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This paper gives a new way of characterizing L-space 3-manifolds by using orderability of quandles. Hence, this answers a question of Clay et al. (Question 1.1 of Can Math Bull 59(3):472–482, 2016). We also investigate both the orderability and circular orderability of dynamical extensions of orderable quandles. We give conditions under which the conjugation quandle on a group, as an extension of the conjugation of a bi-orderable group by the conjugation of a right orderable group, is right orderable. We also study the right circular orderability of link quandles. We prove that the n-quandle Q n (L) of the link quandle of a link L in the 3-sphere is not right circularly orderable and hence it is not right orderable. But on the other hand, we show that there are infinitely many links for which the p-enveloping group of the link quandle is right circularly orderable for any prime integer p. [ABSTRACT FROM AUTHOR]

Several distinct Garside monoids having torus knot groups as groups of fractions are known. For n , m ≥ 2 two coprime integers, we introduce a new Garside monoid ℳ (n , m) having as Garside group the (n , m) -torus knot group, thereby generalizing to all torus knot groups a construction that we previously gave for the (n , n + 1) -torus knot group. As a byproduct, we obtain new Garside structures for the braid groups of a few exceptional complex reflection groups of rank two. Analogous Garside structures are also constructed for a few additional braid groups of exceptional complex reflection groups of rank two which are not isomorphic to torus knot groups, namely, for G 1 3 and for dihedral Artin groups of even type. [ABSTRACT FROM AUTHOR]

We have the generating function which determines the number of 2 -bridge knot groups admitting epimorphisms onto the knot group of a given 2 -bridge knot, in terms of crossing number. In this paper, we will refine this formula by taking into account genus as well as crossing number. Next, we determine the number of epimorphisms between fibered 2 -bridge knot groups. Moreover, we discuss degree one maps and 2 -bridge knots with unknotting number one. [ABSTRACT FROM AUTHOR]

Given a knot K , we may construct a group G n (K) from the fundamental group of K by adjoining an n th root of the meridian that commutes with the corresponding longitude. For n ≥ 2 these "generalized knot groups" determine K up to reflection. The second author has shown that for n ≥ 2 , the generalized knot groups of the square and granny knots can be distinguished by counting homomorphisms into a suitably chosen finite group. We extend this result to certain generalized knot groups of square and granny knot analogues S K a , b = T a , b # T − a , b , G K a , b = T a , b # T a , b , constructed as connected sums of (a , b) -torus knots of opposite or identical chiralities. More precisely, for coprime a , b ≥ 2 and n satisfying a coprimality condition with a and b , we construct an explicit finite group G (depending on a , b and n) such that G n (S K a , b) and G n (G K a , b) can be distinguished by counting homomorphisms into G. The coprimality condition includes all n ≥ 2 coprime to a b. The result shows that the difference between these two groups can be detected using a finite group. [ABSTRACT FROM AUTHOR]

In this paper we give an example of a linear group such that its tensor square is not linear. Also, we formulate some sufficient conditions for the linearity of non-abelian tensor products G ⊗H and tensor squares G ⊗ G. Using these results we prove that tensor squares of some groups with one relation and some knot groups are linear. We prove that the Peiffer square of a finitely generated linear group is linear. At the end we construct faithful linear representations for the non-abelian tensor square of a free group and free nilpotent group. [ABSTRACT FROM AUTHOR]

A knot group has weight one, so is normally generated by a single element called a weight element of the knot group. A meridian is a typical weight element, but some knot groups admit other weight elements. We show that for some infinite classes of three-strand pretzel knots and all prime knots with up to eight crossings, the knot groups admit weight elements that are not automorphic images of meridians. [ABSTRACT FROM AUTHOR]

Let K , K ′ be two-bridge knots of genus k , k ′ respectively. We show the necessary and sufficient condition of k in terms of k ′ that there exists an epimorphism from the knot group of K onto that of K ′ . [ABSTRACT FROM AUTHOR]

