We introduce a new construction of a surface link in the 4-space. We construct a surface link as a branched covering over the standard torus, which we call a torus-covering link. We show that a certain torus-covering $T^2$-link is equivalent to the split union of spun $T^2$-links and turned spun $T^2$-links. We show that a certain torus-covering $T^2$-link has a non-classical link group. We give a certain class of ribbon torus-covering $T^2$-links. We present the quandle cocycle invariant of a certain torus-covering $T^2$-link obtained from a classical braid, by using the quandle cocycle invariants of the closure of the braid.
We consider a surface link in the 4-space which can be presented by a simple branched covering over the standard torus, which we call a torus-covering link. Torus-covering links include spun $T^2$-knots and turned spun $T^2$-knots. In this paper we braid a torus-covering link over the standard 2-sphere. This gives an upper estimate of the braid index of a torus-covering link. In particular we show that the turned spun $T^2$-knot of the torus $(2,\,p)$-knot has the braid index four.
After showing that a covering space of surface bundles over $S^1$ factors as a `covering of fibers' followed by a `power covering', we prove that, for torus bundles, power coverings do not lower Heegaard genus, and that fiber coverings lower the genus only in special cases.
We consider surface links in the 4-space which are presented by the form of simple branched coverings over the standard torus, which we call torus-covering links. In this paper, we study unknotting numbers of torus-covering links. In some cases, we can determine the unknotting numbers.
It is known that any surface knot can be transformed to an unknotted surface knot or a surface knot which has a diagram with no triple points by a finite number of 1-handle additions. The minimum number of such 1-handles is called the unknotting number or the triple point cancelling number, respectively. In this paper, we give upper bounds and lower bounds of unknotting numbers and triple point cancelling numbers of torus-covering knots, which are surface knots in the form of coverings over the standard torus $T$. Upper bounds are given by using $m$-charts on $T$ presenting torus-covering knot…
Let $t_{α,β}\subset S^2\times S^1$ be an ordinary fiber of a Seifert fibering of $S^2\times S^1$ with two exceptional fibers of order $α$. We show that any Seifert manifold with Euler number zero is a branched covering of $S^2\times S^1$ with branching $t_{α,β}$ if $α\geq3$. We compute the Seifert invariants of the Abelian covers of $S^2\times S^1$ branched along a $t_{α,β}$. We also show that $t_{2,1}$, a non-trivial torus knot in $S^2\times S^1$, is not universal.
We study tori which are cyclic covers of the standard torus, that is, the deck transformation group of the covering map is cyclic. These covering tori can be parametrized in a natural way and we show that being cyclic is equivalent to certain arithmetic condition on these parameters. There is a natural $\mathrm{SL}(2,\mathbb{Z})$-action on covering tori and introducing a complete numeric $\mathrm{SL}(2,\mathbb{Z})$-invariant we show that, for $n\in\mathbb{N}$, all $n$-tuple cyclic covers are in the same $\mathrm{SL}(2,\mathbb{Z})$-orbit. We show that cyclic covers are irreducible in a precise …
All knots are known to be hyperbolic, satellite, or torus knots, and one important family is Lorenz links, or T-links, which arise from dynamics. However, it remains difficult to determine the geometric type of a Lorenz link from a description via dynamics or as a T-link. In this paper, we consider those T-links that are torus links. We show that T-links obtained by full twists along torus links can never be torus links, aside from a family of cases. This addresses a question of Birman and Kofman.
We compute the triply graded Khovanov-Rozansky homology of a family of links, including positive torus links and $\operatorname{Sym}^l$-colored torus knots.
The present paper is an addendum to "Spherical structures on torus knots and links", arXiv:1008.0312, and concerns more general case of torus knot and link cone-manifolds.
This article is about a natural distance function induced by smooth cobordisms between links. We show that the cobordism distance of torus links is determined by the profiles of their signature functions, up to a constant factor.
We compute Cayley graphs and automorphism groups for all finite $n$-quandles of two-bridge and torus knots and links, as well as torus links with an axis.
