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We show that the equivariant and non-equivariant non-orientable 4-genus of [Formula: see text]-periodic knots may differ, for any choice of [Formula: see text]. Similar results have previously been obtained for the smooth 4-genus and non-orientable 3-genus of a periodic knot. These stand in contrast to Edmonds’ acclaimed result by which the equivariant and non-equivariant Seifert genus of a periodic knot agree.

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 [Formula: see text]. 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 Khovanov–Lipshitz–Sarkar stable homotopy type and our Steenrod square of links in thickened surfaces with genus [Formula: see text] are stronger than the homotopical Khovanov homology of links in thickened surfaces with genus [Formula: see text]. It is the first meaningful Khovanov–Lipshitz–Sarkar stable homotopy type of links in 3-manifolds other than the 3-sphere. We point out that our theory has a different feature in the torus case.

In this paper, we introduce an algebraic structure called a diquandle which is a set equipped with two quandle operations satisfying the right distributive laws. We discuss various examples and some properties of diquandles and also show that a diquandle enables us to distinguish oriented dichromatic links by telling that their coloring sets are different when their arcs are colored by elements of the diquandle.

Knotoids are open ended knot diagrams regarded up to Reidemeister moves and isotopies. The notion is introduced by Turaev in 2012. Two most important numeric characteristics of a knotoid are the crossing number and the height. The latter is the least number of intersections between a diagram and an arc connecting its endpoints, where the minimum is taken over all representative diagrams and all such arcs which are disjoint from crossings. In the paper, we answer the question: are there any relations between the crossing number and the height of a knotoid. We prove that the crossing number of a knotoid is greater than or equal to twice the height of the knotoid. Combining the inequality with known lower bounds of the height we obtain a lower bounds of the crossing number of a knotoid via the extended bracket polynomial, the affine index polynomial and the arrow polynomial of the knotoid. As an application of our result we prove an upper bound for the length of a bridge in a minimal diagram of a classical knot: the number of crossings in a minimal diagram of a knot is greater than or equal to three times the length of a longest bridge in the diagram.

In this paper, we give a combinatorial description of the concordance invariant $\varepsilon$ defined by Hom in \cite{hom2011knot}, prove some properties of this invariant using grid homology techniques. We also compute $\varepsilon$ of $(p,q)$ torus knots and prove that $\varepsilon(\mathbb{G}_+)=1$ if $\mathbb{G}_+$ is a grid diagram for a positive braid. Furthermore, we show how $\varepsilon$ behaves under $(p,q)$-cabling of negative torus knots., Some additional figures, further details added to the definitions, accepted to Journal of Knot Theory and Its Ramifications

A string figure is topologically a trivial knot lying on an imaginary plane orthogonal to the fingers with some crossings. The fingers prevent cancellation of these crossings. As a mathematical model of string figure we consider a knot diagram on the $xy$-plane in $xyz$-space missing some straight lines parallel to the $z$-axis. These straight lines correspond to fingers. We study minimal number of crossings of these knot diagrams under Reidemeister moves missing these lines., 6 pages, 6 figures, minor revision has done, to appear in Journal of Knot Theory and its Ramifications

A finitely generated group [Formula: see text] acting on a tree with infinite cyclic edge and vertex stabilizers is called a generalized Baumslag–Solitar group (GBS group). We prove that a one-knot group [Formula: see text] is a GBS group if and only if [Formula: see text] is a torus knot group, and describe all n-knot GBS groups for [Formula: see text].

We call a link (knot) [Formula: see text] to be strongly Jones (respectively, Homfly) undetectable, if there are infinitely many links which are not isotopic to [Formula: see text] but share the same Jones (respectively, Homfly) polynomial as [Formula: see text]. We reconstruct Kanenobu’s knot [Kanenobu, Infinitely many knots with the same polynomial invariant, Proc. Amer. Math. Soc. 97(1) (1986), 158–162] and give two new constructions. Using these constructions, we give some examples of strongly Jones undetectable: [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] ([Formula: see text] is the mirror image of [Formula: see text]) and etc. For some special cases, these constructions will be shown to be strongly Jones undetectable and strongly Homfly undetectable.

We generalize Yasuda’s examples of ribbon 2-knots of 1-fusion with different ribbon presentations.