We study the twisted knot module for the universal deformation of an SL2-representation of a knot group and introduce an associated L-function, which may be seen as an analogue of the algebraic p-adic L-function associated to the Selmer module for the universal deformation of a Galois representation. We then investigate two problems proposed by Mazur: Firstly we show the torsion property of the twisted knot module over the universal deformation ring under certain conditions. Secondly we compute the L-function by some concrete examples for 2-bridge knots. [ABSTRACT FROM AUTHOR]

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

Let M be a non-abelian semi-direct product of a cyclic group Z/n and an elementary abelian p-group A = ⊕k(Z/p) of order pk, p being a prime and gcd(n, p) = 1. Suppose that the knot group G(K) of a knot K in the 3-sphere is represented on M. Then we conjectured (and later proved) that the twisted Alexander polynomial Δγ,K(t) associated to γ : G(K)→ M→ GL(pk, Z) is of the form: ΔK(t) 1−t F(tn), where ΔK(t) is the Alexander polynomial of K and F(tn) is an integer polynomial in tn. In this paper, we present a proof of the following. For a 2-bridge knot K(r) in H(p), if n = 2 and k = 1, then F(t2) is written as f(t)f(t-1), where H(p) is the set of 2-bridge knots whose knot groups map on that of K(1/p) with p odd. [ABSTRACT FROM AUTHOR]

If is an orientable, strongly minimal -complex and has one end, then it has no nontrivial locally finite normal subgroup. Hence, if is a 2-knot group, then (a) if is virtually solvable, then either has two ends or , with presentation , or is torsion-free and polycyclic of Hirsch length 4 (b) either has two ends, or has one end and the center is torsion-free, or has infinitely many ends and is finite, and (c) the Hirsch-Plotkin radical is nilpotent. [ABSTRACT FROM AUTHOR]

We give explicit formulas for the volumes of hyperbolic cone-manifolds of double twist knots, a class of two-bridge knots which includes twist knots and two-bridge knots with Conway notation . We also study the Riley polynomial of a class of one-relator groups which includes two-bridge knot groups. [ABSTRACT FROM AUTHOR]

In this paper, we show that if 0-surgery along a ribbon knot and 0-surgery along another knot give diffeomorphic 3-manifolds then has to be a slice knot. Moreover, they have diffeomorphic slice disk exteriors for some ribbon disk and slice disk , where and are tubular neighborhoods of and , respectively. [ABSTRACT FROM AUTHOR]

We provide explicit quadrature rules for spaces of C 1 quintic splines with uniform knot sequences over finite domains. The quadrature nodes and weights are derived via an explicit recursion that avoids numerical solvers. Each rule is optimal, that is, requires the minimal number of nodes, for a given function space. For each of n subintervals, generically, only two nodes are required which reduces the evaluation cost by 2 / 3 when compared to the classical Gaussian quadrature for polynomials over each knot span. Numerical experiments show fast convergence, as n grows, to the “two-third” quadrature rule of Hughes et al. (2010) for infinite domains. [ABSTRACT FROM AUTHOR]

Every classical knot is band-pass equivalent to the unknot or the trefoil. The band-pass class of a knot is a concordance invariant. Every ribbon knot, for example, is band-pass equivalent to the unknot. Here we introduce the long virtual knot concordance group . It is shown that for every concordance class , there is a that is not band-pass equivalent to and an that is not band-pass equivalent to either the long unknot or any long trefoil. This is accomplished by proving that is a band-pass invariant but not a concordance invariant of long virtual knots, where and generate the degree two Polyak group for long virtual knots. [ABSTRACT FROM AUTHOR]

We introduce dual graph diagrams representing oriented knots and links. We use these combinatorial structures to define corresponding algebraic structures which we call biquasiles whose axioms are motivated by dual graph Reidemeister moves, generalizing the Dehn presentation of the knot group analogously to the way quandles and biquandles generalize the Wirtinger presentation. We use these structures to define invariants of oriented knots and links and provide examples. [ABSTRACT FROM AUTHOR]

