Using a combinatorial argument, we prove the well-known result that the Wirtinger and Dehn presentations of a link in 3-space describe isomorphic groups. The result is not true for links $\ell$ in a thickened surface $S \times [0,1]$. Their precise relationship, as given in the 2012 thesis of R.E. Byrd, is established here by an elementary argument. When a diagram in $S$ for $\ell$ can be checkerboard shaded, the Dehn presentation leads naturally to an abelian "Dehn coloring group," an isotopy invariant of $\ell$. Introducing homological information from $S$ produces a stronger invariant, $\cal C$, a module over the group ring of $H_1(S; {\mathbb Z})$. The authors previously defined the Laplacian modules ${\cal L}_G,{ \cal L}_{G^*}$ and polynomials $Δ_G, Δ_{G^*}$ associated to a Tait graph $G$ and its dual $G^*$, and showed that the pairs $\{{\cal L}_G, {\cal L}_{G^*}\}$, $\{Δ_G, Δ_{G^*}\}$ are isotopy invariants of $\ell$. The relationship between $\cal C$ and the Laplacian modules is described and used to prove that $Δ_G$ and $Δ_{G^*}$ are equal when $S$ is a torus.
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…
A group invariant for links in thickened closed orientable surfaces is studied. Associated polynomial invariants are defined. The group detects nontriviality of a virtual link and determines its virtual genus.
Menasco showed that a closed surface in the complement of a non-split prime alternating link in $S^3$ contains a circle isotopic in the link complement to a meridian of the links. This result is known as the meridian lemma for alternating links. We give a meridian lemma for the class of fully alternating links in the thickened orientable surfaces of positive genus.
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.
We construct an infinite family of homology theories of framed links in thickened surfaces, as well as a homology theory whose graded Euler characteristic is exactly the Kauffman bracket of the link in the surface. Both theories are based on ideas coming from Asaeda, Przytycki and Sikora's categorification of the Kauffman bracket skein module of I-bundles over surfaces. This is accomplished by borrowing ideas from Bar-Natan's Khovanov homology theory for tangles and cobordisms and using embedded surfaces to generate the chain groups, instead of diagrams.
Menasco showed that a non-split, prime, alternating link that is not a 2-braid is hyperbolic in $S^3$. We prove a similar result for links in closed thickened surfaces $S \times I$. We define a link to be fully alternating if it has an alternating projection from $S\times I$ to $S$ where the interior of every complementary region is an open disk. We show that a prime, fully alternating link in $S\times I$ is hyperbolic. Similar to Menasco, we also give an easy way to determine primeness in $S\times I$. A fully alternating link is prime in $S\times I$ if and only if it is "obviously prime". Fur…
Two link diagrams on compact surfaces are strongly equivalent if they are related by Reidemeister moves and orientation preserving homeomorphisms of the surfaces. They are stably equivalent if they are related by the two previous operations and adding or removing handles. Turaev and Turner constructed a link homology for each stable equivalence class by applying an unoriented topological quantum field theory (TQFT) to a geometric chain complex similar to Bar-Natan's one. In this paper, by using an unoriented homotopy quantum field theory (HQFT), we construct a link homology for each strong equ…
We define a variant of Khovanov homology for links in thickened disks with multiple punctures. This theory is distinct from the one previously defined by Asaeda, Przytycki, and Sikora, but is related to it by a spectral sequence. Additionally, we show that there are spectral sequences induced by embeddings of thickened surfaces, which recover the spectral sequence from annular Khovanov homology to Khovanov homology as a special case.
