 branched over the closed braid. Every (null homologous) genus one fibered knot in a 3-manifold may be obtained in this way. Using this perspective we answer a question of Morimoto about the number of genus one fibered knots in lens spaces. We determine the number of genus one fibered knots up to homeomorphism in any given lens space. This number is 3 in the case of the lens space L(4,1), 2 for the lens spaces L(m,1) with m>0, and at most 1 otherwise.

共形块的半单性

Semisimplicity of conformal blocks

本文证明编织融合范畴的辫群表示和模融合范畴的映射类群表示总是半单的，并进一步证明它们保持非退化埃尔米特形式且可在CM数域上定义。