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For a virtual knot K and an integer r with r ≥ 2, we introduce a method of constructing an r-component virtual link L(K; r), which we call the r-multiplexing of K. Every invariant of L(K; r) is an invariant of K. We give a way of calculating three kinds of invariants of L(K; r) using invariants of K. As an application of our method, we also show that Manturov’s virtual n-colorings for K can be interpreted as certain classical n-colorings for L(K; 2). [ABSTRACT FROM AUTHOR]

We prove a simple necessary and sufficient condition for a two-bridge knot K (p , q) to be quasipositive, based on the continued fraction expansion of p / q. As an application, coupled with some classification results in contact and symplectic topology, we give a new proof of the fact that smoothly slice two-bridge knots are non-quasipositive. Another proof of this fact using methods within the scope of knot theory is presented in Appendix A, by Stepan Orevkov. [ABSTRACT FROM AUTHOR]

The 3-loop invariant (or, the 3-loop polynomial) of a knot is a rational form (or, a polynomial) presenting the 3-loop part of the Kontsevich invariant of knots. In this paper, we calculate the 3-loop polynomial of knots obtained by plumbing the doubles of two knots; this class of knots includes untwisted Whitehead doubles. We construct the 3-loop invariant by calculating the rational version of the Aarhus integral of a surgery presentation. As a consequence, we obtain an explicit presentation of the 3-loop polynomial for the knots. [ABSTRACT FROM AUTHOR]

For an L-space knot, the formal semigroup is defined from its Alexander polynomial. It is not necessarily a semigroup. That is, it may not be closed under addition. There exists an infinite family of hyperbolic L-space knots whose formal semigroups are semigroups generated by three elements. In this paper, we give the first infinite family of hyperbolic L-space knots whose formal semigroups are semigroups generated by five elements. [ABSTRACT FROM AUTHOR]

An old and challenging conjecture proposed by Fox in 1962 states that the coefficients of the Alexander polynomial of any alternating knot are trapezoidal. In other words, these coefficients, increase, stabilize then decrease in a symmetrical way. This curious behavior of the Alexander polynomial has been confirmed in several special cases of alternating knots. The purpose of this paper is to prove that the conjecture holds for some families of alternating knots of braid index 3. [ABSTRACT FROM AUTHOR]

In this paper, we obtain the crosscap number of any alternating knots by using our recently-introduced diagrammatic knot invariant (Theorem 1). The proof is given by properties of chord diagrams (Kindred proved Theorem 1 independently via other techniques). For non-alternating knots, we give Theorem 2 that generalizes Theorem 1. We also improve known formulas to obtain upper bounds of the crosscap number of knots (alternating or non-alternating) (Theorem 3). As a corollary, this paper connects crosscap numbers and our invariant with other knot invariants such as the Jones polynomial, twist number, crossing number, and hyperbolic volume (Corollaries 1–7). In Appendix A, using Theorem 1, we complete giving the crosscap numbers of the alternating knots with up to 11 crossings including those of the previously unknown values for 1 9 3 knots (Tables A.1). [ABSTRACT FROM AUTHOR]

Ito–Takimura recently defined a splice-unknotting number u − (D) for knot diagrams. They proved that this number provides an upper bound for the crosscap number of any prime knot, asking whether equality holds in the alternating case. We answer their question in the affirmative. (Ito has independently proven the same result.) As an application, we compute the crosscap numbers of all prime alternating knots through at least 13 crossings, using Gauss codes. [ABSTRACT FROM AUTHOR]

Niebrzydowski 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 introduce tribracket modules analogous to quandle/biquandle/rack modules and use these structures to enhance the tribracket counting invariant. We provide examples to illustrate the computation of the invariant and show that the enhancement is proper. [ABSTRACT FROM AUTHOR]

In this paper, we give a definition of ℤ -valued functions from the ambient isotopy classes of spherical/plane curves derived from chord diagrams, denoted by ∑ i α i x i . Then, we introduce certain elements of the free ℤ -module generated by the chord diagrams with at most l chords, called relators of Type (I) ((SI I), (WI I), (SI I I), or (WI I I), respectively), and introduce another function ∑ i α i x ̃ i derived from ∑ i α i x i . The main result (Theorem 1) shows that if ∑ i α i x ̃ i vanishes for the relators of Type (I) ((SI I), (WI I), (SI I I), or (WI I I), respectively), then ∑ i α i x i is invariant under the Reidemeister move of type RI (strong RI I, weak RI I, strong RI I I, or weak RI I I, respectively) that is defined in [N. Ito and Y. Takimura, (1 , 2) and weak (1 , 3) homotopies on knot projections, J. Knot Theory Ramifications22 (2013) 1350085 14 pp]. [ABSTRACT FROM AUTHOR]

