Gordian complex of knots was defined by Hirasawa and Uchida as the simplicial complex whose vertices are knot isotopy classes in 𝕊3. Later Horiuchi and Ohyama defined Gordian complex of virtual knots using v-move and forbidden moves. In this paper, we discuss Gordian complex of knots by region crossing change and Gordian complex of virtual knots by arc shift move. Arc shift move is a local move in

Champanerkar and Kofman [Twisting quasi-alternating links, Proc. Amer. Math. Soc.137(7) (2009) 2451–2458] introduced an interesting way to construct new examples of quasi-alternating links from existing ones. Actually, they proved that replacing a quasi-alternating crossing c in a quasi-alternating link by a rational tangle of same type yields a new quasi-alternating link. This construction has been

We introduce labeled singular knots and equivalently labeled 4-valent rigid vertex spatial graphs. Labeled singular knots are singular knots with labeled singularities. These knots are considered subject to isotopies preserving the labelings. We provide a topological invariant schema similar to that of Henrich and Kauffman in [A. Henrich and L. H. Kauffman, Tangle insertion invariants for pseudoknots

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

We show that Dehn filling on the manifold v2503 results in a non-orderable space for all rational slopes in the interval (−∞,−1). This is consistent with the L-space conjecture, which predicts that all fillings will result in a non-orderable space for this manifold.

It is known that there exists a surjective map from the set of weak (1, 3) homotopy classes of knot projections to the set of positive knots [N. Ito and Y. Takimura, (1, 2) and weak (1, 3) homotopies on knot projections, J. Knot Theory Ramifications22 (2013) 1350085]. An interesting question whether this map is also injective, which question was formulated independently by S. Kamada and Y. Nakanishi

A finitely generated group G 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 G is a GBS group if and only if G is a torus knot group, and describe all n-knot GBS groups for n≥3.

In [V. O. Manturov, An almost classification of free knots, Dokl. Math.88(2) (2013) 556–558.] the second author constructed an invariant which in some sense generalizes the quantum sl(3) link invariant of Kuperberg to the case of free links. In this paper, we generalize this construction to free graph-links. As a result, we obtain an invariant of free graph-links with values in linear combinations

In the 1950s Milnor defined a family of higher-order invariants generalizing the linking number. Even the first of these new invariants, the triple linking number, has received fruitful study since its inception. In the case that a link L has vanishing pairwise linking numbers, this triple linking number gives an integer-valued invariant. When the linking numbers fail to vanish, the triple linking

We study groups of some virtual knots with small number of crossings and prove that there is a virtual knot with long lower central series which, in particular, implies that there is a virtual knot with residually nilpotent group. This gives a possibility to construct invariants of virtual knots using quotients by terms of the lower central series of knot groups. Also, we study decomposition of virtual

In 2003, Ozsváth and Szabó defined the concordance invariant τ for knots in oriented 3-manifolds as part of the Heegaard Floer homology package. In 2011, Sarkar gave a combinatorial definition of τ for knots in S3 and a combinatorial proof that τ gives a lower bound for the slice genus of a knot. Recently, Harvey and O’Donnol defined a relatively bigraded combinatorial Heegaard Floer homology theory

Ng constructed an invariant of knots in ℝ3, a combinatorial knot contact homology. Extending his study, we construct an invariant of surface-knots in ℝ4 using marked graph diagrams.

A twisted torus knot is a torus knot with some consecutive strands twisted. More precisely, a twisted torus knot T(p,q,r,s) is a torus knot T(p,q) with r consecutive strands s times fully twisted. We determine which twisted torus knots T(p,q,p−kq,−1) are a torus knot.

For any classical knot k1, we can construct a ribbon 2-knot spun(k1) by spinning an arc removed a small segment from k1 about R2 in R4. A ribbon 2-knot is an embedded 2-sphere in R4. If k1 has an n-crossing presentation, by spinning this, we can naturally construct a ribbon presentation with n ribbon crossings for spun(k1). Thus, we can define naturally a notion on ribbon 2-knots corresponding to the

We extend results of Positive Twist Knots and Thickenings [W. George, Positive twist knots and thickenings, J. Knot Theory Ramifications22 (2013) 1350046] to show that positive twist knots 𝒦m, for m≥3, satisfy the Uniform Thickness Property.