We prove a Kauffman-Murasugi-Thistlethwaite theorem for alternating links in thickened surfaces. It states that any reduced alternating diagram of a link in a thickened surface has minimal crossing number, and any two reduced alternating diagrams of the same link have the same writhe. This result is proved more generally for link diagrams that are adequate, and the proof involves a two-variable generalization of the Jones polynomial for surface links defined by Krushkal. The main result is used to establish the first and second Tait conjectures for links in thickened surfaces and for virtual l…
In the present paper, we construct an invariant for virtual knots in the thickened sphere with g handles; this invariant is a Laurent polynomial in 2g+3 variables. To this end, we use a modification of the Wirtinger presentation of the knot group and the concept of parity introduced by V.O.Manturov. The section 4 of the paper is devoted to an enhancement of the invariant (construction of the invariant module) by using the parity hierarchy concept suggested by V.O.Manturov. Namely, we discriminate between odd crossings and two types of even crossings; the latter two types depend on whether an e…
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.…
We extend the Gordon-Litherland pairing to links in thickened surfaces, and use it to define signature, determinant, and nullity invariants for links that bound (unoriented) spanning surfaces. The invariants are seen to depend only on the $S^*$-equivalence class of the spanning surface. We prove a duality result relating the invariants from one $S^*$-equivalence class of spanning surfaces to the restricted invariants of the other. Using Kuperberg's theorem, these invariants give rise to well-defined invariants of checkerboard colorable virtual links. The determinants can be applied to determin…
A cobordism between links in thickened surfaces consists of a surface $ S $ and a $3$-manifold $M $, with $ S $ properly embedded in $ M \times I $. We show that there exist links in thickened surfaces such that if $(S,M) $ is a cobordism between them in which $ S $ is simple, then $ M $ must be complex. That is, there are cases in which low complexity of the surface does not imply low complexity of the $3$-manifold. Specifically, we show that there exist concordant links in thickened surfaces between which a concordance can only be realised by passing through thickenings of higher genus surfa…
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 knot…
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 …
In the prequel of this paper, Kauffman and Ogasa introduced new topological quantum invariants of compact oriented 3-manifolds with boundary where the boundary is a disjoint union of two identical surfaces. The invariants are constructed via surgery on manifolds of the form $F \times I$ where $I$ denotes the unit interval. Since virtual knots and links are represented as links in such thickened surfaces, we are able also to construct invariants in terms of virtual link diagrams (planar diagrams with virtual crossings). These invariants are new, nontrivial, and calculable examples of quantum in…
We define additional gradings on two generalisations of Khovanov homology (one due to the first author, the other due to the second), and use them to define invariants of various kinds of embeddings. These include invariants of links in thickened surfaces and of surfaces embedded in thickened $3$-manifolds. In particular, the invariants of embedded surfaces are expressed in terms of certain diagrams related to the thickened $3$-manifold, so that we refer to them as picture-valued invariants. This paper contains the first instance of such invariants for $2$-dimensional objects. The additional g…
We prove a Torres-like formula for the $L^2$-Alexander torsions of links, as well as formulas for connected sums and cablings of links. Along the way we compute explicitly the $L^2$-Alexander torsions of torus links inside the three-sphere, the solid torus and the thickened torus.
Twisted links are a generalization of classical links and correspond to stably equivalence classes of links in thickened surfaces. In this paper we introduce twisted intersection colorings of a diagram and construct two invariants of a twisted link using such colorings. As an application, we show that there exist infinitely many pairs of twisted links such that for each pair the two twisted links are not equivalent but their double coverings are equivalent. We also introduce a method of constructing a pair of twisted links whose double coverings are equivalent.
Twisted links are a generalization of virtual links. As virtual links correspond to abstract links on orientable surfaces, twisted links correspond to abstract links on (possibly non-orientable) surfaces. In this paper, we introduce the notion of the double covering of a twisted link. It is defined by considering the orientation double covering of an abstract link or alternatively by constructing a diagram called a double covering diagram. We also discuss links in thickened surfaces, their diagrams and their stable equivalence classes. Bourgoin's twisted knot group is understood as the virtual…
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…
We prove that any arc-presentation of the unknot admits a monotonic simplification by elementary moves; this yields a simple algorithm for recognizing the unknot. We obtain similar results for split links and composite links.
Habegger and Lin gave a classification of the link-hmotopy classes of links as the link-homotopy classes of string links modulo the actions of conjugations and partial conjugations for string links. In this paper, we calculated the actions of the partial conjugations and the conjugations explicitly for 4- and 5-component string links which gave classifications (presentations) of the link-homotopy classes of 4- and 5-component links. As an application, we can run Habegger and Lin's algorithm which determines whether given two links are link-homotopoic or not for 4- and 5-component links.
