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Virtual concordance and the generalized Alexander polynomialMar 20 2019We 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 ... More

Non left-orderable surgeries on L-space twisted torus knotsMar 17 2019We show that if $K$ is an L-space twisted torus knot $T^{l,m}_{p,pk \pm 1}$ with $p \ge 2$, $k \ge 1$, $m \ge 1$ and $1 \le l \le p-1$, then the fundamental group of the $3$-manifold obtained by $\frac{r}{s}$-surgery along $K$ is not left-orderable whenever ... More

Biquandle Module Invariants of Oriented Surface-LinksMar 16 2019We define invariants of oriented surface-links by enhancing the biquandle counting invariant using \textit{biquandle modules}, algebraic structures defined in terms of biquandle actions on commutative rings analogous to Alexander biquandles. We show that ... More

Torsion in thin regions of Khovanov homologyMar 13 2019In the integral Khovanov homology of links, the presence of odd torsion is rare. Homologically thin links, that is links whose Khovanov homology is supported on two adjacent diagonals, are known to only contain $\mathbb{Z}_2$ torsion. In this paper, we ... More

Multi-tribracketsMar 05 2019We introduce multi-tribrackets, algebraic structures for region coloring of diagrams of knots and links with different operations at different kinds of crossings. In particular we consider the case of component multi-tribrackets which have different tribracket ... More

Khovanov homology and ribbon concordanceMar 04 2019We show that a ribbon concordance between two links induces an injective map on Khovanov homology.

Artin's braids, Braids for three space, and groups $Γ_{n}^{4}$ and $G_{n}^{k}$Feb 28 2019Mar 08 2019We construct a group $\Gamma_{n}^{4}$ corresponding to the motion of points in $\mathbb{R}^{3}$ from the point of view of Delaunay triangulations. We study homomorphisms from pure braids on $n$ strands to the product of copies of $\Gamma_{n}^{4}$. We ... More

Artin's braids, Braids for three space, and groups $Γ_{n}^{4}$ and $G_{n}^{k}$Feb 28 2019Mar 01 2019We construct a group $\Gamma_{n}^{4}$ corresponding to the motion of points in $\mathbb{R}^{3}$ from the point of view of Delaunay triangulations. We study homomorphisms from pure braids on $n$ strands to the product of copies of $\Gamma_{n}^{4}$. We ... More

Artin's braids, Braids for three space, and groups $Γ_{n}^{4}$ and $G_{n}^{k}$Feb 28 2019We construct a group $\Gamma_{n}^{4}$ corresponding to the motion of points in $\mathbb{R}^{3}$ from the point of view of Delaunay triangulations. We study homomorphisms from pure braids on $n$ strands to the product of copies of $\Gamma_{n}^{4}$. We ... More

Milnor invariants, $2n$-moves and $V^{n}$-moves for welded string linksFeb 28 2019In a previous paper, the authors proved that Milnor link-homotopy invariants modulo $n$ classify classical string links up to $2n$-move and link-homotopy. As analogues to the welded case, in terms of Milnor invariants, we give here two classifications ... More

On the 2-head of the colored Jones polynomial for pretzel knotsFeb 19 2019In this paper, we prove a formula for the 2-head of the colored Jones polynomial for an infinite family of pretzel knots. Following Hall, the proof utilizes skein-theoretic techniques and a careful examination of higher order stability properties for ... More

Classification of string links up to $2n$-moves and link-homotopyFeb 16 2019Two string links are equivalent up to $2n$-moves and link-homotopy if and only if their all Milnor link-homotopy invariants are congruent modulo $n$. Moreover, the set of the equivalence classes forms a finite group generated by elements of order $n$. ... More

Knot Floer homology obstructs ribbon concordanceFeb 11 2019We prove that the map on knot Floer homology induced by a ribbon concordance is injective. As a consequence, we prove that the Seifert genus is monotonic under ribbon concordance. We also generalize a theorem of Gabai about the super-additivity of the ... More

Polynomial Invariants, Knot Homologies, and Higher Twist Numbers of Weaving Knots $W(3,n)$Feb 05 2019In this paper we first compute the signature for a family of knots $W(k,n)$, the weaving knots of type $(k,n)$. Specializing to knots $W(3,n)$ we develop recursive formulas for elements in the Hecke algebras arising from representations of $W(3,n)$ as ... More

