平面完全增强链接由其补集确定

加厚环面中的全增广链环

Fully Augmented Links in the Thickened Torus

本文推广了 $S^3$ 中全增广链环的研究，分类了加厚环面中的双曲全增广链环，证明其补空间可分解为规范边的理想直角多面体，并证明了全增广链环的体积密度猜想。

Hyperbolicity of Augmented Links in the Thickened Torus

For a hyperbolic link K in the thickened torus with no bigons, we show that there is a decomposition of the complement of a link L, obtained from augmenting K, into torihedra. We further decompose the torihedra into angled pyramids and finally angled tetrahedra. These fit into an angled structure on a triangulation of the link complement, and thus by [5], this shows that L is hyperbolic.

Symmetries and hidden symmetries of $(ε, d_L)$-twisted knot complements

In this paper we analyze symmetries, hidden symmetries, and commensurability classes of $(ε, d_L)$-twisted knot complements, which are the complements of knots that have a sufficiently large number of twists in each of their twist regions. These knot complements can be constructed via long Dehn fillings on fully augmented links complements. We show that such knot complements have no hidden symmetries, which implies that there are at most two other knot complements in their respective commensurability classes. Under mild additional hypotheses, we show that these knots have at most four (orienta…

局部平坦子流形的补集是有限CW复形

Complements of locally flat submanifolds are finite CW complexes

本文证明紧致拓扑流形中局部平坦子流形的补集同伦等价于有限CW复形，给出直接证明，避免依赖大量拓扑流形理论。

On multiply twisted knots that are Seifert fibered or toroidal

We consider knots whose diagrams have a high amount of twisting of multiple strands. By encircling twists on multiple strands with unknotted curves, we obtain a link called a generalized augmented link. Dehn filling this link gives the original knot. We classify those generalized augmented links that are Seifert fibered, and give a torus decomposition for those that are toroidal. In particular, we find that each component of the torus decomposition is either "trivial", in some sense, or homeomorphic to the complement of a generalized augmented link. We show this structure persists under high D…

模链环的新视角：从旧视角出发

A new twist on modular links from an old perspective

本文从旧视角出发，揭示了算术模链环与增强链环补集的同胚关系，并展示了其作为白头链环补集的n-fold循环覆盖结构。

Invariants of tangles with flat connections in their complements

Let G be a simple complex algebraic group. By using a notion of a G-category we define invariants of tangles with flat G-connections in their complements. We also show that quantized universal enveloping algebras at roots of unity provide examples of G-categories.

Groups of locally-flat disk knots and non-locally-flat sphere knots

The classical knot groups are the fundamental groups of the complements of smooth or piecewise-linear (PL) locally-flat knots. For PL knots that are not locally-flat, there is a pair of interesting groups to study: the fundamental group of the knot complement and that of the complement of the ``boundary knot'' that occurs around the singular set, the set of points at which the embedding is not locally-flat. If a knot has only point singularities, this is equivalent to studying the groups of a PL locally-flat disk knot and its boundary sphere knot; in this case, we obtain a complete classificat…

A classification of links of the flat plumbing basket numbers 4 or less

Flat plumbing basket surfaces of links were introduced to study the geometry of the complement of the links. In present article, we study links of the flat plumbing basket numbers $4$ or less using a special presentation of the flat plumbing basket surfaces. We find a complete classification theorem of links of the flat plumbing basket numbers 4 or less.

Some non-trivial PL knots whose complements are homotopy circles

We show that there exist non-trivial piecewise-linear (PL) knots with isolated singularities $S^{n-2}\subset S^n$, $n\geq 5$, whose complements have the homotopy type of a circle. This is in contrast to the case of smooth, PL locally-flat, and topological locally-flat knots, for which it is known that if the complement has the homotopy type of a circle, then the knot is trivial.

具有单连通补空间的纽结曲面

Knotted surfaces with simply-connected complements

研究边界为固定纽结 K 且嵌入单连通四维流形的特定曲面，证明若此类曲面同调，则在相对于边界的拓扑环境同痕下是同痕的。

链甲链环与非左序性

Chainmail links and non-left-orderability

本文证明了平坦全增广链甲链环的交错手术产生全L空间，并利用序检测研究了与L空间结连通和手术的非左序性。

The complete list of prime knots whose flat plumbing basket numbers are 6 or less

Flat plumbing basket surfaces of links were introduced to study the geometry of the complement of the links. These flat plumbing basket surface can be presented by a sequential presentation known as flat plumbing basket code first found by Furihata, Hirasawa and Kobayashi. The minimum number of flat plumbings to obtain a flat plumbing basket surfaces of a link is defined to be the flat plumbing basket number of the given link. In present article, we use these sequential presentations to find the complete classification theorem of prime knots whose flat plumbing basket number $6$ or less. As ap…

The volume of hyperbolic alternating link complements

If a hyperbolic link has a prime alternating diagram D, then we show that the link complement's volume can be estimated directly from D. We define a very elementary invariant of the diagram D, its twist number t(D), and show that the volume lies between v_3(t(D) - 2)/2 and v_3(16t(D) - 16), where v_3 is the volume of a regular hyperbolic ideal 3-simplex. As a consequence, the set of all hyperbolic alternating and augmented alternating link complements is a closed subset of the space of all complete finite volume hyperbolic 3-manifolds, in the geometric topology. The appendix by Ian Agol and Dy…