We directly connect topological changes that can occur in mathematical three-space via surgery, with black hole formation, the formation of wormholes and new generalizations of these phenomena. This work widens the bridge between topology and natural sciences and creates a new platform for exploring geometrical physics.

We obtain information on torsion in Khovanov cohomology by performing calculations directly over ℤ/pkℤ for p prime and k≥2. In particular, we get that the torus knots T(9,10) and T(9,11) contain torsion of order 9 and 27 in their Khovanov cohomology.

A sequence of F-polynomials {FKn(t,ℓ)}n=1∞ of virtual knots K was defined by Kaur et al. in 2018. These polynomials have been expressed in terms of index value of crossing and n-writhe of K. By the construction, F-polynomials are generalizations of Kauffman’s Affine Index Polynomial, and are invariants of virtual knot K. We present values of F-polynomials of oriented virtual knots having at most four

We prove that a version of the Thurston–Bennequin inequality holds for Legendrian and transverse links in a rational homology contact 3-sphere (M,ξ), whenever ξ is tight. More specifically, we show that the self-linking number of a transverse link T in (M,ξ), such that the boundary of its tubular neighborhood consists of incompressible tori, is bounded by the Thurston norm ∥T∥T of T. A similar inequality

We show an infinite family of satellite knots that can be unknotted by a single band move, but such that there is no band unknotting the knots which is disjoint from the satellite torus.

Laplacian matrices of signed graphs in surfaces S are used to define module and polynomial invariants of ℤ/2-homologically trivial links in S×[0,1]. Information about virtual genus is obtained.

We determine the minimal number of colors for nontrivial ℤ-colorings on the standard minimal diagrams of ℤ-colorable torus links. Also included is a complete classification of such ℤ-colorings, which are shown by using rack colorings on link diagrams.

The warping degree of an oriented knot diagram is the minimal number of crossing changes which are required to obtain a monotone diagram from the diagram. The minimal warping degree of a knot is the minimal value of the warping degree for all oriented minimal diagrams of the knot. In this paper, all prime alternating knots with minimal warping degree two are determined.

Motivated by problem of determining the unknotted routes for the scaffolding strand in DNA origami self-assembly, we examine the existence and knottedness of A-trails in graphs embedded on the torus. We show that any A-trail in a checkerboard-colorable torus graph is unknotted and characterizes the existence of A-trails in checkerboard-colorable torus graphs in terms of pairs of quasitrees in associated

Reshetikhin–Turaev (a.k.a. Chern–Simons) TQFT is a functor that associates vector spaces to two-dimensional genus g surfaces and linear operators to automorphisms of surfaces. The purpose of this paper is to demonstrate that there exists a Macdonald q,t-deformation — refinement — of these operators that preserves the defining relations of the mapping class groups beyond genus 1. For this, we explicitly

A spatial surface is a compact surface embedded in the 3-sphere. In this paper, we provide several typical examples of spatial surfaces and construct a coloring invariant to distinguish them. The coloring is defined by using a multiple group rack, which is a rack version of a multiple conjugation quandle.

Letting C be a compact Cω-curve embedded in the Euclidean 3-space (Cω means real analyticity), we consider a Cω-cuspidal edge f along C. When C is non-closed, in the authors’ previous works, the local existence of three distinct cuspidal edges along C whose first fundamental forms coincide with that of f was shown, under a certain reasonable assumption on f. In this paper, if C is closed, that is,

We call a link (knot) L to be strongly Jones (respectively, Homfly) undetectable, if there are infinitely many links which are not isotopic to L but share the same Jones (respectively, Homfly) polynomial as L. 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

In this paper, we describe the equivalence classes of simple arcs between the two punctures on a 2-punctured torus Σ1,2 up to isotopy by using the given four generators g1,g2,g3 and g4. Actually, we show that a class of simple arcs is represented by an ordered sequence of four integers. Also, we introduce an algorithm to check whether or not an ordered sequence of four integers represents a class of

Khovanov homology is a categorification of the Jones polynomial, so it may be seen as a kind of quantum invariant of knots and links. Although polynomial quantum invariants are deeply involved with Vassiliev (aka. finite type) invariants, the relation remains unclear in case of Khovanov homology. Aiming at it, in this paper, we discuss a categorified version of Vassiliev skein relation on Khovanov

For any alternating prime knot K, it is expected that the canonical genus of its Whitehead double is equal to the crossing number of K. We introduce local moves of alternating knots, and prove that these local moves preserve the canonical genus of its Whitehead double under a certain condition. By this result, we give a new family of knots which satisfy this conjecture.