State graphs and fibered state surfacesFeb 05 2019Associated to every state surface for a knot or link is a state graph, which embeds as a spine of the state surface. A state graph can be decomposed along cut-vertices into graphs with induced planar embeddings. Associated with each such planar graph ... More

The boundary of the Milnor fibre of certain non-isolated singularitiesFeb 04 2019Let $ \Phi: ({\mathbb C}^2, 0) \to ( {\mathbb C}^3, 0) $ be a finitely determined complex analytic germ and let $(\{f=0\},0)$ be the reduced equation of its image, a non-isolated hypersurface singularity. We provide the plumbing graph of the boundary ... More

Symmetric union diagrams and Alexander idealsJan 29 2019Eisermann and Lamm introduced a notion of symmetric equivalence among symmetric union diagrams and studied it using a refined form of the Jones polynomial. We introduced invariants of symmetric equivalence via refined versions of topological spin models ... More

Characterization of affine links in the projective spaceJan 23 2019A projective link is a smooth closed 1-submanifold of the real projective space of dimension three. A projective link is said to be affine if it is isotopic to a link, which does not intersect some projective plane. The main result: a projective link ... More

Band Number and the Double Slice GenusJan 22 2019We study the double slice genus of a knot, a natural generalization of slice genus. We define a notion called band number, a natural generalization of band unknotting number, and prove it is an upper bound on double slice genus. Our bound is based on ... More

A parity for 2-colourable linksJan 22 2019We introduce the 2-colour parity. It is a theory of parity for a large class of virtual links, defined using the interaction between orientations of the link components and a certain type of colouring. The 2-colour parity is an extension of the Gaussian ... More

On the topology of elliptic singularitiesJan 18 2019For any elliptic normal surface singularity with rational homology sphere link we consider a new elliptic sequence, which differs from the one introduced by Laufer and S. S.-T. Yau. However, we show that their length coincide. Using the properties of ... More

Twist Number and the Alternating Volume of KnotsJan 08 2019It was previously shown by the second author that every knot in $S^3$ is ambient isotopic to one component of a two-component, alternating, hyperbolic link. In this paper, we define the alternating volume of a knot $K$ to be the minimum volume of any ... More

A surgery formula for knot Floer homologyJan 08 2019Let $K$ be a rationally null-homologous knot in a $3$-manifold $Y$, equipped with a nonzero framing $\lambda$, and let $Y_\lambda(K)$ denote the result of $\lambda$-framed surgery on $Y$. Ozsv\'ath and Szab\'o gave a formula for the Heegaard Floer homology ... More

Two-solvable and two-bipolar knots with large four-generaJan 07 2019For every integer g, we construct a 2-solvable and 2-bipolar knot whose topological 4-genus is greater than g. Note that 2-solvable knots are in particular algebraically slice and have vanishing Casson-Gordon obstructions. Similarly all known smooth 4-genus ... More

Cohomology jump loci of 3-manifoldsJan 05 2019The cohomology jump loci of a space $X$ are of two basic types: the characteristic varieties, defined in terms of homology with coefficients in rank one local systems, and the resonance varieties, constructed from information encoded in either the cohomology ... More

Alternating KnotsJan 03 2019This is a short expository article on alternating knots and is to appear in the Concise Encyclopedia of Knot Theory.

Relationship between quandle shadow cocycle invariants and Vassiliev invariants of linksDec 30 2018In this study, we deduce Vassiliev invariants from quandle shadow cocycle invariants using the Alexander quandle of links. First, we relate the quandle (shadow) cocycle invariants and Vassiliev invariants of links. Second, we obtain the relation between ... More

Extremal Khovanov homology of Turaev genus one linksDec 29 2018The Turaev genus of a link can be thought of as a way of measuring how non-alternating a link is. A link is Turaev genus zero if and only if it is alternating, and in this viewpoint, links with large Turaev genus are very non-alternating. In this paper, ... More

A Reidemeister type theorem for petal diagrams of knotsDec 21 2018We study petal diagrams of knots, which provide a method of describing knots in terms of permutations in a symmetric group $S_{2n+1}$. We define two classes of moves on such permutations, called trivial petal additions and crossing exchanges, which do ... More

