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Recently, the first named author defined a 2-parametric family of groups Gnk [V. O. Manturov, Non-reidemeister knot theory and its applications in dynamical systems, geometry and topology, preprint (2015), arXiv:1501.05208]. Those groups may be regarded as analogues of braid groups. Study of the connection between the groups Gnk and dynamical systems led to the discovery of the following fundamental principle: "If dynamical systems describing the motion of n particles possess a nice codimension one property governed by exactly kk particles, then these dynamical systems admit a topological invariant valued in Gnk". The Gnk groups have connections to different algebraic structures, Coxeter groups, Kirillov-Fomin algebras, and cluster algebras, to name three. Study of the Gnk groups led to, in particular, the construction of invariants, valued in free products of cyclic groups. All generators of the Gnk groups are reflections which make them similar to Coxeter groups and not to braid groups. Nevertheless, there are many ways to enhance Gnk groups to get rid of this 22-torsion. Later the first and the fourth named authors introduced and studied the second family of groups, denoted by Γnk, which are closely related to triangulations of manifolds. The spaces of triangulations of a given manifolds have been widely studied. The celebrated theorem of Pachner [P.L. homeomorphic manifolds are equivalent by elementary shellings, Europ. J. Combin.12(2) (1991) 129-145] says that any two triangulations of a given manifold can be connected by a sequence of bistellar moves or Pachner moves. See also [I. M. Gelfand, M. M. Kapranov and A. V. Zelevinsky, Discriminants, Resultants, and Multidimensional Determinants (Birkhäuser, Boston, 1994); A. Nabutovsky, Fundamental group and contractible closed geodesics, Comm. Pure Appl. Math.49(12) (1996) 1257-1270]; the Γnk naturally appear when considering the set of triangulations with the fixed number of points. There are two ways of introducing the groups Γnk: the geometrical one, which depends on the metric, and the topological one. The second one can be thought of as a "braid group" of the manifold and, by definition, is an invariant of the topological type of manifold; in a similar way, one can construct the smooth version. In this paper, we give a survey of the ideas lying in the foundation of the Gnk and Γnk theories and give an overview of recent results in the study of those groups, manifolds, dynamical systems, knot and braid theories. [ABSTRACT FROM AUTHOR]

We provide a physical definition of new homological invariants ℋ a (M 3) of 3-manifolds (possibly, with knots) labeled by abelian flat connections. The physical system in question involves a 6d fivebrane theory on M 3 times a 2-disk, D 2 , whose Hilbert space of BPS states plays the role of a basic building block in categorification of various partition functions of 3d 𝒩 = 2 theory T [ M 3 ] : D 2 × S 1 half-index, S 2 × S 1 superconformal index, and S 2 × S 1 topologically twisted index. The first partition function is labeled by a choice of boundary condition and provides a refinement of Chern–Simons (WRT) invariant. A linear combination of them in the unrefined limit gives the analytically continued WRT invariant of M 3 . The last two can be factorized into the product of half-indices. We show how this works explicitly for many examples, including Lens spaces, circle fibrations over Riemann surfaces, and plumbed 3-manifolds. [ABSTRACT FROM AUTHOR]

We construct a complete invariant of oriented connected closed surfaces in 𝕊 3 , which generalizes the notion of peripheral system of a knot group. As an application, we define two computable invariants to investigate handlebody knots and bi-knotted surfaces with homeomorphic complements. In particular, we obtain an alternative proof of inequivalence of Ishii, Kishimoto, Moriuchi and Suzuki's handlebody knots 5 1 and 6 4 . [ABSTRACT FROM AUTHOR]

The Chen–Yang volume conjecture [Q. Chen and T. Yang, Volume conjectures for the Reshetikhin–Turaev and the Turaev–Viro invariants, preprint (2015), arXiv:1503.02547] states that the growth rate of the Turaev–Viro invariants of a compact oriented 3 -manifold determines its simplicial volume. In this paper, we prove that the Chen–Yang conjecture is stable under (2 n + 1 , 2) -cabling. [ABSTRACT FROM AUTHOR]

