We introduce stable equivalence classes of oriented links in orientable three-manifolds that are orientation $I$-bundles over closed but not necessarily orientable surfaces. We call these twisted links, and show that they subsume the virtual knots introduced by L. Kauffman, and the projective links introduced by Yu. Drobotukhina. We show that these links have unique minimal genus three-manifolds. We use link diagrams to define an extension of the Jones polynomial for these links, and show that this polynomial fails to distinguish two-colorable links over non-orientable surfaces from non-two-colorable virtual links.
We determine the first homology group with coefficients in $H_1(N;\mathbb{Z})$ for the mapping class group of a non-orientable surface $N$ of genus three with two boundary components.
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 construction of integer linking numbers of closed curves in a three-dimensional manifold usually appeals to the orientation of this manifold. We discuss how to avoid it constructing similar homotopy invariants of links in non-orientable manifolds.
We compute the $p$-primary components of the linking pairings of orientable 3-manifolds admitting a fixed-point free $S^1$-action. Using this, we show that any non-singular linking pairing on a finite abelian group with homogeneous 2-primary summand is realized by such a manifold. However, some pairings on inhomogeneous 2-groups are not realizable by any Seifert fibred 3-manifold.
A virtual link can be understood as a link in a trivial I-bundle over an orientable compact surface with genus. A twisted virtual link is a link in a trivial I-bundle over a not-necessarily orientable compact surface. A twisted virtual birack is an algebraic structure with axioms derived from the twisted virtual Reidemeister moves. We extend a method previously used with racks and biracks to the twisted case to define computable invariants of twisted virtual links using finite twisted virtual biracks with birack rank $N\ge 1$. As an application, we classify twist structures on the virtual Hopf…
We apply the techniques of totally twisted Khovanov homology to the constructions by M. Asaeda, J. Przytycki, and A. Sikora of Khovanov type homologies for links and tangles in I-bundles over (orientable) surfaces. As a result we describe an invariant chain complex built out of resolutions with only non-contractible circles. We use these to understand the δ-graded homology for links with alternating projection to the surface.
We introduce twisted Alexander norms of a compact connected orientable 3-manifold with first Betti number bigger than one generalizing norms of McMullen and Turaev. We show that twisted Alexander norms give lower bounds on the Thurston norm of a 3-manifold. Using these we completely determine the Thurston norm of many 3-manifolds which can not be determined by norms of McMullen and Turaev.
Virtual knot theory is a generalization of knot theory which is based on Gauss chord diagrams and link diagrams on closed oriented surfaces. A twisted knot is a generalization of a virtual knot, which corresponds to a link diagram on a possibly non-orientable surface. In this paper, we discuss an invariant of twisted links which is obtained from the JKSS invariant of virtual links by use of double coverings. We also discuss some properties of double covering diagrams.
A virtual link is a generalization of a classical link that is defined as an equivalence class of certain diagrams, called virtual link diagrams. It is further generalized to a twisted link. Twisted links are in one-to-one correspondence with stable equivalence classes of links in oriented thickenings of (possibly non-orientable) closed surfaces. By definition, equivalent virtual links are also equivalent as twisted links. In this paper, we discuss when two virtual links are equivalent as twisted links, and give a necessary and sufficient condition for this to be the case.
We prove that the Whitehead link complement and the (-2, 3, 8) pretzel link complement are the minimal volume orientable hyperbolic 3-manifolds with two cusps, with volume 3.66... = 4 x Catalan's constant. We use topological arguments to establish the existence of an essential surface which provides a lower bound on volume and strong constraints on the manifolds that realize that lower bound.
We introduce two kinds of structures, called v-structures and t-structures, on biquandles. These structures are used for colorings of diagrams of virtual links and twisted links such that the numbers of colorings are invariants. Given a biquandle or a quandle, we give a method of constructing a biquandle with these structures. Using the numbers of colorings, we show that Bourgoin's twofoil and non-orientable virtual $m$-foils do not represent virtual links.