Invariants of Spatial GraphsDec 20 2018This is a short review article on invariants of spatial graphs, written for "A Concise Encyclopedia of Knot Theory" (ed. Adams et. al.). The emphasis is on combinatorial and polynomial invariants of spatial graphs, including the Alexander polynomial, ... More

Khovanov homology for links in $\#^r(S^2\times S^1)$Dec 17 2018We revisit Rozansky's construction of Khovanov homology for links $L$ in $S^2\times S^1$, extending it to define Khovanov homology $Kh(L)$ for links in $M^r=\#^r(S^2\times S^1)$ for any $r$. The graded Euler characteristic of $Kh(L)$ recovers the evaluation ... More

Traversing three-manifold triangulations and spinesDec 06 2018Dec 20 2018A celebrated result concerning triangulations of a given closed 3-manifold is that any two triangulations with the same number of vertices are connected by a sequence of so-called 2-3 and 3-2 moves. A similar result is known for ideal triangulations of ... More

Diagrammatic representations of knots and links as closed braidsNov 28 2018This is an expository article on diagrammatic representations of knots and links in various settings via braids.

The Strong Slope Conjecture for twisted generalized Whitehead doublesNov 28 2018The Slope Conjecture proposed by Garoufalidis asserts that the degree of the colored Jones polynomial determines a boundary slope, and its refinement, the Strong Slope Conjecture proposed by Kalfagianni and Tran asserts that the linear term in the degree ... More

A survey on knotoids, braidoids and their applicationsNov 28 2018This paper is a survey on the theory of knotoids and braidoids. Knotoids are open ended knot diagrams in surfaces and braidoids are geometric objects analogous to classical braids, forming a counterpart theory to the theory of knotoids in the plane. We ... More

A survey on knotoids, braidoids and their applicationsNov 28 2018Mar 04 2019This paper is a survey on the theory of knotoids and braidoids. Knotoids are open ended knot diagrams in surfaces and braidoids are geometric objects analogous to classical braids, forming a counterpart theory to the theory of knotoids in the plane. We ... More

A note on coverings of virtual knotsNov 27 2018For a virtual knot $K$ and an integer $r\geq 0$, the $r$-covering $K^{(r)}$ is defined by using the indices of chords on a Gauss diagram of $K$. In this paper, we prove that for any finite set of virtual knots $J_0,J_2,J_3,\dots,J_m$, there is a virtual ... More

Double branched covers of knotoidsNov 22 2018By using double branched covers, we prove that there is a 1-1 correspondence between the set of knotoids in the 2-sphere, up to orientation reversion and rotation, and knots with a strong inversion, up to conjugacy. This correspondence allows us to study ... More

Detecting and visualizing 3-dimensional surgeryNov 20 2018Topological surgery in dimension $3$ is intrinsically connected with the classification of $3$-manifolds and with patterns of natural phenomena. In this expository paper, we present two different approaches for understanding and visualizing the process ... More

Intercusp Geodesics and Cusp Shapes of Fully Augmented LinksNov 18 2018We study the geometry of fully augmented link complements in $S^3$ by looking at their link diagrams. We extend the method introduced by Thistlethwaite and Tsvietkova to fully augmented links and define a system of algebraic equations in terms of parameters ... More

Structural aspects of twin and pure twin groupsNov 09 2018Jan 23 2019The twin group $T_n$ is a Coxeter group generated by $n-1$ involutions and the pure twin group $PT_n$ is the kernel of the natural surjection of $T_n$ onto the symmetric group on $n$ letters. In this paper, we investigate structural aspects of twin and ... More

Holonomy perturbations of the Chern-Simons functional for lens spacesNov 05 2018We describe a scheme for constructing generating sets for Kronheimer and Mrowka's singular instanton knot homology for the case of knots in lens spaces. The scheme involves Heegaard-splitting a lens space containing a knot into two solid tori. One solid ... More

Rational cobordisms and integral homologyNov 04 2018We consider the question of when a rational homology 3-sphere is rational homology cobordant to a connected sum of lens spaces. We prove that every rational homology cobordism class in the subgroup generated by lens spaces is represented by a unique connected ... More