We study new skein invariants of links based on a procedure where we first apply a given skein relation only to crossings of distinct components, so as to produce collections of unlinked knots. We then evaluate the resulting knots using the given invariant. A skein invariant can be computed on each link solely by the use of skein relations and a set of initial conditions. The new procedure, remarkably, leads to generalizations of the known skein invariants. We make skein invariants of classical links, H [ R ] , K [ Q ] and D [ T ] , based on the invariants of knots, R , Q and T , denoting the regular isotopy version of the Homflypt polynomial, the Kauffman polynomial and the Dubrovnik polynomial. We provide skein theoretic proofs of the well-definedness of these invariants. These invariants are also reformulated into summations of the generating invariants (R , Q , T) on sublinks of a given link L , obtained by partitioning L into collections of sublinks. These summations exhibit the tight and surprising relationship between our generalized skein-theoretic procedure and the structure of sublinks of a given link. [ABSTRACT FROM AUTHOR]

Let ν be either the Ozsváth–Szabó τ–invariant or the Rasmussen s–invariant, suitably normalized. For a knot K, Livingston and Naik defined the invariant tν(K) to be the minimum of k for which ν of the k–twisted positive Whitehead double of K vanishes. They proved that tν(K) is bounded above by −TB(−K), where TB is the maximal Thurston–Bennequin number. We use a blowing-up process to find a crossing change formula and a new upper bound for tν in terms of the unknotting number. As an application, we present infinitely many knots K such that the difference between Livingston–Naik’s upper bound −TB(−K) and tν(K) can be arbitrarily large. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a family of transverse invariants arising from the deformations of Khovanov homology. This family includes the invariants introduced by Plamenevskaya and by Lipshitz, Ng, and Sarkar. Then, we investigate the invariants arising from Bar-Natan's deformation. These invariants, called β -invariants, are essentially equivalent to Lipshitz, Ng, and Sarkar's invariants ψ ± . From the β -invariants, we extract two non-negative integers which are transverse invariants (the c -invariants). Finally, we give several conditions which imply the non-effectiveness of the c -invariants, and use them to prove several vanishing criteria for the Plamenevskaya invariant [ ψ ] , and the non-effectiveness of the vanishing of [ ψ ] , for all prime knots with less than 12 crossings. [ABSTRACT FROM AUTHOR]

We consider a quiver structure on the set of quandle colorings of an oriented knot or link diagram. This structure contains a wealth of knot and link invariants and provides a categorification of the quandle counting invariant in the most literal sense, i.e. giving the set of quandle colorings the structure of a small category which is unchanged by Reidemeister moves. We derive some new enhancements of the counting invariant from this quiver structure and show that the enhancements are proper with explicit examples. [ABSTRACT FROM AUTHOR]

We prove that every Legendrian knot in the tight contact structure of the 3 -sphere is determined by the contactomorphism type of its exterior. Moreover, by giving counterexamples we show this to be not true for Legendrian links in the tight 3 -sphere. On the way a new user-friendly formula for computing the Thurston–Bennequin invariant of a Legendrian knot in a surgery diagram is given. [ABSTRACT FROM AUTHOR]

We introduce two sequences of two-variable polynomials { L K n (t , ℓ) } n = 1 ∞ and { F K n (t , ℓ) } n = 1 ∞ , expressed in terms of index value of a crossing and n -dwrithe value of a virtual knot K , where t and ℓ are variables. Basing on the fact that n -dwrithe is a flat virtual knot invariant, we prove that L K n and F K n are virtual knot invariants containing Kauffman affine index polynomial as a particular case. Using L K n we give sufficient conditions when virtual knot does not admit cosmetic crossing change. [ABSTRACT FROM AUTHOR]

In the paper [13], for an arbitrary virtual link L , three groups G 1 , r (L) , r > 0 , G 2 (L) and G 3 (L) were defined. In the present paper, these groups for the virtual trefoil are investigated. The structure of these groups are found out and the fact that some of them are not isomorphic to each other is proved. Also, we prove that G 3 distinguishes the Kishino knot from the trivial knot. The fact that these groups have the lower central series which does not stabilize on the second term is noted. Hence, we have a possibility to study these groups using quotients by terms of the lower central series and to construct representations of these groups in rings of formal power series. It allows to construct an invariants for virtual knots. [ABSTRACT FROM AUTHOR]

The knot coloring polynomial defined by Eisermann for a finite pointed group is generalized to an infinite pointed group as the longitudinal mapping invariant of a knot. In turn, this can be thought of as a generalization of the quandle 2-cocycle invariant for finite quandles. If the group is a topological group, then this invariant can be thought of as a topological generalization of the 2-cocycle invariant. The longitudinal mapping invariant is based on a meridian–longitude pair in the knot group. We also give an interpretation of the invariant in terms of quandle colorings of a 1-tangle for generalized Alexander quandles without use of a meridian–longitude pair in the knot group. The invariant values are concretely evaluated for the torus knots T (2 , n) , their mirror images, and the figure eight knot for the group SU(2). [ABSTRACT FROM AUTHOR]

