Spaces of knots in the solid torus, knots in the thickened torus, and links in the 3-sphere
Andrew Havens, Robin Koytcheff
备注：
Enlarged the scope of some content to cover split links as well as irreducible ones. Added some references for context. Other minor revisions. Final pre-publication version. Accepted for publication in Geometriae Dedicata. Now 62 pages, 33 figures
We recursively determine the homotopy type of the space of any irreducible framed link in the 3-sphere, modulo rotations. This leads us to the homotopy type of the space of any knot in the solid torus, thus answering a question posed by Arnold. We similarly study spaces of unframed links in the 3-sphere, modulo rotations, and spaces of knots in the thickened torus. The subgroup of meridional rotations splits as a direct factor of the fundamental group of the space of any framed link except the unknot. Its generators can be viewed as generalizations of the Gramain loop in the space of long knots. Taking the quotient by certain such rotations relates the spaces we study. All of our results generalize previous work of Hatcher and Budney. We provide many examples and explicitly describe generators of fundamental groups.
A knot in a thickened surface $K$ is a smooth embedding $K:S^1 \rightarrow Σ\times [0,1]$, where $Σ$ is a closed, connected, orientable surface. There is a bijective correspondence between knots in $S^2 \times [0,1]$ and knots in $S^3$, so one can view the study of knots in thickened surfaces as an extension of classical knot theory. An immediate question is if other classical definitions, concepts, and results extend or generalize to the study of knots in a thickened surface. One such famous result is the Fox Milnor Theorem, which relates the Alexander polynomials of concordant knots. We prov…
The parity hierarchy and new invariants of knots in thickened surfaces
在本文中，我们构造了一个不变量的虚拟纽结加厚球与g手柄，这个不变量是一个Laurent多项式在2g+3个变量。为此，我们使用修改的Wirtinger介绍的纽结群和宇称的概念介绍了V. O. Manturov。第四节利用V. O. Manturov提出的奇偶性层次概念对不变式进行了改进（构造不变模）。也就是说，我们区分奇数交叉和两种类型的偶数交叉;后两种类型取决于是否偶数交叉仍然是偶数/奇数后，所有奇数交叉的图被删除。不变量的构造也适用于虚拟节点。
In the present paper, we address the problem how to get a map from knots in the cylinder and on the thickened torus to some (generalisation of) virtual knots called virtual-flat knots. The main construction takes a diagram on a cylinder (torus) and adds some ``invisible'' crossings which gives rise to a diagram which can be formally immersed but not embedded (drawn) on the cylinder (torus) and living comfortably in thickened surfaces of higher genera. This allows one to ``pull back'' invariants of virtual theory to the theory of knots in the thickened cylinder (torus) where the parity bracket …
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.
Using the Gordon-Litherland pairing, one can define invariants (signature, nullity, determinant) for ${\mathbb Z}/2$ null-homologous links in thickened surfaces. In this paper, we study the concordance properties of these invariants. For example, if $K \subset Σ\times I$ is ${\mathbb Z}/2$ null-homologous and slice, we show that its signatures vanish and its determinants are perfect squares. These statements are derived from a cobordism result for closed unoriented surfaces in certain 4-manifolds. The Brown invariants are defined for ${\mathbb Z}/2$ null-homologous links in thickened surfaces.…
Crossings of knot diagrams can be divided into classes (tribes) compatible with Reidemeister moves. Tribes can be considered as localization of the notion of weak chord index introduced by M. Xu. In the article we describe tribes of crossings for tangles in a fixed surface and show that crossings are differentiated by their component, order and homotopy types. As a consequence, we conclude that there are no nontrivial indices on diagrams of classical knots.
Suppose $Σ$ is a compact oriented surface (possibly with boundary) that has genus zero, and L is a link in the interior of $(-1,1)\timesΣ$. We prove that the Asaeda-Przytycki-Sikora (APS) homology of L has rank 2 if and only if L is isotopic to an embedded knot in $\{0\}\timesΣ$. As a consequence, the APS homology detects the unknot in $(-1,1)\timesΣ$. This is the first detection result for generalized Khovanov homology that is valid on an infinite family of manifolds, and it partially solves a conjecture in arxiv:2005.12863. Our proof is different from the previous detection results obtained …
We use the Bar-Natan Zh-correspondence to identify the generalized Alexander polynomial of a virtual knot with the Alexander polynomial of a two component welded link. We show that the Zh-map is functorial under concordance, and also that Satoh's Tube map (from welded links to ribbon knotted tori in $S^4$) is functorial under concordance. In addition, we extend classical results of Chen, Milnor, and Hillman on the lower central series of link groups to links in thickened surfaces. Our main result is that the generalized Alexander polynomial vanishes on any knot in a thickened surface which is …
We discuss links in thickened surfaces. We define the Khovanov-Lipshitz-Sarkar stable homotopy type and the Steenrod square for the homotopical Khovanov homology of links in thickened surfaces with genus$>1$. A surface means a closed oriented surface unless otherwise stated. Of course, a surface may or may not be the sphere. A thickened surface means a product manifold of a surface and the interval. A link in a thickened surface (respectively, a 3-manifold) means a submanifold of a thickened surface (respectively, a 3-manifold) which is diffeomorphic to a disjoint collection of circles. Our Kh…
Twisted torus knots are torus knots with some full twists added along some number of adjacent strands. There are infinitely many known examples of twisted torus knots which are actually torus knots. We give eight more infinite families of such twisted torus knots with a single negative twist.
Milnor's $\barμ$-invariants of links in the $3$-sphere $S^3$ vanish on any link concordant to a boundary link. In particular, they are trivial on any knot in $S^3$. Here we consider knots in thickened surfaces $Σ\times [0,1]$, where $Σ$ is closed and oriented. We construct new concordance invariants by adapting the Chen-Milnor theory of links in $S^3$ to an extension of the group of a virtual knot. A key ingredient is the Bar-Natan $\textit{Zh}$ map, which allows for a geometric interpretation of the group extension. The group extension itself was originally defined by Silver-Williams. Our ext…
It is known that connected sums of positive torus knots are not concordant to $L$-space knots. Here we consider differences of torus knots. The main result states that the subgroup of the concordance group generated by two positive torus knots contains no nontrivial $L$-space knots other than the torus knots themselves. Generalizations to subgroups generated by more than two torus knots are also considered.
The aim of this paper is to realise the techniques of picture-valued invariants and invariants valued in free groups for long knots in the full torus. Such knots and links are of a particular interest because of their relation to Legendrian knots, knotoids, 3-manifolds and many other objects. Invariants constructed in the paper are powerful and easy to compare. This paper is a sequel of [6]. Long knots naturally appear in the study of classical knots [1, 8].
We consider the question, asked by Friedl, Livingston and Zentner, of which sums of torus knots are concordant to alternating knots. After a brief analysis of the problem in its full generality, we focus on sums of two torus knots. We describe some effective obstructions based on Heegaard Floer homology.
The twisted torus knots K(p, q; r, s) are obtained by performing a sequence of s full twists on r adjacent strands of (p, q)-torus knots. Morimoto asked whether all twisted torus knots with essential tori in the exterior fit into one of two families. We prove that the answer to this question is no, by finding two different new families of toroidal twisted torus knots.
We show that the triple-crossing number of any knot is greater or equal to twice its (canonical) genus and we show an even stronger bound in the case of links. As an application we show that this bound is strong enough to obtain the triple-crossing numbers of all torus knots, and of many more knots and their connected sums.