Arithmeticity and Hidden Symmetries of Fully Augmented Pretzel Link ComplementsNov 01 2018This paper examines number theoretic and topological properties of fully augmented pretzel link complements. In particular, we determine exactly when these link complements are arithmetic and exactly which are commensurable with one another. We show these ... More

An unoriented skein relation via bordered-sutured Floer homologyOct 31 2018We show that the bordered-sutured Floer invariant of the complement of a tangle in an arbitrary 3-manifold $Y$, with minimal conditions on the bordered-sutured structure, satisfies an unoriented skein exact triangle. This generalizes a theorem by Manolescu ... More

On the potential functions for a link diagramOct 22 2018For an oriented diagram of a link $L$ in the 3-sphere, Cho and Murakami defined the potential function whose critical point, slightly different from the usual sense, corresponds to a boundary parabolic $\mathrm{PSL}(2,\mathbb{C})$-representation of $\pi_1(S^3 ... More

Recent advances on the non-coherent band surgery model for site-specific recombinationOct 20 2018Site-specific recombination is an enzymatic process where two sites of precise sequence and orientation along a circle come together, are cleaved, and the ends are recombined. Site-specific recombination on a knotted substrate produces another knot or ... More

Character varieties of even classical pretzel knotsOct 18 2018We determine the ${\rm SL}(2,\mathbb{C})$-character variety of each even classical pretzel knot $P(2k_1+1,2k_2+1,2k_3)$.

Character varieties of even classical pretzel knotsOct 18 2018Mar 11 2019For each even classical pretzel knot $P(2k_1+1,2k_2+1,2k_3)$, we determine the character variety of irreducible ${\rm SL}(2,\mathbb{C})$-representations, and clarify the steps of computing its A-polynomial.

On the construction of knots and links from Thompson's groupsOct 14 2018We review recent developments in the theory of Thompson group representations related to knot theory.

Quotients of Definite Periodic Knots are DefiniteOct 02 2018A knot $K$ is definite if $|\sigma(K)| = 2g(K)$. We prove that the quotient of a definite periodic knot is definite by considering equivariant minimal genus Seifert surfaces.

Heegaard Floer invariants of contact structures on links of surface singularitiesSep 28 2018Let a contact 3-manifold $(Y, \xi_0)$ be the link of a normal surface singularity equipped with its canonical contact structure $\xi_0$. We prove a special property of such contact 3-manifolds of "algebraic" origin: the Heegaard Floer invariant $c^+(\xi_0)\in ... More

Local biquandles and Niebrzydowski's tribracket theorySep 25 2018We introduce a new algebraic structure called \textit{local biquandles} and show how colorings of oriented classical link diagrams and of broken surface diagrams are related to tribracket colorings. We define a (co)homology theory for local biquandles ... More

Local biquandles and Niebrzydowski's tribracket theorySep 25 2018Feb 17 2019We introduce a new algebraic structure called \textit{local biquandles} and show how colorings of oriented classical link diagrams and of broken surface diagrams are related to tribracket colorings. We define a (co)homology theory for local biquandles ... More

An alternative basis for the Kauffman bracket skein module of the Solid Torus via braidsSep 22 2018In this paper we give an alternative basis, $\mathcal{B}_{\rm ST}$, for the Kauffman bracket skein module of the solid torus, ${\rm KBSM}\left({\rm ST}\right)$. The basis $\mathcal{B}_{\rm ST}$ is obtained with the use of the Tempereley--Lieb algebra ... More

Upsilon invariants from cyclic branched coversSep 21 2018We extend the construction of upsilon-type invariants to null-homologous knots in rational homology three-spheres. By considering $m$-fold cyclic branched covers with $m$ a prime power, this extension provides new knot concordance invariants $\Upsilon_m^C ... More

Multi-Skein Invariants for Welded and Extended Welded Knots and LinksSep 16 2018The theory of welded and extended welded knots is a generalization of classical knot theory. Welded (resp. extended welded) knot diagrams include virtual crossings (resp. virtual crossings and wen marks) and are equivalent under an extended set of Reidemeister-type ... More

${\rm SL}(3,\mathbb{C})$-representations of twist knot groupsSep 13 2018For each twist knot, we give a parametrization for an open subset of the ${\rm SL}(3,\mathbb{C})$-character variety, showing it to be an affine algebraic curve.