In this paper, a regional knot invariant is constructed. Like the Wirtinger presentation of a knot group, each planar region contributes a generator, and each crossing contributes a relation. The invariant is called a tridle of the link. As in the quandle theory, one can define Alexander quandle and get Alexander polynomial from it. For link diagram, one can also define a linear tridle and its presentation matrix. A polynomial invariant can be derived from the matrix just like the Alexander polynomial case. [ABSTRACT FROM AUTHOR]

We extend the Wada representations of the classical braid group to the virtual and welded braid groups. Using the resulting representations, we construct the groups of virtual links and prove that they are link invariants. We give some examples of calculating the groups of torus (virtual) links. [ABSTRACT FROM AUTHOR]

In this paper, we give sufficient conditions on group presentations, with two generators and one relator, in order to be the group of a virtual knot diagram. Although those conditions are not enough, we use them to determine, completely, whether or not a Baumslag-Solitar group is the group of a two-bridge virtual knot. Moreover, we present a combinatorial proof of the fact that these groups are not two-bridge classical knot groups. [ABSTRACT FROM AUTHOR]

A knot is called minimal if its knot group admits epimorphisms onto the knot groups of only the trivial knot and itself. In this paper, we determine which two-bridge knot is minimal where or . [ABSTRACT FROM AUTHOR]

We show that some hyperbolic 3-manifolds which are tessellated by copies of the regular ideal hyperbolic tetrahedron are geodesically embedded in a complete, finite volume, hyperbolic 4-manifold. This allows us to prove that the complement of the figure-eight knot geometrically bounds a complete, finite volume hyperbolic 4-manifold. This is the first example of geometrically bounding hyperbolic knot complements and, amongst known examples of geometrically bounding manifolds, the one with the smallest volume. [ABSTRACT FROM AUTHOR]

In the present paper, a new representation of the virtual braid group into the automorphism group of free product of the free group and free abelian group is constructed. This representation generalizes the previously constructed ones. The fact that the previously known representations are not faithful for is verified. Using representations of , a virtual link group is defined. Also representations of the welded braid group are constructed and the welded link group is defined. [ABSTRACT FROM AUTHOR]

The volume density of a hyperbolic link is defined as the ratio of hyperbolic volume to crossing number. We study its properties and a closely-related invariant called the determinant density. It is known that the sets of volume densities and determinant densities of links are dense in the interval . We construct sequences of alternating knots whose volume and determinant densities both converge to any . We also investigate the distributions of volume and determinant densities for hyperbolic rational links, and establish upper bounds and density results for these invariants. [ABSTRACT FROM AUTHOR]

We define a family of virtual knots generalizing the classical twist knots. We develop a recursive formula for the Alexander polynomial (as defined by Silver and Williams [Polynomial invariants of virtual links, J. Knot Theory Ramifications 12 (2003) 987-1000]) of these virtual twist knots. These results are applied to provide evidence for a conjecture that the odd writhe of a virtual knot can be obtained from . [ABSTRACT FROM AUTHOR]

We show that a 2-knot group discovered in the course of a census of 4-manifolds with small triangulations is an HNN extension with finite base and proper associated subgroups, and has the smallest base among such knot groups. [ABSTRACT FROM AUTHOR]

We introduce the even index polynomial of virtual link diagrams and some even state sum polynomials which are the even Jones-Kauffman, even Miyazawa and even arrow polynomials. By simple computation, we can distinguish virtual links with these even polynomial invariants. Also, we give examples so that these invariants are different from the original ones. [ABSTRACT FROM AUTHOR]

We show that if the fundamental group of the complement of a rationally homologically fibered knot in a rational homology 3-sphere is bi-orderable, then its Alexander polynomial has at least one positive real root. Our argument can be applied for a finitely generated group which is an HNN extension with certain properties. [ABSTRACT FROM AUTHOR]