Knots and links in 3-manifolds are studied by applying intersection invariants to singular concordances. The resulting link invariants generalize the Arf invariant, the mod 2 Sato-Levine invariants, and Milnor's triple linking numbers. Besides fitting into a general theory of Whitney towers, these invariants provide obstructions to the existence of a singular concordance which can be homotoped to an embedding after stabilization by connected sums with $S^2\times S^2$. Results include classifications of stably slice links in orientable 3-manifolds, stable knot concordance in products of an orie…
We introduce the notion of a quandle with a good involution and its homology groups. Carter et al. defined quandle cocycle invariants for oriented links and oriented surface-links. By use of good involutions, quandle cocyle invariants can be defined for links and surface-links which are not necessarily oriented or orientable. The invariants can be used in order to estimate the minimal triple point numbers of non-orientable surface-links.
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.
Macbeath gave a formula for the number of fixed points for each non-identity element of a cyclic group of automorphisms of a compact Riemann surface in terms of the universal covering transformation group of the cyclic group. We observe that this formula generalizes to determine the fixed-point set of each non-identity element of a cyclic group of automorphisms acting on a closed non-orientable surface with one exception; namely, when this element has order 2. In this case the fixed-point set may have simple closed curves (called ovals) as well as fixed points. In this note we extend Macbeath'…
We compute the A-polynomial 2-tuple of twisted Whitehead links. As applications, we determine canonical components of twisted Whitehead links and give a formula for the volume of twisted Whitehead link cone-manifolds.
It follows from earlier work of Silver-Williams and the authors that twisted Alexander polynomials detect the unknot and the Hopf link. We now show that twisted Alexander polynomials also detect the trefoil and the figure-8 knot, that twisted Alexander polynomials detect whether a link is split and that twisted Alexander modules detect trivial links.
A triangulated $d$-manifold $K$, satisfies the inequality $\binom{f_0(K)-d-1}{2}\geq \binom{d+2}{2}β_1(K;\mathbb{Z}_2)$ for $d\geq 3$. The triangulated $d$-manifolds that meet the bound with equality are called {\em tight neighborly}. In this paper, we present tight neighborly triangulations of 4-manifolds on 15 vertices with $\mathbb{Z}_3$ as automorphism group. One such example wasconstructed by Bagchi and Datta in 2011. We show that there are exactly 12 such triangulations up to isomorphism, 10 of which are orientable.
We prove that fundamental groups of non-orientable 3-manifolds have a solvable conjugacy problem, and construct an algorithm. Together with our earlier work on the conjugacy problem in groups on orientable geometrizable 3-manifolds, all $π_1$ of (geometrizable) 3-manifolds have a solvable conjugacy problem. In corollaries, both the twisted conjugacy problem in closed surface groups and the conjugacy problem in closed surface-by-cyclic groups, are solvable.
We decompose the twisted index obstruction $θ(M)$ against positive scalar curvature metrics for oriented manifolds with spin universal cover into a pairing of a twisted $K$-homology with a twisted $K$-theory class and prove that $θ(M)$ does not vanish if $M$ is an orientable enlargeable manifold with spin universal cover.
Checkerboard framings are an extension of checkerboard colorings for virtual links. According to checkerboard framings, in 2017, Dye obtained an independent invariant of virtual links: the cut point number. Checkerboard framings and cut points can be used as a tool to extend other classical invariants to virtual links. We prove that one of the conjectures in Dye's paper is correct. Moreover, we analyze the connection and difference between checkerboard framing obtained from virtual link diagram by adapting cut points and twisted link diagram obtained from virtual link diagram by introducing ba…
We use the knot homology of Khovanov and Lee to construct link concordance invariants generalizing the Rasmussen $s$-invariant of knots. The relevant invariant for a link is a filtration on a vector space of dimension $2^{|L|}$. The basic properties of the $s$-invariant all extend to the case of links; in particular, any orientable cobordism $Σ$ between links induces a map between their corresponding vector spaces which is filtered of degree $χ(Σ)$. A corollary of this construction is that any component preserving orientable cobordism from a $\Kh$-thin link to a link split into $k$ components …
In recent years, twisted Alexander polynomial has been playing an important role in low-dimensional topology. For Montesinos links, we develop an efficient method to compute the twisted Alexander polynomial associated to any linear representation. In particular, formulas for multi-variable Alexander polynomials of these links can be easily derived.