A region crossing change at a region of a spatial-graph diagram is a transformation changing every crossing on the boundary of the region. In this paper, it is shown that every spatial graph consisting of theta-curves can be unknotted by region crossing changes., 13 pages, 12 figures

Joyce showed that for a classical knot [Formula: see text], the order of the involutory medial quandle is [Formula: see text]. Generalizing Joyce’s result, we show that for a classical link [Formula: see text] of [Formula: see text] components, the order of the involutory medial quandle is [Formula: see text]. In particular, [Formula: see text] is infinite if and only if [Formula: see text]. We also relate [Formula: see text] to several other link invariants.

O’Hara’s energies, introduced by Jun O’Hara, were proposed to answer the question what the canonical shape in a given knot type is, and were configured so that the less the energy value of a knot is, the “better” its shape is. The existence and regularity of minimizers has been well studied. In this paper, we calculate the first and second variational formulae of the [Formula: see text]-O’Hara energies and show absolute integrability, uniform boundedness, and continuity properties. Although several authors have already considered the variational formulae of the [Formula: see text]-O’Hara energies, their techniques do not seem to be applicable to the case [Formula: see text]. We obtain the variational formulae in a novel manner by extracting a certain function from the energy density. All of the [Formula: see text]-energies are made from this function, and by analyzing it, we obtain not only the variational formulae but also the estimates in several function spaces.

Every link in the 3-sphere has a projection to the plane where the only singularities are pairwise transverse triple points. The associated diagram, with height information at each triple point, is a triple-crossing diagram of the link. We give a set of diagrammatic moves on triple-crossing diagrams analogous to the Reidemeister moves on ordinary diagrams. The existence of n-crossing diagrams for every n>1 allows the definition of the n-crossing number. We prove that for any nontrivial, nonsplit link, other than the Hopf link, its triple-crossing number is strictly greater than its quintuple-crossing number., 21 pages, 13 figures

For a knot, the ascending number is the minimum number of crossing changes which are needed to obtain an ascending diagram. We study the ascending number of a knot by analyzing the Conway polynomial. In this paper, we give a sharper lower bound of the ascending number of a knot and newly determine the ascending number for 26 prime knots up to 10 crossings.

Using Blanchfield pairings, we show that two Alexander polynomials cannot be realized by a pair of matrices with algebraic Gordian distance one if a corresponding quadratic equation does not have an integer solution. We also give an example of how our results help in calculating the Gordian distances, algebraic Gordian distances and polynomial distances.

We show that if a braid [Formula: see text] can be parametrized in a certain way, then the previous work (B. Bode and M. R. Dennis, Constructing a polynomial whose nodal set is any prescribed knot or link, arXiv:1612.06328 ) can be extended to a construction of a polynomial [Formula: see text] with the closure of [Formula: see text] as the link of an isolated singularity of [Formula: see text], showing that the closure of [Formula: see text] is real algebraic. In particular, we prove that closures of squares of strictly homogeneous braids and certain lemniscate links are real algebraic. We also show that the constructed polynomials satisfy the strong Milnor condition, providing an explicit fibration of the complement of the closure of [Formula: see text] over [Formula: see text].

Let [Formula: see text] be a hyperbolic knot in the [Formula: see text]-sphere. If a [Formula: see text]-Dehn surgery on [Formula: see text] produces manifold with an embedded Klein bottle or essential [Formula: see text]-torus, then we prove that [Formula: see text], where [Formula: see text] is the genus of [Formula: see text]. We obtain different upper bounds according to the production of a Klein bottle, a non-separating [Formula: see text]-torus, or an essential and separating [Formula: see text]-torus. The well known examples which are the figure eight knot and the pretzel knot [Formula: see text] reach the given upper bounds. We study this problem considering null-homologous hyperbolic knots in compact, orientable and closed [Formula: see text]-manifolds.

The number [Formula: see text] of colorings of a knot [Formula: see text] by a finite quandle [Formula: see text] has been used in the literature to distinguish between knot types. In this paper, we suggest a refinement [Formula: see text] to this knot invariant involving any computable functor [Formula: see text] from finitely presented groups to finitely generated abelian groups. We are mainly interested in the functor [Formula: see text] that sends each finitely presented group [Formula: see text] to its abelianization [Formula: see text]. We describe algorithms needed for computing the refined invariant and illustrate implementations that have been made available as part of the HAP package for the GAP system for computational algebra. We use these implementations to investigate the performance of the refined invariant on prime knots with [Formula: see text] crossings.

It is known that every nontrivial knot has at least two quadrisecants. Given a knot, we mark each intersection point of each of its quadrisecants. Replacing each subarc between two nearby marked points with a straight line segment joining them, we obtain a polygonal closed curve which we will call the quadrisecant approximation of the given knot. We show that for any hexagonal trefoil knot, there are only three quadrisecants, and the resulting quadrisecant approximation has the same knot type., Comment: 8 pages

By using Rasmussen's s-invariant derived from Khovanov homology, we give a combinatorial proof for the existence of countably many Casson handles.