Bott and Taubes used integrals over configuration spaces to produce finite-type a.k.a. Vassiliev knot invariants. Cattaneo, Cotta-Ramusino and Longoni then used these methods together with graph cohomology to construct "Vassiliev classes" in the real cohomology of spaces of knots in higher-dimensional Euclidean spaces, as first promised by Kontsevich. Here we construct integer-valued cohomology classes in spaces of knots and links in ℝ d for d > 3. We construct such a class for any integer-valued graph cocycle, by the method of gluing compactified configuration spaces. Our classes form the integer lattice among the previously discovered real cohomology classes. Thus we obtain nontrivial classes from trivalent graph cocycles. Our methods generalize to yield mod- p classes out of mod- p graph cocycles, which need not be reductions of classes over the integers. [ABSTRACT FROM AUTHOR]

In the setting of finite type invariants for null-homologous knots in rational homology 3-spheres with respect to null Lagrangian-preserving surgeries, there are two candidates to be universal invariants, defined, respectively, by Kricker and Lescop. In a previous paper, the second author defined maps between spaces of Jacobi diagrams. Injectivity for these maps would imply that Kricker and Lescop invariants are indeed universal invariants; this would prove in particular that these two invariants are equivalent. In the present paper, we investigate the injectivity status of these maps for degree 2 invariants, in the case of knots whose Blanchfield modules are direct sums of isomorphic Blanchfield modules of ℚ -dimension two. We prove that they are always injective except in one case, for which we determine explicitly the kernel. [ABSTRACT FROM AUTHOR]

From the viewpoint of contact round surgery, the Lutz twist and the Giroux torsion on contact 3 -manifolds are studied in this paper. It is proved that the Lutz twist along a transverse knot and that along a pre-Lagrangian torus are realized by contact round surgeries of index  1 and  2 of contact 3 -manifolds. This construction gives an important perspective on the Lutz twists. Both can be regarded as fiberwise connected sums with the compactified Lutz twist elements. [ABSTRACT FROM AUTHOR]

We introduce an unknotting-type number of knot projections that gives an upper bound of the crosscap number of knots. We determine the set of knot projections with the unknotting-type number at most two, and this result implies classical and new results that determine the set of alternating knots with the crosscap number at most two. [ABSTRACT FROM AUTHOR]

We determine the SL (2 , ℂ) -character variety for each odd classical pretzel knot P (2 k 1 + 1 , 2 k 2 + 1 , 2 k 3 + 1) , and present a method for computing its A-polynomial. [ABSTRACT FROM AUTHOR]

We introduce a method of computing biquandle brackets of oriented knots and links using a type of decorated trivalent spatial graphs we call trace diagrams. We identify algebraic conditions on the biquandle bracket coefficients for moving strands over and under traces and identify a new stop condition for the recursive expansion. In the case of monochromatic crossings we show that biquandle brackets satisfy a Homflypt-style skein relation and we identify algebraic conditions on the biquandle bracket coefficients to allow pass-through trace moves. [ABSTRACT FROM AUTHOR]

In his famous Princeton Notes, Thurston introduced the so-called gluing equations defining the deformation variety. Later, Kashaev defined a noncommutative ring from H-triangulations of 3-manifolds and observed that for trefoil and figure-eight knot complements the abelianization of this ring is isomorphic to the ring of regular functions on the deformation variety, Kashaev, -groupoids in knot theory, Geom. Dedicata 150(1) (2010) 105-130; Kashaev, Noncommutative teichmüller spaces and deformation varieties of knot completeness; Kashaev, Delta-groupoids and ideal triangulation in Chern-Simons gauge theory: 20 Years After, AMS/IP Studies Advanced Mathematics, Vol. 50 (American Mathematical Society, RI, 2011). In this paper, we prove that this is true for any knot complement in a homology sphere. We also analyze some examples on other manifolds. [ABSTRACT FROM AUTHOR]

We study knots of order in the grope filtration and the solvable filtration of the knot concordance group. We show that, for any integer , there are knots generating a subgroup of . Considering the solvable filtration, our knots generate a subgroup of distinct from the subgroup generated by the previously known -torsion knots of Cochran, Harvey, and Leidy. We also present a result on the -torsion part in the Cochran, Harvey, and Leidy's primary decomposition of the solvable filtration. [ABSTRACT FROM AUTHOR]

Twisted Alexander invariants of knots are well defined up to multiplication of units. We get rid of this multiplicative ambiguity via a combinatorial method and define normalized twisted Alexander invariants. We then show that the invariants coincide with sign-determined Reidemeister torsion in a normalized setting, and refine the duality theorem. We further obtain necessary conditions on the invariants for a knot to be fibered, and study behavior of the highest degrees of the invariants. [ABSTRACT FROM AUTHOR]

We provide linear lower bounds for the signature of positive braids in terms of the three-genus of their braid closure. This yields linear bounds for the topological slice genus of knots that arise as closures of positive braids. [ABSTRACT FROM AUTHOR]