Let K be a knot in S³ and X its complement. We study deformations of non-abelian, metabelian, reducible representations of the knot group π1(X) into SL(n, C) which are associated to a simple root of the Alexander polynomial. We prove that some of these metabelian reducible representations are smooth points of the SL(n, C)-representation variety and that they have irreducible deformations. [ABSTRACT FROM AUTHOR]

When the signed weighted resolution set (or were-set) was defined as an invariant of pseudoknots, it was unknown whether this invariant was complete. Using the Gauss-diagrammatic invariants of pseudoknots introduced by Dorais et al., we show that the signed were-set cannot distinguish all non-equivalent pseudoknots. This goal is achieved through studying the effects of a flype-like local move on a pseudodiagram. [ABSTRACT FROM AUTHOR]

Twisted links are a generalization of virtual links. As virtual links correspond to abstract links on orientable surfaces, twisted links correspond to abstract links on (possibly non-orientable) surfaces. In this paper, we introduce the notion of the double covering of a twisted link. It is defined by considering the orientation double covering of an abstract link or alternatively by constructing a diagram called a double covering diagram. We also discuss links in thickened surfaces, their diagrams and their stable equivalence classes. Bourgoin's twisted knot group is understood as the virtual knot group of the double covering. [ABSTRACT FROM AUTHOR]

We give a new interpretation of the Alexander polynomial for virtual knots due to Sawollek [On Alexander-Conway polynomials for virtual knots and Links, preprint (2001), arXiv:math/9912173] and Silver and Williams [Polynomial invariants of virtual links, J. Knot Theory Ramifications 12 (2003) 987-1000], and use it to show that, for any virtual knot, determines the writhe polynomial of Cheng and Gao [A polynomial invariant of virtual links, J. Knot Theory Ramifications 22(12) (2013), Article ID: 1341002, 33pp.] (equivalently, Kauffman's affine index polynomial [An affine index polynomial invariant of virtual knots, J. Knot Theory Ramifications 22(4) (2013), Article ID: 1340007, 30pp.]). We also use it to define a second-order writhe polynomial, and give some applications. [ABSTRACT FROM AUTHOR]

It is well known that any knot group is torsion-free, but it may admit a generalized torsion element. We show that the knot group of any negative twist knot admits a generalized torsion element. This is a generalization of the same claim for the knot 52, which is the (-2)-twist knot, by Naylor and Rolfsen. [ABSTRACT FROM AUTHOR]

We give an extension of Fox's formula of the Alexander polynomial for a double branched cover over the three-sphere. Our formula provides the Reidemeister torsion of the double branched cover along a knot for a non-trivial 1-dimensional representation. In our formula, the Reidemeister torsion is given by the product of two factors derived from the knot group. One of the factors is determined by the twisted Alexander polynomial and the other is determined by a rational function on the character variety of the knot group. As an application, we show that these products distinguish the isotopy classes of two-bridge knots up to their mirror images. [ABSTRACT FROM AUTHOR]

We show that any parabolic generating pair of a genus-one hyperbolic 2-bridge knot group is equivalent to the upper or lower meridian pair. As an application, we obtain a complete classification of the epimorphisms from 2-bridge knot groups to genus-one hyperbolic 2-bridge knot groups. [ABSTRACT FROM AUTHOR]

In this paper, we study stable equivalence of exotically knotted surfaces in 4-manifolds, surfaces that are topologically isotopic but not smoothly isotopic. We prove that any pair of embedded surfaces in the same homology class become smoothly isotopic after stabilizing them by handle additions in the ambient 4-manifold, which can, moreover, be assumed to be attached in a standard way (locally and unknottedly) in many favorable situations. In particular, any exotically knotted pair of surfaces with cyclic fundamental group complements become smoothly isotopic after a same number of standard stabilizations, analogous to C.T.C. Wall's celebrated result on the stable equivalence of simply connected 4-manifolds.We, moreover, show that all constructions of exotic knottings of surfaces we are aware of, which display a good variety of techniques and ideas, produce surfaces that become smoothly isotopic after a single stabilization. [ABSTRACT FROM AUTHOR]