Twisted torus knots and links are given by twisting adjacent strands of a torus link. They are geometrically simple and contain many examples of the smallest volume hyperbolic knots. Many are also Lorenz links. We study the geometry of twisted torus links and related generalizations. We determine upper bounds on their hyperbolic volumes that depend only on the number of strands being twisted. We exhibit a family of twisted torus knots for which this upper bound is sharp, and another family with volumes approaching infinity. Consequently, we show there exist twisted torus knots with arbitrarily…
The structure of the Khovanov homology of $(n,m)$ torus links has been extensively studied. In particular, Marko Stosic proved that the homology groups stabilize as $m\rightarrow\infty$. We show that the Khovanov homotopy types of $(n,m)$ torus links, as constructed by Robert Lipshitz and Sucharit Sarkar, also become stably homotopy equivalent as $m\rightarrow\infty$. We provide an explicit bound on values of $m$ beyond which the stabilization begins. As an application, we give new examples of torus links with non-trivial $Sq^2$ action.
We introduce the notion of rational links in the solid torus. We show that rational links in the solid torus are fully characterized by rational tangles, and hence by the continued fraction of the rational tangle. Furthermore, we generalize this by giving an infinite family of ambient isotopy invariants of colored diagrams in the Kauffman bracket skein module of an oriented surface.
In the present paper we extend the definition of slice-torus invariant to links. We prove a few properties of the newly-defined slice-torus link invariants: the behaviour under crossing change, a slice genus bound, an obstruction to strong sliceness, and a combinatorial bound. Furthermore, we provide an application to the computation of the splitting number. Finally, we use the slice-torus link invariants, and the Whitehead doubling to define new strong concordance invariants for links, which are proven to be independent from the corresponding slice-torus link invariant.
We determine the minimal number of colors for non-trivial $\mathbb{Z}$-colorings on the standard minimal diagrams of $\mathbb{Z}$-colorable torus links. Also included are complete classifications of such $\mathbb{Z}$-colorings and of such $\mathbb{Z}$-colorings by only four colors, which are shown by using rack colorings on link diagrams.
In this paper we give an explicit formula for the twisted Alexander polynomial of any torus link and show that it is a locally constant function on the $SL(2, \mathbb C)$-character variety. We also discuss similar things for the higher dimensional twisted Alexander polynomial and the Reidemeister torsion.
The family of negative torus links $T_{p,q}$ over a fixed number of strands $p$ admits a stable limit in reduced Khovanov homology as $q$ grows to infinity. In this paper, we endow this stable space with a bi-graded commutative algebra structure. We describe these algebras explicitly for $p=2,3,4$. As an application, we compute the homology of two families of links, and produce a lower bound for the width of the homology of any $4$-stranded torus link.
In classical knot theory, Markov's theorem gives a way of describing all braids with isotopic closures as links in $\mathbb{R}^3$. We present a version of Markov's theorem for extended loop braids with closure in $B^3 \times S^1$, as a first step towards a Markov's theorem for extended loop braids and ribbon torus-links in $\mathbb{R}^4$.
A quandle coloring quiver is a quiver structure, introduced by Karina Cho and Sam Nelson, which is defined on the set of quandle colorings of an oriented knot or link by a finite quandle. We study quandle coloring quivers of (p, 2)-torus knots and links with respect to dihedral quandles.
In this article we study the Heegaard Floer link homology of $(n, n)$-torus links. The Alexander multigradings which support non-trivial homology form a string of $n-1$ unit hypercubes in $\mathbb{R}^{n}$, and we compute the ranks and gradings of the homology in nearly all Alexander gradings. We also conjecture a complete description of the link homology and provide some support for this conjecture. This article is taken from the author's 2007 Ph.D. thesis and contains several open questions.
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.
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.
Among all torus links, we characterise those arising as links of simple plane curve singularities by the property that their fibre surfaces admit only a finite number of cutting arcs that preserve fibredness. The same property allows a characterisation of Coxeter-Dynkin trees (i.e., $A_n$, $D_n$, $E_6$, $E_7$ and $E_8$) among all positive tree-like Hopf plumbings.