We introduce a notion of a chain of Yang–Baxter like operations. This is a sequence of solutions of an asymmetric variant of the Yang–Baxter equation and is a multi-operator generalization of (bi)rack/quandles. We discuss knot and link invariants coming from a chain of Yang–Baxter like operations, and give potential applications. Among them, we define a cocycle invariant for transverse links. [ABSTRACT FROM AUTHOR]

We investigate some algebraic structures called quasi-trivial quandles and we use them to study link-homotopy of pretzel links. Precisely, a necessary and sufficient condition for a pretzel link with at least two components being trivial under link-homotopy is given. We also generalize the quasi-trivial quandle idea to the case of biquandles and consider enhancement of the quasi-trivial biquandle cocycle counting invariant by quasi-trivial biquandle cocycles, obtaining invariants of link-homotopy type of links analogous to the quasi-trivial quandle cocycle invariants in Inoue's paper [A. Inoue, Quasi-triviality of quandles for link-homotopy, J. Knot Theory Ramifications22(6) (2013) 1350026, doi:10.1142/S0218216513500260, MR3070837]. [ABSTRACT FROM AUTHOR]

The warping sum e (K) of a knot K is the minimal value of the sum of the warping degrees of a minimal diagram of K with both orientations. In this paper, knots K with e (K) ≤ 3 are characterized, and some knots K with e (K) = 4 are given. [ABSTRACT FROM AUTHOR]

In this article we will construct a knot invariant which is complementary to HOMFLY knot invariant by proving that the skein relation is not compatible with HOMFLY’s. [ABSTRACT FROM AUTHOR]

We give presentations of the asymptotic expansions of the Kashaev invariant of the knots with 6 crossings. In particular, we show the volume conjecture for these knots, which states that the leading terms of the expansions present the hyperbolic volume and the Chern--Simons invariant of the complements of the knots. As higher coefficients of the expansions, we obtain a new series of invariants of these knots. A non-trivial part of the proof is to apply the saddle point method to calculate the asymptotic expansion of an integral which presents the Kashaev invariant. A key step of this part is to give a concrete homotopy of the (real 3-dimensional) domain of the integral in ℂ3 in such a way that the boundary of the domain always stays in a certain domain in ℂ3 given by the potential function of the hyperbolic structure. [ABSTRACT FROM AUTHOR]

We study character varieties of symmetric knots and their reductions mod p. We observe that the varieties present a different behaviour according to whether the knots admit a free or periodic symmetry. [ABSTRACT FROM AUTHOR]

The ordinary unknotting number of 1-dimensional knots, which is defined by using the crossing-change, is a very basic and important invariant. Let n be a positive integer. It is very natural to consider the "unknotting number" associated with other local moves on n -dimensional knots. In this paper, we prove the following. For the ribbon-move on 2-knots, which is a local move on knots, we have the following: There is a 2-knot which is changed into the unknot by two times of the ribbon-move not by one time. The "unknotting number" associated with the ribbon-move is unbounded. For the pass-move on 1-knots, which is a local move on knots, we have the following: There is a 1-knot such that it is changed into the unknot by two times of the pass-move not by one time and such that the ordinary unknotting number is 4. For any positive integer n , there is a 1-knot whose "unknotting number" associated with the pass-move is > n and whose ordinary unknotting number is 4 n. Let p and q be positive integers. For the (p , q) -move on (p + q − 1) -knots, which is a local move on knots, we have the following: Let k be a non-negative integer. There is a (4 k + 2) -knot which is changed into the unknot by two times of the (2 k + 1 , 2 k + 2) -move not by one time. The "unknotting number" associated with the (2 k + 1 , 2 k + 2) -move is unbounded. There is a (4 k + 1) -knot which is changed into the unknot by two times of the (2 k + 1 , 2 k + 1) -move not by one. The "unknotting number" associated with the (2 k + 1 , 2 k + 1) -move is unbounded. We prove the following: For any positive integer n and any positive integer m < n , there is a (4 k + 1) -knot which is changed into the unknot by n times of the twist-move not by m times. [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]