Self-avoiding polygons in the cubic lattice are models of ring polymers in dilute solution. The conformational entropy of a ring polymer is a dominant factor in its physical and chemical properties, and this is modeled by the large number of conformations of lattice polygons. Cubic lattice polygons are embeddings of the circle in three space and may be used as a model of knotting in ring polymers. In this paper we study the effects of knotting on the conformational entropy of lattice polygons and so determine the relative fraction of polygons of different knot types at large lengths. More precisely, we consider the number of cubic lattice polygons of n edges with knot type K, pn(K). Numerical evidence strongly suggests that [Formula: see text] as n → ∞, where μ0 is the growth constant of unknotted lattice polygons, α is the entropic exponent of lattice polygons, and NK is the number of prime knot components in the knot type K (see the paper [Asymptotics of knotted lattice polygons, J. Phys. A: Math. Gen.31 (1998) 5953–5967]). Determining the exact value of pn(K) is far beyond current techniques for all but very small values of n. Instead we use the GAS algorithm (see the paper [Generalised atmospheric sampling of self-avoiding walks, J. Phys. A: Math. Theor.42 (2009) 335001–335030]) to enumerate pn (K) approximately. We then extrapolate ratios [pn(K)/pn(L)] to larger values of n for a number of given knot types. We give evidence that for the unknot 01 and the trefoil knot 31, there exists a number M01, 31 ≈170000 such that pn (01) > pn (31) if n < M01, 31 and pn (01) ≤pn (31) if n ≥M01, 31. In addition, the asymptotic relative frequencies for a variety of knot types are determined. For example, we find that [pn(31)/pn(41)] → 27.0 ± 2.2, implying that there are approximately 27 polygons of the trefoil knot type for every polygon of knot of type 41 (the figure eight knot), in the asymptotic limit. Finally, we examine the dominant knot types at moderate values of n and conjecture that the most frequent knot types in polygons of any given length n are of the form [Formula: see text] (or its chiral partner), where [Formula: see text] are right- and left-handed trefoils, and N increases with n.

In this paper following the scheme of Massey–Milnor invariant theory [C. C. Hsieh, Combinatoric and diagrammatic studies in knot theory J. Knot Theory Ramifications16 (2007) 1235–1253; C. C. Hsieh, Massey-Milnor linking = Chern-Simons-Witten graphs, J. Knot Theory Ramifications17 (2008) 877–903; C. C. Hsieh and S. W. Yang, Chern-Simons-Witten configuration space integrals in knot theory, J. Knot Theory Ramifications14 (2005) 689–711], we studied the first non-vanishing linkings of knot theory in ℝ3 and also derived the combinatorial formulae from which we could read out the invariants directly from the knot diagrams. Though the theme is calculus, the idea comes from perturbative quantum field theory.

It is known by few that a trivial knot can be transformed into a lattice knot whose three shadows are all trees (here, three shadows mean the projections of the lattice knot to the directions of ordinary orthogonal axes). Since a knot itself is a loop in the space, this fact may be rather astonishing. It will be interesting to ask whether a non-trivial knot has such a transformation or not. The purpose of this paper is to show that any two-bridge torus knot or link has a transformation into a lattice knot whose three shadows are all trees. The algorithm to construct such a position will be demonstrated.

Diagrams and Reidemeister moves for links in a twisted S^1-bundle over an unorientable surface are introduced. Using these diagrams, we compute the Kauffman Bracket Skein Module (KBSM) of the connected sum of two projective spaces. In particular, we show that it has torsion. We also present a new computation of the KBSM of S^1 x S^2 and the lens spaces L(p,1)., 22 pages, 21 figures

We give a simple argument to show that every polynomial f(t) ∈ ℤ[t] such that f(1) = 1 is the Alexander polynomial of some ribbon 2-knot whose group is a 1-relator group, and we extend this result to links.

Diagrams and Reidemeister moves for links in orientable Seifert manifolds are introduced. Using these diagrams, we compute the Kauffman bracket skein modules of prism manifolds with first homology of order 4. In particular, we show that these modules are free.