A knot projection is an image of a generic immersion from a circle into a two-dimensional sphere. We can find homotopies between any two knot projections by local replacements of knot projections of three types, called Reidemeister moves. This paper defines an equivalence relation for knot projections called weak (1, 2, 3) homotopy, which consists of Reidemeister moves of type 1, weak type 2, and weak type 3. This paper defines the first nontrivial invariant under weak (1, 2, 3) homotopy. We use this invariant to show that there exist an infinite number of weak (1, 2, 3) homotopy equivalence classes of knot projections. By contrast, all equivalence classes of knot projections consisting of the other variants of a triple type, i.e. Reidemeister moves of (1, strong type 2, strong type 3), (1, weak type 2, strong type 3), and (1, strong type 2, weak type 3), are contractible. [ABSTRACT FROM AUTHOR]

Applying the concept of braiding sequences and the inequality between the signature and number of roots of the Alexander polynomial on the unit circle, we prove that only finitely many special alternating knots are (even algebraically) concordant, in that their concordance class determines their Alexander polynomial. We discuss some extensions of this result to positive and almost positive knots, and links. [ABSTRACT FROM AUTHOR]

We study the twisted Alexander polynomial from the viewpoint of the SL(2, ℂ)-character variety of nonabelian representations of a knot group. It is known that if a knot is fibered, then the twisted Alexander polynomials associated with nonabelian SL(2, ℂ)-representations are all monic. In this paper, we show that for a 2-bridge knot there exists a curve component in the SL(2, ℂ)-character variety such that if the knot is not fibered then there are only finitely many characters in the component for which the associated twisted Alexander polynomials are monic. We also show that for a 2-bridge knot of genus g, in the above curve component for all but finitely many characters the associated twisted Alexander polynomials have degree 4g - 2. [ABSTRACT FROM AUTHOR]

A (t, s)-rack is a rack structure defined on a module over the ring $\ddot\Lambda = {\mathbb Z}[t^{\pm 1}, s]/(s^2-(1-t)s)$. We identify necessary and sufficient conditions for two (t, s)-racks to be isomorphic. We define enhancements of the rack counting invariant using the structure of (t, s)-racks and give some computations and examples. As an application, we use these enhanced invariants to obtain obstructions to knot ordering. [ABSTRACT FROM AUTHOR]

It is known that S4 is a union of two fishtails, and S2 × S2 is a union of two cusps (glued along their boundaries). Here we prove that for any choice of knots K, L ⊂ S3 the knot surgery operations $S^{4} \rightsquigarrow S^4_{K, L}$ and S2 × S2 ⇝ (S2 × S2)K, L along both of these fishtails and cusps, respectively, do not change the diffeomorphism type of these manifolds. [ABSTRACT FROM AUTHOR]

It is known that S4 is a union of two fishtails, and S2 × S2 is a union of two cusps (glued along their boundaries). Here we prove that for any choice of knots K, L ⊂ S3 the knot surgery operations $S^{4} \rightsquigarrow S^4_{K, L}$ and S2 × S2 ⇝ (S2 × S2)K, L along both of these fishtails and cusps, respectively, do not change the diffeomorphism type of these manifolds. [ABSTRACT FROM AUTHOR]

Let G be a group of type Fn with n ≥ 2, i.e. G admits a finite n-dimensional (n - 1)-aspherical CW-complex X such that π1(X) is isomorphic to G. We introduce the notion of the n-deficiency dn(G) which is a generalized notion of deficiency of G and present various results about the relation between dn(G) and the L2-Betti numbers of G. We also give some partial results about the Eilenberg-Ganea conjecture and the Whitehead conjecture. These results correspond to a generalization of several earlier results [Quart. J. Math. Oxford (2)38 (1987) 35-44; J. Pure Appl. Algebra7 (1976) 241-248; Quart. J. Math. Oxford (2)32 (1981) 45-55; 2-Knots and Their Groups (Cambridge University Press, 1989)]. [ABSTRACT FROM AUTHOR]

In this paper, the notion of the generic knots in contact 3-manifolds is introduced as an extension of that of the transversal knots. We show that any generic knots in contact 3-manifolds are constructed by perturbations of some Legendrian knots. As for the classification problem, the situation is completely different from the cases of the transversal and Legendrian knots. Two generic knots in a contact 3-manifold are generically isotopic if and only if they have the same number of non-transversal points and belong to the same topological knot class. We treat, in this paper, not only trivial knots in tight contact 3-manifolds but also non-trivial knots and those in overtwisted contact 3-manifolds. [ABSTRACT FROM AUTHOR]

We introduce the beta function of a knot in Euclidean three-space. This is a meromorphic function of a complex variable which we prove admits a Bernstein type functional equation. We determine the first residues. [ABSTRACT FROM AUTHOR]