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.
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…
Our earlier twisted-face-pairing construction showed how to modify an arbitrary orientation-reversing face-pairing on a faceted 3-ball in a mechanical way so that the quotient is automatically a closed, orientable 3-manifold. The modifications were, in fact, parametrized by a finite set of positive integers, arbitrarily chosen, one integer for each edge class of the original face-pairing. This allowed us to find very simple face-pairing descriptions of many, though presumably not all, 3-manifolds. Here we show how to modify the construction to allow negative parameters, as well as positive par…
We prove that the 8^4_2 link complement is the minimal volume orientable hyperbolic manifold with 4 cusps. Its volume is twice of the volume V_8 of the ideal regular octahedron, i.e. 7.32... = 2V_8. The proof relies on Agol's argument used to determine the minimal volume hyperbolic 3-manifolds with 2 cusps. We also need to estimate the volume of a hyperbolic 3-manifold with totally geodesic boundary which contains an essential surface with non-separating boundary.
We extend our previous work from arXiv:1903.06863 on biquandle module invariants of oriented surface-links to the case of unoriented surface-links using bikei modules. The resulting infinite family of enhanced invariants proves be effective at distinguishing unoriented and especially non-orientable surface-links; in particular, we show that these invariants are more effective than the bikei homset cardinality invariant alone at distinguishing non-orientable surface-links. Moreover, as another application we note that our previous biquandle modules which do not satisfy the bikei module axioms a…
We use Reidemeister torsion to study a twisted Alexander polynomial, as defined by Turaev, for links in the projective space. Using sign-refined torsion we derive a skein relation for a normalized form of this polynomial.
The twisted torsion of a 3-manifold is well-known to be zero whenever the corresponding twisted Alexander module is non-torsion. Under mild extra assumptions we introduce a new twisted torsion invariant which is always non-zero. We show how this torsion invariant relates to the twisted intersection form of a bounding 4-manifold, generalizing a theorem of Milnor. Using this result, we give new obstructions to 3-manifolds being homology cobordant and to links being concordant. These obstructions are sufficiently strong to detect that the Bing double of the figure eight knot is not slice.
We determine lens surgeries (i.e.\ Dehn surgery yielding a lens space) along the $n$-twisted Whitehead link. To do so, we first give necessary conditions to yield a lens space from the Alexander polynomial of the link as: (1) $n=1$ (i.e. the Whitehead link), and (2) one of surgery coefficients is 1, 2 or 3. Our interests are not only lens surgery itself but also how to apply the Alexander polynomial for this kind of problems.
There are many studies about twisted Alexander invariants for knots and links, but calculations of twisted Alexander invariants for spatial graphs, handlebody-knots, and surface-links have not been demonstrated well. In this paper, we give some remarks to calculate the twisted Alexander ideals for spatial graphs, handlebody-knots and surface-links, and observe their behaviors. For spatial graphs, we calculate the invariants of Suzuki's theta-curves and show that the invariants are nontrivial for Suzuki's theta-curves whose Alexander ideals are trivial. For handlebody-knots, we give a remark on…
Twisted links are obtained from a base link by starting with a $n$-braid representation, choosing several ($m$) adjacent strands, and applying one or more twists to the set. Various restrictions may be applied, e.g. the twists may be required to be positive or full twists, or the base braid may be required to have a certain form. The Jones polynomial of full $m$-twisted links have some interesting properties. It is known that when sufficiently many full $m$-twists are added that the coefficients break up into disjoint blocks which are independent of the number of full twists. These blocks are …
Using Kirby Calculus, we explicitly pass from Berge's R-R descriptions of ten families of knots with lens space surgeries to surgery descriptions on the minimally twisted five chain link (MT5C). Since the MT5C admits a strong involution, we also give the corresponding tangle descriptions.