New presentations of a link and a virtual link are introduced and algebraic systems on links and virtual links are constructed respectively. Based on the algebraic systems, Reduction Crossing Algorithms for them are proposed which are used to reduce the number of crossings in a link and virtual link. For known unknots, one can transform them into a trivial knot in a polynomial time by applying corresponding algorithm. As special consequences, Goeritz's unknot and Thistlethwaite's unknot are unknotted. Moreover, an infinite family of knots $K_{G_{2k,2l}}$ are unknotted in $O(n^2)$ time where $n…
For a link in a thickened annulus $A \times I$, we define a $\mathbb{Z} \oplus \mathbb{Z} \oplus \mathbb{Z}$ filtration on Sarkar-Seed-Szabó's perturbation of the geometric spectral sequence. The filtered chain homotopy type is an invariant of the isotopy class of the annular link. From this, we define a two-dimensional family of annular link invariants and study their behavior under cobordisms. In the case of annular links obtained from braid closures, we obtain a necessary condition for braid quasipositivity and a sufficient condition for right-veeringness, as well as Bennequin-type inequali…
It is well known that surface-links in 4-space can be presented by diagrams on the plane of 4-valent spatial graphs with makers on the vertices, called marked graph diagrams. In this paper we extend the method of presenting surface-links by marked graph diagrams to presenting immersed surface-links. We also give some moves on marked graph diagrams that preserve the ambient isotopy classes of their presenting immersed surface-links.
A virtual link may be defined as an equivalence class of diagrams, or alternatively as a stable equivalence class of links in thickened surfaces. We prove that a minimal crossing virtual link diagram has minimal genus across representatives of the stable equivalence class. This is achieved by constructing a new parity theory for virtual links. As corollaries, we prove that the crossing, bridge, and ascending numbers of a classical link do not decrease when it is regarded as a virtual link. This extends corresponding results in the case of virtual knots due to Manturov and Chernov.
We develop a calculus of surgery data, called bridged links, which involves besides links also pairs of balls that describe one-handle attachements. As opposed to the usual link calculi of Kirby and others this description uses only elementary, local moves(namely modifications and isolated cancellations), and it is valid also on non-simply connected and disconnected manifolds. In particular, it allows us to give a presentation of a 3-manifold by doing surgery on any other 3-manifold with the same boundary. Bridged link presentations on unions of handlebodies are used to give a Cerf-theoretical…
We will discuss a method for visual presentation of knotted surfaces in the four space, by examining a number and a position of its Morse's critical points. Using this method, we will investigate surface-knot with one critical point of index 1. Then we show infinitely many mutually distinct surface-knots that has an embedding with two critical points of index 1. Next we define a long flat for a banded link for any surface-knot and show diagrammatically a long flat form of $n$-twist-spun $(2,t)$-torus knots.
We define braid presentation of edge-oriented spatial graphs as a natural generalization of braid presentation of oriented links. We show that every spatial graph has a braid presentation. For an oriented link it is known that the braid index is equal to the minimal number of Seifert circles. We show that an analogy does not hold for spatial graphs.
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…
Knitted and woven textile structures are examples of doubly periodic structures in a thickened plane made out of intertwining strands of yarn. Factoring out the group of translation symmetries of such a structure gives rise to a link diagram in a thickened torus. Such a diagram on a standard torus is converted into a classical link by including two auxiliary components which form the cores of the complementary solid tori. The resulting link, called a kernel for the structure, is determined by a choice of generators u and v for the group of symmetries. A normalised form of the multi-variable Al…
A virtual knot, which is one of generalizations of knots in $\mathbb{R}^{3}$ (or $S^{3}$), is, roughly speaking, an embedded circle in thickened surface $S_{g} \times I$. In this talk we will discuss about knots in 3 dimensional $S_{g} \times S^{1}$. We introduce basic notions for knots in $S_{g} \times S^{1}$, for example, diagrams, moves for diagrams and so on. For knots in $S_{g} \times S^{1}$ technically we lose over/under information, but we will have information how many times the knot rotates along $S^{1}$. We will discuss the geometric meaning of the rotating information and how to con…
We consider knots equipped with a representation of their knot groups onto a dihedral group D_{2n} (where n is odd). To each such knot there corresponds a closed 3-manifold, the (irregular) dihedral branched covering space, with the branching set over the knot forming a link in it. We report a variety of results relating to the problem of passing from the initial data of a D_{2n}-coloured knot to a surgery presentation of the corresponding branched covering space and covering link. In particular, we describe effective algorithms for constructing such presentations. A by-product of these invest…
The lattice stick number $s_L(L)$ of a link $L$ is defined to be the minimal number of straight line segments required to construct a stick presentation of $L$ in the cubic lattice. Hong, No and Oh found a general upper bound $s_L(K) \leq 3 c(K) +2$. A rational link can be represented by a lattice presentation with exactly 4 $z$-sticks. An $n$-circuit is the disjoint union of $n$ arcs in the lattice plane $\mathbb{Z}^2$. An $n$-circuit presentation is an embedding obtained from the $n$-circuit by connecting each $n$ pair of vertices with one line segment above the circuit. By using a 2-circuit…