Colorings and doubled colorings of virtual doodlesSep 12 2018A virtual doodle is an equivalence class of virtual diagrams under an equivalence relation generated by flat version of classical Reidemesiter moves and virtual Reidemsiter moves such that Reidemeister moves of type 3 are forbidden. In this paper we discuss ... More

Satellites of Infinite Rank in the Smooth Concordance GroupSep 11 2018We conjecture that satellite operations are either constant or have infinite rank in the concordance group. We reduce this to the difficult case of winding number zero satellites, and use $SO(3)$ gauge theory to provide a general criterion sufficient ... More

The Abel map for surface singularities II. Generic analytic structureSep 11 2018We study the analytic and topological invariants associated with complex normal singularities. Our goal is to provide topological formulae for several discrete analytic invariants whenever the analytic structure is generic (with respect to a fixed topological ... More

The Abel map for surface singularities I. Generalities and examplesSep 11 2018Let $(X,o)$ be a complex normal surface singularity. We fix one of its good resolutions $\widetilde{X}\to X$, an effective cycle $Z$ supported on the reduced exceptional curve, and any possible (first Chern) class $l'\in H^2(\widetilde{X},\mathbb{Z})$. ... More

The categorification of the Kauffman bracket skein module of $\mathbb{R}P^3$Sep 10 2018Sep 14 2018Khovanov homology, an invariant of links in $\mathbb{R}^3$, is a graded homology theory that categorifies the Jones polynomial in the sense that the graded Euler characteristic of the homology is the Jones polynomial. Asaeda, Przytycki and Sikora generalized ... More

On the tree-width of knot diagramsSep 06 2018We show that a small tree-decomposition of a knot diagram induces a small sphere-decomposition of the corresponding knot. This, in turn, implies that the knot admits a small essential planar meridional surface or a small bridge sphere. We use this to ... More

On a Nonorientable Analogue of the Milnor ConjectureSep 06 2018Jan 26 2019The nonorientable 4-genus $\gamma_4(K)$ of a knot $K$ is the smallest first Betti number of any nonorientable surface properly embedded in the 4-ball, and bounding the knot $K$. We study a conjecture proposed by Batson about the value of $\gamma_4$ for ... More

Left-orderability for surgeries on twisted torus knotsSep 04 2018We show that the fundamental group of the $3$-manifold obtained by $\frac{p}{q}$-surgery along the $(n-2)$-twisted $(3,3m+2)$-torus knot, with $n,m \ge 1$, is not left-orderable if $\frac{p}{q} \ge 2n + 6m-3$ and is left-orderable if $\frac{p}{q}$ is ... More

The strong slope conjecture for graph knotsSep 04 2018We prove the Strong Slope Conjecture for graph knots. Along the way we propose some variants of the Strong Slope Conjecture and discuss their inheritance under cablings and connected sums.

Rational homology 3-spheres and simply connected definite boundingAug 28 2018For each rational homology 3-sphere $Y$ which bounds simply connected definite 4-manifolds of both signs, we construct an infinite family of irreducible rational homology 3-spheres which are homology cobordant to $Y$ but cannot bound any simply connected ... More

Quantum computing with Bianchi groupsAug 21 2018Dec 13 2018It has been shown that non-stabilizer eigenstates of permutation gates are appropriate for allowing $d$-dimensional universal quantum computing (uqc) based on minimal informationally complete POVMs. The relevant quantum gates may be built from subgroups ... More

Ineffectiveness of homotopical invariants on Nakanishi's 4-move conjectureAug 16 2018Aug 20 2018A $4$-move is a local operation for links consisting in replacing two parallel arcs by four half twists. At the present time, it is not known if this induces an unkotting operation for knots. Studying the Dabkowski-Sahi invariant, we prove that any invariant ... More

Tribracket ModulesAug 13 2018Niebrzydowski tribrackets are ternary operations on sets satisfying conditions obtained from the oriented Reidemeister moves such that the set of tribracket colorings of an oriented knot or link diagram is an invariant of oriented knots and links. We ... More

Higher Order Degrees of Affine Plane Curve ComplementsAug 09 2018We study finiteness (and vanishing) properties of the higher order degrees associated to complements of complex affine plane curves with mild singularities at infinity. Our results impose new obstructions on the class of groups that can be realized as ... More