Either fibered knots supporting the tight contact structure are unique in their smooth concordance class or there exists a fibered counterexample to the Slice-Ribbon Conjecture. [ABSTRACT FROM AUTHOR]

Let L ⊂ S3 denote an alternating link and Σ(L) its branched double-cover. We give a short proof of the fact that the fundamental group of Σ(L) admits a left-ordering iff L is an unlink. This result is originally due to Boyer-Gordon-Watson. [ABSTRACT FROM AUTHOR]

Each pattern P in a solid torus gives a function P : C ? C on the smooth knot concordance group, taking any knot K to its satellite P(K). We give examples of winding number one patterns P and a class of knots K, such that the iterated satellites Pi(K) are distinct in concordance, i.e. if i ≄ j = 0, Pi(K) ≄ Pj (K). This implies that the operators Pi give distinct functions on C, providing further evidence for the (conjectured) fractal nature of C. Our theorem also allows us to construct several sets of examples, such as infinite families of topologically slice knots that are distinct in smooth concordance, infinite families of 2-component links (with unknotted components and linking number one) which are not smoothly concordant to the positive Hopf link, and infinitely many prime knots which have the same Alexander polynomial as an L-space knot but are not themselves L-space knots. [ABSTRACT FROM AUTHOR]

We consider the group of unrestricted virtual braids, describe its structure and explore its relations with fused links. Also, we define the groups of flat virtual braids and virtual Gauss braids and study some of their properties, in particular their linearity. [ABSTRACT FROM AUTHOR]

We introduce the warping crossing polynomial of an oriented knot diagram by using the warping degrees of crossing points of the diagram. Given a closed transversely intersected plane curve, we consider oriented knot diagrams obtained from the plane curve as states to take the sum of the warping crossing polynomials for all the states for the plane curve. As an application, we show that every closed transversely intersected plane curve with even crossing points has two independent canonical orientations and every based closed transversely intersected plane curve with odd crossing points has two independent canonical orientations. [ABSTRACT FROM AUTHOR]

Given a virtual knot K, we introduce a new group-valued invariant VGK called the virtual knot group, and we use the elementary ideals of VGK to define invariants of K called the virtual Alexander invariants. For instance, associated to the zeroth ideal is a polynomial HK(s, t, q) in three variables which we call the virtual Alexander polynomial, and we show that it is closely related to the generalized Alexander polynomial GK(s, t) introduced by Sawollek; Kauffman and Radford; and Silver and Williams. We define a natural normalization of the virtual Alexander polynomial and show it satisfies a skein formula. We also introduce the twisted virtual Alexander polynomial associated to a virtual knot K and a representation ϱ : VGK → GLn(R), and we define a normalization of the twisted virtual Alexander polynomial. As applications we derive bounds on the virtual crossing numbers of virtual knots from the virtual Alexander polynomial and twisted virtual Alexander polynomial. [ABSTRACT FROM AUTHOR]

Let G = 〈x, t | w〉 be a one-relator group, where w is a word in x, t. If w is a product of conjugates of x then, associated with w, there is a polynomial Aw(X) over the integers, which in the case when G is a knot group, is the Alexander polynomial of the knot. We prove, subject to certain restrictions on w, that if all roots of Aw(X) are real and positive then G is bi-orderable, and that if G is bi-orderable then at least one root is real and positive. This sheds light on the bi-orderability of certain knot groups and on a question of Clay and Rolfsen. One of the results relies on an extension of work of G. Baumslag on adjunction of roots to groups, and this may have independent interest. [ABSTRACT FROM AUTHOR]