Suciu constructed infinitely many ribbon 2-knots in S4 whose knot groups are isomorphic to the trefoil knot group. They are distinguished by the second homotopy groups. We classify these knots by using SL(2,ℂ)-representations of the fundamental groups of the 2-fold branched covering spaces.

In the paper, we give an equivalent description of the lens space L(p,q) with p prime in terms of any corresponding Heegaard diagrams as follows: Let M be a closed orientable 3-manifold with π1(M)≠1, and U∪FV a Heegaard splitting of genus n for M with an associated Heegaard diagram (U;J1,…,Jn). Assume p is a prime integer. Then M is homeomorphic to the lens space L(p,q) if and only if there exists

The twin group Tn is a right angled Coxeter group generated by n−1 involutions and having only far commutativity relations. These groups can be thought of as planar analogues of Artin braid groups. In this paper, we study some properties of twin groups whose analogues are well known for Artin braid groups. We give an algorithm for two twins to be equivalent under individual Markov moves. Further, we

This paper is dedicated to cabling on virtual braids. This construction gives a new generating set for the virtual pure braid group VPn. We define simplicial group VP∗ and its simplicial subgroup T∗ which is generated by VP2. Consequently, we describe VP4 as HNN-extension and find presentation of T2 and T3. As an application to classical braids, we find a new presentation of the Artin pure braid group

In this paper, we discuss how to define a chord index via smoothing a real crossing point of a virtual knot diagram. Several polynomial invariants of virtual knots and links can be recovered from this general construction. We also explain how to extend this construction from virtual knots to flat virtual knots.

In this paper, we study the singular pure braid group SPn for n=2,3. We find generators, defining relations and the algebraical structure of these groups. In particular, we prove that SP3 is a semi-direct product SP3=Ṽ3⋋ℤ, where Ṽ3 is an HNN-extension with base group ℤ2∗ℤ2 and cyclic associated subgroups. We prove that the center Z(SP3) of SP3 is a direct factor in SP3.

The knotting probability of an arc diagram is defined as the quadruplet of four kinds of finner knotting probabilities which are invariant under a reasonable deformation containing an isomorphism on an arc diagram. In a separated paper, it is shown that every oriented spatial arc admits four kinds of unique arc diagrams up to isomorphisms determined from the spatial arc and the projection, so that

The d-fold (d≥3) branched coverings on a disk give an infinite family of nongeometric embeddings of braid groups into mapping class groups. We, in this paper, give new explicit expressions of these braid group representations into automorphism groups of free groups in terms of the actions on the generators of free groups. We also give a systematic way of constructing and expressing these braid group

In this paper, we study the minimal symplectic elliptic surfaces E(k) with homologically trivial symplectic finite group actions, and get a rigidity theorem under some restriction.

We formulate large N duality of U(N) refined Chern–Simons theory with a torus knot/link in S3. By studying refined BPS states in M-theory, we provide the explicit form of low-energy effective actions of Type IIA string theory with D4-branes on the Ω-background. This form enables us to relate refined Chern–Simons invariants of a torus knot/link in S3 to refined BPS invariants in the resolved conifold

A string-net model associates a vector space to a surface in terms of graphs decorated by objects and morphisms of a pivotal fusion category modulo local relations. String-net models are usually considered for spherical fusion categories, and in this case, the vector spaces agree with the state spaces of the corresponding Turaev–Viro topological quantum field theory. In the present work, some effects

We study the effects of certain local moves on Homflyptand Kauffman polynomials. We show that all Homflypt(or Kauffman) polynomials are equal in a certain nontrivial quotient of the Laurent polynomial ring. As a consequence, we discover some new properties of these invariants.