A spinning construction for virtual 1-knots and 2-knots, and the fiberwise and welded equivalence of virtual 1-knotsAug 09 2018We succeed to generalize spun knots of classical 1-knots to the virtual 1-knot case by using the `spinning construction'. That, is, we prove the following: Let $Q$ be a spun knot of a virtual 1-knot $K$ by our method. The embedding type $Q$ in $S^4$ depends ... More

On definite lattices bounded by integer surgeries along knots with slice genus at most 2Jul 31 2018Feb 13 2019We classify the positive definite intersection forms that arise from smooth 4-manifolds with torsion-free homology bounded by positive integer surgeries on the right-handed trefoil. A similar, slightly less complete classification is given for the (2,5)-torus ... More

Quandle Coloring QuiversJul 27 2018Oct 06 2018We consider a quiver structure on the set of quandle colorings of an oriented knot or link diagram. This structure contains a wealth of knot and link invariants and provides a categorification of the quandle counting invariant in the most literal sense, ... More

On the knot quandle of a fibered knot, finiteness and equivalence of knot quandlesJul 24 2018We show that the structure of a fibered knot, as a fiber bundle, is reflected in its knot quandle. As an application, we discuss finiteness and equivalence of knot quandles of concrete fibered 2-knots.

Skein relations for spin networks, modifiedJul 19 2018An alternative framework underlying connection between tensor ${\rm sl}_2$-calculus and spin networks is suggested. New sign convention for the inner product in the dual spinor space leads to a simpler and direct set of initial rules for the diagrammatic ... More

Geometric realization of the almost-extreme Khovanov homology of semiadequate linksJul 10 2018We introduce the notion of partial presimplicial set and construct its geometric realization. We show that any semiadequate diagram yields a partial presimplicial set leading to a geometric realization of the almost-extreme Khovanov homology of the diagram. ... More

Realisation of groups as automorphism groups in categoriesJul 02 2018Oct 15 2018It is shown that in various categories, including many consisting of maps or hypermaps, oriented or unoriented, of a given hyperbolic type, every countable group $A$ is isomorphic to the automorphism group of uncountably many non-isomorphic objects, infinitely ... More

The leading coefficient of the $L^2$-Alexander torsionJun 28 2018We give upper and lower bounds on the leading coefficients of the $L^2$-Alexander torsions of a $3$-manifold $M$ in terms of hyperbolic volumes and of relative $L^2$-torsions of sutured manifolds obtained by cutting $M$ along certain surfaces. We prove ... More

Slice-torus concordance invariants and Whitehead doubles of linksJun 27 2018In 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, ... More

Lattice stick number of spatial graphsJun 25 2018The lattice stick number of knots is defined to be the minimal number of straight sticks in the cubic lattice required to construct a lattice stick presentation of the knot. We similarly define the lattice stick number $s_{L}(G)$ of spatial graphs $G$ ... More

Bisected vertex leveling of plane graphs: braid index, arc index and delta diagramsJun 25 2018In this paper, we introduce a bisected vertex leveling of a plane graph. Using this planar embedding, we present elementary proofs of the well-known upper bounds in terms of the minimal crossing number on braid index $b(L)$ and arc index $\alpha(L)$ for ... More

Stick number of spatial graphsJun 25 2018For a nontrivial knot $K$, Negami found an upper bound on the stick number $s(K)$ in terms of its crossing number $c(K)$ which is $s(K) \leq 2 c(K)$. Later, Huh and Oh utilized the arc index $\alpha(K)$ to present a more precise upper bound $s(K) \leq ... More

Symplectic dynamics and the 3-sphereJun 22 2018Given a knot in a closed connected orientable 3-manifold we prove that if the exterior of the knot admits an aperiodic contact form that is Euclidean near the boundary, then the 3-manifold is diffeomorphic to the 3-sphere and the knot is the unknot.

Symplectic dynamics and the 3-sphereJun 22 2018Mar 12 2019Given a knot in a closed connected orientable 3-manifold we prove that if the exterior of the knot admits an aperiodic contact form that is Euclidean near the boundary, then the 3-manifold is diffeomorphic to the 3-sphere and the knot is the unknot.