We prove that in the complement of a highly twisted link, all closed, essential, meridionally incompressible surfaces must have high genus. The genus bound is proportional to the number of crossings per twist region. A similar result holds for surfaces with meridional boundary: such a surface either has large negative Euler characteristic, or is an n-punctured sphere visible in the diagram.
A. Ishii and K. Oshiro introduced the notion of an $f$-twisted Alexander matrix. This notion is a quandle version of the twisted Alexander matrix which was introduced by M. Wada. They showed that the twisted Alexander matrix of a pair of a link group and a group representation can be recovered from the $f$-twisted Alexander matrix of a pair of a link quandle and a quandle homomorphism. In this paper, we study a relationship between the $f$-twisted Alexander matrix and the quandle cocycle invariant. As an application, we show that an $f$-twisted Alexander invariant of knot quandles is a really …
In this paper we apply the twisted Alexander polynomial to study the fibering and genus detecting problems for oriented links. In particular we generalize a conjecture of Dunfield, Friedl and Jackson on the torsion polynomial of hyperbolic knots to hyperbolic links, and confirm it for an infinite family of hyperbolic 2-bridge links. Moreover we consider a similar problem for parabolic representations of 2-bridge link groups.
The first examples of totally geodesic Seifert surfaces are constructed for hyperbolic knots and links, including both free and totally knotted surfaces. Then it is proved that two bridge knot complements cannot contain totally geodesic orientable surfaces.
We prove that every finitely presentable group G arises as the fundamental group of an orientable 3-complex obtained from a hyperbolic link complement, by coning each boundary torus of the link exterior to a distinct point. We define the closed-link-genus, clg(G), of any finitely presentable group G, which completely characterizes fundamental groups of closed orientable 3-manifolds: clg(G)=0 if and only if G is the fundamental group of a closed orientable 3-manifold. Moreover clg(G) gives an upper bound for the concept `genus(G)' of genus defined earlier by Aitchison and Reeves, and in turn is…
In the present paper we give a simple proof of the fact that the set of virtual links with orientable atoms is closed. More precisely, the theorem states that if two virtual diagrams $K$ and $K'$ have orientable atoms and they are equivalent by Reidemeister moves, then there is a sequence of diagrams $K = K_1 \to...\to K_n=K'$ all having orientable atoms where $K_i$ is obtained from $K_{i-1}$ by a Reidemeister move. The initial proof heavily relies on the topology of virtual links and was published in \cite{IM}. Our proof is based on the notion of parity which was introduced by the second name…
Checkerboard surfaces in alternating link complements are used frequently to determine information about the link. However, when many crossings are added to a single twist region of a link diagram, the geometry of the link complement stabilizes (approaches a geometric limit), but a corresponding checkerboard surface increases in complexity with crossing number. In this paper, we generalize checkerboard surfaces to certain immersed surfaces, called twisted checkerboard surfaces, whose geometry better reflects that of the alternating link in many cases. We describe the surfaces, show that they a…
Let $Δ_{L,ρ_n}(t)$ be the twisted Alexander polynomial with respect to the representation given by the composition of the lift of the holonomy representation of a certain hyperbolic link $L$ and the $n$-dimensional irreducible complex representation of $\text{SL}(2,\mathbb C)$. We consider a sequence of $Δ_{L,ρ_n}(t)$ and extract the volume of the complement of $L$ from the asymptotic behaviour of the sequence obtained by evaluating $t=1$ or $t=-1$.
Let $M_1$ and $M_2$ be closed connected orientable $3$-manifolds. We classify the sets of smooth and piecewise linear isotopy classes of embeddings $M_1\sqcup M_2\rightarrow S^6$.