We give a description of Rasmussen’s s-invariant from the divisibility of Lee’s canonical class. More precisely, given any link diagram D, for any choice of an integral domain R and a non-zero, non-invertible element c∈R, we define the c-divisibility kc(D) of Lee’s canonical class of D, and prove that a combination of kc(D) and some elementary properties of D yields a link invariant s̄c. Each s̄c possesses

A virtualn-string is a chord diagram with n core circles and a collection of arrows between core circles. We consider virtual n-strings up to virtual homotopy, compositions of flat virtual Reidemeister moves on chord diagrams. Given a virtual 1-string α, Turaev associated a based matrix that encodes invariants of the virtual homotopy class of α. We generalize Turaev’s method to associate a multistring

Cohomological invariants of twisted wild character varieties as constructed by Boalch and Yamakawa are derived from enumerative Calabi–Yau geometry and refined Chern–Simons invariants of torus knots. Generalizing the untwisted case, the present approach is based on a spectral correspondence for meromorphic Higgs bundles with fixed conjugacy classes at the marked points. This construction is carried

In their paper entitled “Quantum Enhancements and Biquandle Brackets”, Nelson, Orrison, and Rivera introduced biquandle brackets, which are customized skein invariants for biquandle-colored links. We prove herein that if a biquandle bracket (A,B) is the pointwise product of the pair of functions (A′,B′) with a function ϕ, then (A′,B′) is also a biquandle bracket if and only if ϕ is a biquandle 2-cocycle

We show that determining the crossing number of a link is NP-hard. For some weaker notions of link equivalence, we also show NP-completeness.

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

We investigate representations of mapping class groups of surfaces that arise from the untwisted Drinfeld double of a finite group G, focusing on surfaces without marked points or with one marked point. We obtain concrete descriptions of such representations in terms of finite group data. This allows us to establish various properties of these representations. In particular, we show that they have

We construct an infinite tower of covering spaces over the configuration space of n−1 distinct nonzero points in the complex plane. This results in an action of the braid group 𝔹n on the set of n-adic integers ℤn for all natural numbers n≥2. We study some of the properties of these actions such as continuity and transitivity. The construction of the actions involves a new way of associating to any

We explore the relationship between two knot invariants, unknotting number and crosscap number. Specifically, we present two infinite families of knots with a specific crossing change that lowers the unknotting number while raising the crosscap number. One of these families is hyperbolic. This is a non-orientable parallel of Scharlemann and Thompson’s 1988 result for orientable surfaces.

We introduce some polynomial invariants for flat virtual links which are similar to the Jones–Kauffman polynomial, the Miyazawa polynomial and the arrow polynomial for virtual link diagrams, and we give several properties and examples.

We describe an effective method for simultaneously computing d-invariants of infinite families of Brieskorn spheres Σ(p,q,r) with pq+pr−qr=1.

We use Kauffman’s bracket polynomial to define a complex-valued invariant of virtual rational tangles that generalizes the well-known fraction invariant for classical rational tangles. We provide a recursive formula for computing the invariant, and use it to compute several examples.

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.

We examine the relationship between the oriented cube of resolutions for knot Floer homology and HOMFLY-PT homology. By using a filtration induced by additional basepoints on the Heegaard diagram for a knot K, we see that the filtered complex decomposes as a direct sum of HOMFLY-PT complexes of various subdiagrams. Applying Jaeger’s composition product formula for knot polynomials, we deduce that the

We investigate embeddings from the set of long flat virtual knot diagrams to the set of long virtual knot diagrams so that we can construct invariants for long flat virtual knots. Also, we give properties and examples of several invariants for long flat virtual knots via these embeddings and invariants for long virtual knots.

In this paper, we prove an algorithmical solvability of exponential-Diophantine equations in rings represented by matrices over fields of positive characteristic. Consider the system of exponential-Diophantine equations ∑i=1sPij(n1,…,nt)bij0aij1n1bij1…aijtntbijt=0 where bijk,aijk are constants from matrix ring of characteristic p, ni are indeterminates. For any solution (n1,…,nt) of the system we construct

Joyce showed that for a classical knot K, the order of the involutory medial quandle is |detK|. Generalizing Joyce’s result, we show that for a classical link L of μ≥1 components, the order of the involutory medial quandle is μ|detL|/2μ−1. In particular, IMQ(L) is infinite if and only if detL=0. We also relate IMQ(L) to several other link invariants.