In a recent work "Arc-presentation of links: Monotonic simplification", Dynnikov shows that each rectangular diagram of the unknot, composite link, or split link can be monotonically simplified into a trivial, composite, or split diagram, respectively. The following natural question arises: Is it always possible to simplify monotonically a rectangular diagram of a satellite knot or link into one where the satellite structure is seen? Here we give a negative answer to that question both for knot and link cases. An example of a torus embedding that cannot be obtained from ordinary "thin" torus by methods of paper [2] is also constructed.

A 2-universal knot is a universal knot having all branching indices 1 or 2. Here we construct the first example of a 2-universal knot which could be a hyperbolic knot (SnapPea). Hence, we could have an example of a universal group generated by 180° rotations.

The knot coloring polynomial defined by Eisermann for a finite pointed group is generalized to an infinite pointed group as the longitudinal mapping invariant of a knot. In turn, this can be thought of as a generalization of the quandle 2-cocycle invariant for finite quandles. If the group is a topological group, then this invariant can be thought of as a topological generalization of the 2-cocycle invariant. The longitudinal mapping invariant is based on a meridian–longitude pair in the knot group. We also give an interpretation of the invariant in terms of quandle colorings of a 1-tangle for generalized Alexander quandles without use of a meridian–longitude pair in the knot group. The invariant values are concretely evaluated for the torus knots [Formula: see text], their mirror images, and the figure eight knot for the group SU(2).

We introduce a notion of a chain of Yang–Baxter like operations. This is a sequence of solutions of an asymmetric variant of the Yang–Baxter equation and is a multi-operator generalization of (bi)rack/quandles. We discuss knot and link invariants coming from a chain of Yang–Baxter like operations, and give potential applications. Among them, we define a cocycle invariant for transverse links.

Knot mosaics are used to model physical quantum states. The mosaic number of a knot is the smallest integer [Formula: see text] such that the knot can be represented as a knot [Formula: see text]-mosaic. In this paper, we establish an upper bound for the crossing number of a knot in terms of the mosaic number. Given an [Formula: see text]-mosaic and any knot [Formula: see text] that is represented on the mosaic, its crossing number [Formula: see text] is bounded above by [Formula: see text] if [Formula: see text] is odd, and by [Formula: see text] if [Formula: see text] is even. In the process, we develop a useful new tool called the mosaic complement.

Energy minimizing smooth knot configurations have long been approximated by finding knotted polygons that minimize discretized versions of the given energy. However, for most knot energy functionals, the question remains open on whether the minimum polygonal energies are "close" to the minimum smooth energies. In this paper, we determine an explicit bound between the Minimum-Distance Energy of a polygon and the Möbius Energy of a piecewise-C2 knot inscribed in the polygon. This bound is written in terms of the ropelength and the number of edges and can be used to determine an upper bound for the minimum Möbius Energy for different knot types.

In 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 [Formula: see text] and arc index [Formula: see text] for any knot or non-split link [Formula: see text], which are [Formula: see text] and [Formula: see text]. We also find a quadratic upper bound of the minimal crossing number of delta diagrams of [Formula: see text].

Let K be a two-bridge knot or link in S3. Then K is also denoted as the four-plat, b(p, q) to indicate its association with some rational number p/q. The lens space L = L(p, q) admits an isometry τ of order two, such that the quotient space L modulo the involution τ is an orbifold whose exceptional set is K and whose underlying space is S3. In this paper, the mapping class groups of these orbifolds are classified. While these groups can be found as a result of Mostow's Rigidity Theorem, this paper calculates the generators and relations of the groups and the proof does not rely on this strong theorem for the majority of cases.

We consider the problem of classification of links up to (2, 2)-moves. Our motivation comes from the theory of skein modules, more specifically from the skein module of S3 based on the deformation of (2, 2)-move. As it was proved in D-P-2, not every link can be reduced to a trivial link by (2, 2)-moves, for instance, the closure of (σ1σ2)6. In this paper, we classify 3-braids up to (2, 2)-moves and, we show how the Harikae–Nakanishi–Uchida conjecture can be modified to hold for closed 3-braids. As an important step in the classification we prove the conjecture for 2-algebraic links and classify (2, 2)-equivalence classes for links up to nine crossings. We also analyze an action of (2, 2)-move on Kei (involutive quandle) associated to a link diagram. We define Burnside Kei, Q(m, n), and ask for which values of m and n, is Q(m, n) finite. This question is motivated by classical Burnside problem.

A surface-knot is a closed oriented surface smoothly embedded in 4-space and a surface-knot diagram is a projected image of a surface-knot under the orthogonal projection in 3-space with crossing information. Every surface-knot diagram induces a rectangular-cell complex. In this paper, we introduce a covering diagram over a surface-knot diagram. the covering map induces a covering of the rectangular-cell complexes. As an application, a lower bound of triple point numbers for a family of surface-knots is obtained.

