加厚曲面中交错链环的八面体分解

关于周期编织的分类与加厚曲面中链环的万有覆叠

On the classification of periodic weaves and universal cover of links in thickened surfaces

针对加厚曲面中链环的周期编织分类难题，引入双曲周期编织推广双周期编织，拓展并证明适用于极小既约交错编织基序的Tait猜想，为周期编织分类提供新的理论工具。

加厚曲面中的直角链环

Right-angled links in thickened surfaces

本文研究加厚曲面上的交错链环，定义并刻画了直角广义完全可实现（RGCR）性质，利用投影图限制对其分类，给出了数量界限并找到 RGCR 纽结，同时回答了关于环面交错投影链环的遗留问题。

加厚曲面中链环的端本质支撑曲面

End-essential spanning surfaces for links in thickened surfaces

本文证明了无赘余交叉的闭曲面胞腔交错链环图的棋盘曲面具有 $π_1$-本质性且无 $\partial$-平行本质闭曲线，为证明交错虚拟链环的 Tait 翻转猜想提供技术支撑并探讨了 Turaev 曲面的应用。

增厚曲面中的充分链环与广义Tait猜想

Adequate links in thickened surfaces and the generalized Tait conjectures

本文应用Kauffman括号辫代数发展增厚曲面中的辫充分链环理论，证明了交错链环图是辫充分的，并由此建立第一和第二Tait猜想，推广了关于辫括号跨度与交叉点数的经典不等式及等号判定结果。

曲面中的突变、曲面图与交错链环

Mutation, surface graphs, and alternating links in surfaces

本文通过 Tait 图上的整数流格研究增厚曲面中的交错链环，证明了流格在圆盘突变下的不变性，并发现 Gordon-Litherland 链接型是比流格更强的突变不变量。

交错虚拟链环的素性

Primeness of alternating virtual links

本文利用 lassos 工具建立闭曲面胞腔链环图与虚拟链环图的对应关系，揭示了虚拟链环素性概念更严格，并扩展 Menasco 经典结果，证明增厚曲面上的交错链环在更严格意义下是素的。

交替虚拟链环的体积型定理

A volumish theorem for alternating virtual links

本文将 Dasbach-Lin 的体积型定理推广到曲面上的交替链环图解，用约化 Jones-Krushkal 多项式系数给出加厚曲面中链环双曲体积的界，并揭示 4 变量 Krushkal 多项式系数与约化 Tait 图圈秩的关系。

Guts, volume and Skein Modules of 3-manifolds

We consider hyperbolic links that admit alternating projections on surfaces in compact, irreducible 3-manifolds. We show that, under some mild hypotheses, the volume of the complement of such a link is bounded below in terms of a Kauffman bracket function defined on link diagrams on the surface. In the case that the 3-manifold is a thickened surface, this Kauffman bracket function leads to a Jones-type polynomial that is an isotopy invariant of links. We show that coefficients of this polynomial provide 2-sided linear bounds on the volume of hyperbolic alternating links in the thickened surfac…

Alternating links on surfaces and volume bounds

Weakly generalised alternating knots are knots with an alternating projection onto a closed surface in a compact irreducible 3-manifold, and they share many hyperbolic geometric properties with usual alternating knots. For example, usual alternating knots have volume bounded above and below by the twist number of the alternating diagram due to Lackenby. Howie and Purcell showed that a similar lower bound holds for weakly generalised alternating knots. In this paper, we show that a generalisation of the upper volume bound does not hold, by producing a family of weakly generalised alternating kn…

Geometry of biperiodic alternating links

A biperiodic alternating link has an alternating quotient link in the thickened torus. In this paper, we focus on semi-regular links, a class of biperiodic alternating links whose hyperbolic structure can be immediately determined from a corresponding Euclidean tiling. Consequently, we determine the exact volumes of semi-regular links. We relate their commensurability and arithmeticity to the corresponding tiling, and assuming a conjecture of Milnor, we show there exist infinitely many pairwise incommensurable semi-regular links with the same invariant trace field. We show that only two semi-r…

Generalized bipyramids and hyperbolic volumes of alternating $k$-uniform tiling links

We present explicit geometric decompositions of the hyperbolic complements of alternating $k$-uniform tiling links, which are alternating links whose projection graphs are $k$-uniform tilings of $S^2$, $\mathbb{E}^2$, or $\mathbb{H}^2$. A consequence of this decomposition is that the volumes of spherical alternating $k$-uniform tiling links are precisely twice the maximal volumes of the ideal Archimedean solids of the same combinatorial description, and the hyperbolic structures for the hyperbolic alternating tiling links come from the equilateral realization of the $k$-uniform tiling on $\mat…

TG-Hyperbolicity of Virtual Links

We extend the theory of hyperbolicity of links in the 3-sphere to tg-hyperbolicity of virtual links, using the fact that the theory of virtual links can be translated into the theory of links living in closed orientable thickened surfaces. When the boundary surfaces are taken to be totally geodesic, we obtain a tg-hyperbolic structure with a unique associated volume. We prove that all virtual alternating links are tg-hyperbolic. We further extend tg-hyperbolicity to several classes of non-alternating virtual links. We then consider bounds on volumes of virtual links and include a table for vol…

虚拟翻转定理

The virtual flyping theorem

本文将翻转定理扩展至增厚曲面上的交错链环与交错虚拟链环，并扩展了 Gordon--Litherland 的经典结果，建立了跨越曲面配对与4维流形相交形式的同构。

Generalized bipyramids and hyperbolic volumes of tiling links

We present explicit geometric decompositions of the complement of tiling links, which are alternating links whose projection graphs are uniform tilings of the 2-sphere, the Euclidean plane or the hyperbolic plane. This requires generalizing the angle structures program of Casson and Rivin for triangulations with a mixture of finite, ideal, and truncated (i.e. ultra-ideal) vertices. A consequence of this decomposition is that the volumes of spherical tiling links are precisely twice the maximal volumes of the ideal Archimedean solids of the same combinatorial description. In the case of hyperbo…

Matrix Integrals and the Generation and Counting of Virtual Tangles and Links

Virtual links are generalizations of classical links that can be represented by links embedded in a ``thickened'' surface $Σ\times I$, product of a Riemann surface of genus $h$ with an interval. In this paper, we show that virtual alternating links and tangles are naturally associated with the $1/N^2$ expansion of an integral over $N\times N$ complex matrices. We suggest that it is sufficient to count the equivalence classes of these diagrams modulo ordinary (planar) flypes. To test this hypothesis, we use an algorithm coding the corresponding Feynman diagrams by means of permutations that gen…