We consider Milnor invariants for certain covering links as a generalization of covering linkage invariants formulated by R. Hartley and K. Murasugi. A set of Milnor invariants for covering links is a cobordism invariant of a link, and that this invariant can distinguish some links for which the ordinary Milnor invariants coincide. Moreover, for a Brunnian link $L$, the first non-vanishing Milnor invariants of $L$ is modulo-$2$ congruent to a sum of Milnor invariants of covering links. As a consequence, a sum of linking numbers of ' iterated' covering links gives the first non-vanishing Milnor…
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…
The Turaev genus and dealternating number of a link are two invariants that measure how far away a link is from alternating. We determine the Turaev genus of a torus knot with five or fewer strands either exactly or up to an error of at most one. We also determine the dealternating number of a torus knot with five or fewer strand up to an error of at most two. Additional bounds are given on the Turaev genus and dealternating number of torus links with five or fewer strands and on some infinite families of torus links on six strands.
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.
For an arbitrary positive integer $n$ and a pair $(p, q)$ of coprime integers, consider $n$ copies of a torus $(p,q)$ knot placed parallel to each other on the surface of the corresponding auxiliary torus: we call this assembly a torus $n$-link. We compute economical presentations of knot groups for torus links using the groupoid version of the Seifert--van Kampen theorem. Moreover, the result for an individual torus $n$-link is generalized to the case of multiple "nested" torus links, where we inductively include a torus link in the interior (or the exterior) of the auxiliary torus correspond…
We introduce a new method for computing triply graded link homology, which is particularly well-adapted to torus links. Our main application is to the (n,n)-torus links, for which we give an exact answer for all n. In several cases, our computations verify conjectures of Gorsky et al relating homology of torus links with Hilbert schemes.
We describe a strategy for constructing reduced Khovanov homology for links in lens spaces by generalizing a symplectic interpretation of reduced Khovanov homology for links in $S^3$ due to Hedden, Herald, Hogancamp, and Kirk. The strategy relies on a partly conjectural description of the Fukaya category of the traceless $SU(2)$ character variety of the 2-torus with two punctures. From a diagram of a 1-tangle in a solid torus, we construct a corresponding object $(X,δ)$ in the $A_\infty$ category of twisted complexes over this Fukaya category. The homotopy type of $(X,δ)$ is an isotopy invaria…
A whittled complex for the Khovanov homology of torus links
California State University, Fresno（弗雷斯诺加州州立大学）；The University of British Columbia（不列颠哥伦比亚大学）；Texas State University（德克萨斯州立大学）；North Carolina State University（北卡罗来纳州立大学）
A virtual link diagram is called mod $m$ almost classical if it admits an Alexander numbering valued in integers modulo $m$, and a virtual link is called mod $m$ almost classical if it has a mod $m$ almost classical diagram as a representative. In this paper, we introduce a method of constructing a mod $m$ almost classical virtual link diagram from a given virtual link diagram, which we call an $m$-fold cyclic covering diagram. The main result is that $m$-fold cyclic covering diagrams obtained from two equivalent virtual link diagrams are equivalent. Thus we have a well-defined map from the se…
We show that all nontrivial embeddings of planar graphs on the torus contain a nontrivial knot or a nonsplit link. This is equivalent to showing that no minimally knotted planar spatial graphs on the torus exist that contain neither a nontrivial knot nor a nonsplit link all of whose components are unknots.
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.
It is a well-known procedure for constructing a torus knot or link that first we prepare an unknotted torus and meridian disks in the complementary solid tori of it, and second smooth the intersections of the boundary of meridian disks uniformly. Then we obtain a torus knot or link on the unknotted torus and its Seifert surface made of meridian disks. In the present paper, we generalize this procedure by a closed fake surface and show that the resultant two surfaces obtained by smoothing triple points uniformly are essential. We also show that a knot obtained by this procedure satisfies the Ne…