We completely determine the splitting number of augmented links arising from knot and link diagrams in which each twist region has an even number of crossings. In the case of augmented links obtained from knot diagrams, we show that the splitting number is given by the size of a maximal collection of Boromean sublinks, any two of which have one component in common. The general case is stablished by considering the linking numbers between components of the augmented links. We also discuss the case when the augmented link arises from a link diagram in which twist regions may have an odd number of crossings.

We use curvature techniques from geometric group theory to produce examples of virtual knot groups that are not classical knot groups.

The purpose of this note is to show that classical cobordism arguments, which go back to the pioneering works of Mandelbaum and Moishezon, provide quick and unified proofs of any knot surgered compact simply-connected 4-manifold [Formula: see text] becoming diffeomorphic to [Formula: see text] after a single stabilization by connected summing with [Formula: see text] or [Formula: see text], and almost complete decomposability of [Formula: see text] for many almost completely decomposable [Formula: see text], such as the elliptic surfaces.

We present a generalization of the Fox formula for twisted Alexander invariants associated to representations of knot groups over rings of [Formula: see text]-integers of [Formula: see text], where [Formula: see text] is a finite set of finite primes of a number field [Formula: see text]. As an application, we give the asymptotic growth of twisted homology groups.

We work with a generalization of knot theory, in which one diagram is reachable from another via a finite sequence of moves if a fixed condition regarding the existence of certain morphisms in an associated category is satisfied for every move of the sequence. This conditional setting leads to a possibility of irreversible moves, terminal states, and to using functors more general than the ones used as knot invariants. Our main focus is the category of diagrams with a binary relation on the set of arcs, indicating which arc can move over another arc. We define homology of binary relations, and merge it with quandle homology, to obtain the homology for partial quandles with a binary relation. This last homology can be used to analyze link diagrams with a binary relation on the set of components.

We will study boundaries of incompressible surfaces properly embedded in graph knot exteriors. We will first show that any bounded two-sided surface is meridional or longitudinal. In particular, iterated torus knot exteriors contain no bounded two-sided essential surface which is meridional or preferred longitudinal but Seifert surfaces. Even if the surface is possibly one-sided, the boundary is not of type (2p, 2q + 1) for any integers p and q.

For any knot K represented as an n-braid (n ≥ 3), we construct an infinite sequence of pairwise non-conjugate (n + 1)-braids {bm, m ∈ ℕ} representing K.

For any positive integer [Formula: see text], we exhibit a prime knot [Formula: see text] with [Formula: see text] crossings whose Jones polynomial [Formula: see text] is equal to [Formula: see text] modulo [Formula: see text]. Our construction rests on a certain [Formula: see text]-crossing prime tangle [Formula: see text] which is undetectable by the Kauffman bracket polynomial pair mod [Formula: see text].

An example of an integral ternary quadratic form [Formula: see text] such that its associated orbifold [Formula: see text] is a manifold is given. Hence, the title is proved.

A ternary, integral quadratic form is called unilateral if its associated orbifold [Formula: see text] is orientable. Examples of unilateral forms [Formula: see text] and their associated orbifolds [Formula: see text] are given. The examples are selected to give credit to the conjecture that every form with square-free, prime determinant has a unilateral [Formula: see text]-cover.

A knot diagram has an associated looped interlacement graph, obtained from the intersection graph of the Gauss diagram by attaching loops to the vertices that correspond to negative crossings. This construction suggests an extension of the Kauffman bracket to an invariant of looped graphs, and an extension of Reidemeister equivalence to an equivalence relation on looped graphs. The graph bracket polynomial can be defined recursively using the same pivot and local complementation operations used to define the interlace polynomial, and it gives rise to a graph Jones polynomial VG(t) that is invariant under the graph Reidemeister moves.

We found a family of infinitely many hyperbolic knot manifolds each member of which has a strongly detected boundary slope with associated root of unity of order 4.

We construct a series of combinatorial quandle-like knot invariants. We color regions of a knot diagram rather than lines and assign a weight to each coloring. Sets of these weights are the invariants we construct (colorings and weights depend on several parameters).Using these invariants, we prove that left and right trefoils are not isotopic using this invariant (in a particular case).

We calculate certain limits of the colored Alexander invariant of the figure-eight knot for some cases and show that this limit is related to the volume of hyperbolic orbifolds whose singular set is the figure-eight knot.