Invariants for trivalent tangles and handlebody-tanglesJun 17 2018Dec 17 2018An enhanced trivalent tangle is a trivalent tangle with some of its edges labeled. We use enhanced trivalent tangles and classical knot theory to provide a recipe for constructing invariants for trivalent tangles, and in particular, for knotted trivalent ... More

Linear independence of cables in the knot concordance groupJun 16 2018We produce infinite families of knots $\{K^i\}_{i\geq 1}$ for which the set of cables $\{K^i_{p,1}\}_{i,p\geq 1}$ is linearly independent in the knot concordance group. We arrange that these examples lie arbitrarily deep in the solvable and bipolar filtrations ... More

On Gauss codes of virtual doodlesJun 15 2018We discuss Gauss codes of virtual diagrams and virtual doodles. The notion of a left canonical Gauss code is introduced and it is shown that oriented virtual doodles are uniquely presented by left canonical Gauss codes.

Torsion in Khovanov homology of homologically thin knotsJun 13 2018We prove that every $\mathbb{Z}_2$H-thin link has no $2^k$-torsion for $k>1$ in its Khovanov homology. Together with previous results by Eun Soo Lee and the author, this implies that integer Khovanov homology of non-split alternating links is completely ... More

A note on band surgery and the signature of a knotJun 06 2018Jul 12 2018Band surgery is an operation which transforms a knot or link in the three-sphere into another knot or link. We prove that if two quasi-alternating knots $K$ and $K'$ of the same square-free determinant are related by a band surgery, then the absolute ... More

Transverse link invariants from the deformations of Khovanov $\mathfrak{sl}_{3}$-homologyJun 03 2018Jun 14 2018In this paper we will make use of the Mackaay-Vaz approach to the universal $\mathfrak{sl}_3$-homology to define a family of cycles (called $\beta_3$-invariants) which are transverse braid invariants. This family includes Wu's $\psi_{3}$-invariant. Furthermore, ... More

Characterization of 3-punctured spheres in non-hyperbolic link exteriorsMay 31 2018In this paper, we characterize non-hyperbolic 3-component links in the 3-sphere whose exteriors contain essential 3-punctured spheres with non-integral boundary slopes. We also show the existence of embeddings of some multibranched surfaces in the 3-sphere ... More

A Survey of Quantum EnhancementsMay 30 2018In this short survey article we collect the current state of the art in the nascent field of \textit{quantum enhancements}, a type of knot invariant defined by collecting values of quantum invariants of knots with colorings by various algebraic objects ... More

On the Hikami-Inoue conjectureMay 30 2018Given a braid presentation $\sigma$ of a hyperbolic knot $K$, Hikami and Inoue consider a system of polynomial equations arising from a sequence of cluster mutations determined by $\sigma$. They show that any solution gives rise to shape variables and ... More

On the Hikami-Inoue conjectureMay 30 2018Feb 21 2019Given a braid presentation $D$ of a hyperbolic knot, Hikami and Inoue consider a system of polynomial equations arising from a sequence of cluster mutations determined by $D$. They show that any solution gives rise to shape parameters and thus determines ... More

Automorphism groups of maps, hypermaps and dessinsMay 24 2018A detailed proof is given of a theorem describing the centraliser of a transitive permutation group, with applications to automorphism groups of objects in various categories of maps, hypermaps, dessins, polytopes and covering spaces, where the automorphism ... More

Homfly polynomials for periodic knots via state modelMay 24 2018We give criteria for oriented links to be periodic of prime order using the quantum $\mathrm{SL}(N)$-invariant. The criteria are based upon an observation on the linking number between a periodic knot and its axis of the rotation.

Knot-theoretic ternary groupsMay 20 2018Aug 05 2018We describe various properties and give several characterizations of ternary groups satisfying two axioms derived from the third Reidemeister move in knot theory. Using special attributes of such ternary groups, such as semi-commutativity, we construct ... More

A note on the writhe polynomial and the virtual crossing numberMay 20 2018May 24 2018In this note we give a new lower bound on the virtual crossing number via the writhe polynomial, which refines a result of B. Mellor. The proof is based on a new interpretation of the writhe polynomial. The characterization of the writhe polynomial is ... More