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.

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.

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

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.

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.

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.

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

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.

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 TT of T. A similar inequality

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.

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

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.

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

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,

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.

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

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 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.

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

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 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 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.

Quandle colorings and cocycle invariants are studied for composite knots, and applied to chirality and abelian extensions. The square and granny knots, for example, can be distinguished by quandle colorings, so that a trefoil and its mirror can be distinguished by quandle coloring of composite knots. We investigate this and related phenomena. Quandle cocycle invariants are studied in relation to quandle

A chord diagram consists of a circle, called the backbone, with line segments, called chords, whose endpoints are attached to distinct points on the circle. The genus of a chord diagram is the genus of the orientable surface obtained by thickening the backbone to an annulus and attaching bands to the inner boundary circle at the ends of each chord. Variations of this construction are considered here

We present a set of 26 finite quandles that distinguish (up to reversal and mirror image) by number of colorings, all of the 2977 prime oriented knots with up to 12 crossings. We also show that 1058 of these knots can be distinguished from their mirror images by the number of colorings by quandles from a certain set of 23 finite quandles. We study the colorings of these 2977 knots by all of the 431

A formula for the Alexander polynomial of a 2-bridge knot or link given by Hartley and also by Minkus has a beautiful interpretation as a walk on the integers. We extend this to the 2-variable Alexander polynomial of a 2-component, 2-bridge link, obtaining a formula that corresponds to a walk on the 2-dimensional integer lattice.

For a virtual knot K and an integer r≥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 J0,J2,J3,…,Jm, there is a virtual knot K such that K(r)=Jr(r=0 and 2≤r≤m), K(1)=K, and otherwise K(r)=J0.