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A classical theorem of braid theory, dating back to Artin's work, says that two closed braids in a solid torus are ambient isotopic if and only if they represent the same conjugacy class of the braid group. This theorem can be reformulated in the framework of link theory without referring to the group structure. A link in a surface bundle over the circle is transversal whenever it covers the circle. In this terminology, Artin's theorem states that in a solid torus trivially fibered over the circle transversal links are ambient isotopic if and only if they are isotopic in the class of transversal links. We generalize this result by proving that (in the piecewise linear category) transversal links in an arbitrary compact orientable -manifold fibered over the circle with a compact fiber are ambient isotopic if and only if they are isotopic in the class of transversal links. [ABSTRACT FROM AUTHOR]

In this paper, we study representations of G n 3 -like groups. The group G n 3 itself appeared in works of the third named author on non-Reidemeister knot (and braid) theory. This group is closely related to dynamical systems of points and their invariants. Representations of G n 3 -like groups are useful both for the study of those groups themselves, and constructing invariants of knots and braids based on the G n 3 -like group structure. [ABSTRACT FROM AUTHOR]

A twisted torus knot is a knot obtained from a torus knot by twisting adjacent strands by full twists. The twisted torus knots lie in , the genus 2 Heegaard surface for . Primitive/primitive and primitive/Seifert knots lie in in a particular way. Dean gives sufficient conditions for the parameters of the twisted torus knots to ensure they are primitive/primitive or primitive/Seifert. Using Dean's conditions, Doleshal shows that there are infinitely many twisted torus knots that are fibered and that there are twisted torus knots with distinct primitive/Seifert representatives with the same slope in . In this paper, we extend Doleshal's results to show there is a four parameter family of positive twisted torus knots. Additionally, we provide new examples of twisted torus knots with distinct representatives with the same surface slope in . [ABSTRACT FROM AUTHOR]

We give an explicit construction of complex maps whose nodal lines have the form of lemniscate knots. We review the properties of lemniscate knots, defined as closures of braids where all strands follow the same transverse (1, ℓ) Lissajous figure, and are therefore a subfamily of spiral knots generalizing the torus knots. We then prove that such maps exist and are in fact fibrations with appropriate choices of parameters. We describe how this may be useful in physics for creating knotted fields, in quantum mechanics, optics and generalizing to rational maps with application to the Skyrme-Faddeev model. We also prove how this construction extends to maps with weakly isolated singularities. [ABSTRACT FROM AUTHOR]

In [V. O. Manturov, ], the second author defined the -free braid group with strands . These groups appear naturally as groups describing dynamical systems of particles in some 'general position'. Moreover, in [V. O. Manturov and I. M. Nikonov, J. Knot Theory Ramification 24 (2015) 1541009] the second author and Nikonov showed that is closely related to classical braids. The authors showed that there are homomorphisms from the pure braids group on strands to and and they defined homomorphisms from to the free products of . That is, there are invariants for pure free braids by and . On the other hand in [D. A. Fedoseev and V. O. Manturov, J. Knot Theory Ramification 24(13) (2015) 1541005, 12 pages] Fedoseev and the second author studied classical braids with addition structures: parity and points on each strands. The authors showed that the parity, which is an abstract structure, has geometric meaning - points on strands. In [S. Kim, arXiv:submit/1548032], the first author studied with parity and points. the author constructed a homomorphism from to the group with parity. In the present paper, we investigate the groups and extract new powerful invariants of classical braids from . In particular, these invariants allow one to distinguish the non-triviality of Brunnian braids. [ABSTRACT FROM AUTHOR]

In previous papers (see e.g. [V. O. Manturov, Parity in knot theory, Mat. Sbornik 201(5) (2010) 65-110]), the author realized the following principle for many knot theories: if a knot diagram is complicated enough then it reproduces itself, i.e. a subdiagram of any other diagram equivalent to it. This principle is realized by diagram-valued invariants [⋅] of knots such that [K] = K for K complicated enough. It turns out that in the case of free braids, the same principle can be realized in an unexpectedly easy way by a one-term invariant formula. [ABSTRACT FROM AUTHOR]

We introduce a quasitoric virtual braid and show that every virtual link can be obtained by the closure of a quasitoric virtual braid. Also, we show that the set of quasitoric virtual braids with n strands forms a group which is a subgroup of the n-virtual braid group. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this paper we examine the relationship between various types of positivity for knots and the concordance invariant τ discovered by Ozsváth and Szabó and independently by Rasmussen. The main result shows that, for fibered knots, τ characterizes strong quasipositivity. This is quantified by the statement that for K fibered, τ(K) = g(K) if and only if K is strongly quasipositive. A corollary is that any knot admitting a lens space surgery or, more generally, an L-space surgery, is strongly quasipositive. In addition, we survey existing results regarding τ and forms of positivity and highlight several consequences concerning the types of knots which are (strongly) (quasi) positive. [ABSTRACT FROM AUTHOR]

Fix a straight line L in Euclidean 3-space and consider the fibration of the complement of L by half-planes. A generic knot K in the complement of L has neither fiber quadrisecants nor fiber extreme secants such that K touches the corresponding half-plane at 2 points. Both types of secants occur in generic isotopies of knots. We give lower bounds for the number of these fiber secants in all isotopies connecting given isotopic knots. The bounds are expressed in terms of invariants calculable in linear time with respect to the number of crossings. [ABSTRACT FROM AUTHOR]

We review the q-deformed spin network approach to Topological Quantum Field Theory and apply these methods to produce unitary representations of the braid groups that are dense in the unitary groups. Our methods are rooted in the bracket state sum model for the Jones polynomial. We give our results for a large class of representations based on values for the bracket polynomial that are roots of unity. We make a separate and self-contained study of the quantum universal Fibonacci model in this framework. We apply our results to give quantum algorithms for the computation of the colored Jones polynomials for knots and links, and the Witten–Reshetikhin–Turaev invariant of three manifolds. [ABSTRACT FROM AUTHOR]

No abstract received. [ABSTRACT FROM AUTHOR]