In this paper, the ♯ ¯ -move is defined. We show that for any knot K 0 , there exists an infinite family of knots { K 0 , K 1 , … } such that the Gordian distance d (K i , K j) = 1 and pass-move-Gordian distance d p (K i , K j) = 1 for any i ≠ j. We also show that there is another infinite family of knots { K 0 ′ , K 1 ′ , … } (where K 0 ′ = K 0 ) such that the ♯ ¯ -move-Gordian distance d ♯ ¯ (K i ′ , K j ′) = 1 and H (n) -Gordian distance d H (n) (K i ′ , K j ′) = 1 for any i ≠ j and all n ≥ 2. [ABSTRACT FROM AUTHOR]

The Thurston-Bennequin invariant provides one notion of self-linking for any homologically trivial Legendrian curve in a contact three-manifold. Here we discuss related analytic notions of self-linking for Legendrian knots in R 3 . Our definition is based upon reformulation of the elementary Gauss linking integral and is motivated by ideas from supersymmetric gauge theory. We recover the Thurston-Bennequin invariant as a special case. [ABSTRACT FROM AUTHOR]

Abstract: We prove that fibered, −amphicheiral knots with irreducible Alexander polynomials are rationally slice. This contrasts with the result of Miyazaki that ( 2 n , 1 )‐cables of these knots are not ribbon. We also show that the concordance invariants ν + and Υ from Heegaard Floer homology vanish for a class of knots that includes rationally slice knots. In particular, the ν +‐ and Υ‐invariants vanish for these cable knots. [ABSTRACT FROM AUTHOR]

We consider involutory virtual biracks with good involutions, also known as symmetric involutory virtual biracks. Any good involution on an involutory virtual birack defines an enhancement of the counting invariant. We provide examples demonstrating that the enhancement is stronger than the unenhanced counting invariant. [ABSTRACT FROM AUTHOR]

In this paper, we describe a method of making an invariant of virtual knots that is defined in terms of an integer labeling of the flat virtual knot diagram. We give an invariant of flat virtual knots using the invariant above. Moreover, we derive a relation of two invariants. [ABSTRACT FROM AUTHOR]

We construct an enhanced version of knot contact homology, and show that we can deduce from it the group ring of the knot group together with the peripheral subgroup. In particular, it completely determines a knot up to smooth isotopy. The enhancement consists of the (fully noncommutative) Legendrian contact homology associated to the union of the conormal torus of the knot and a disjoint cotangent fiber sphere, along with a product on a filtered part of this homology. As a corollary, we obtain a new, holomorphic-curve proof of a result of the third author that the Legendrian isotopy class of the conormal torus is a complete knot invariant. [ABSTRACT FROM AUTHOR]

We work with a generalization of knot theory, in which one diagram is reachable from another via a finite sequence of moves if a fixed condition regarding the existence of certain morphisms in an associated category is satisfied for every move of the sequence. This conditional setting leads to a possibility of irreversible moves, terminal states, and to using functors more general than the ones used as knot invariants. Our main focus is the category of diagrams with a binary relation on the set of arcs, indicating which arc can move over another arc. We define homology of binary relations, and merge it with quandle homology, to obtain the homology for partial quandles with a binary relation. This last homology can be used to analyze link diagrams with a binary relation on the set of components. [ABSTRACT FROM AUTHOR]

In [Przytyski and Traczyk, Invariants of links of Conway type, Kobe J. Math. 4 (1989) 115-139], Przytyski and Traczyk introduced an algebraic structure, called a Conway algebra, and constructed an invariant of oriented links, which is a generalization of the Homflypt polynomial invariant. On the other hand, in [Kauffman and Lambropoulou, New invariants of links and their state sum models, arXiv:1703.03655v2 [math.GT] 15 Mar 2017], Kauffman and Lambropoulou introduced new 4-variable invariants of oriented links, which are obtained by two computational steps: in the first step, we apply a skein relation on every mixed crossing to produce unions of unlinked knots. In the second step, we apply another skein relation on crossings of the unions of unlinked knots, which introduces a new variable. In this paper, we will introduce a generalization of the Conway algebra b  with two binary operations and we construct an invariant valued in b A by applying those two binary operations to mixed crossings and pure crossing, respectively. The 4-variable invariant of Kauffman and Lambropoulou with a specific condition is derived from the invariant valued in b Â. Moreover, the generalized Conway algebra gives us an invariant of oriented links, which satisfies nonlinear skein relations. [ABSTRACT FROM AUTHOR]

We introduce a new topological invariant ν(X) ∊ N ∪ {0} of compact manifoldswith-boundaries X which is much connected with boundary-unions. A boundary-unionis a kind of decomposition of compact manifolds-with-boundaries. See the body of thepaper for the precise definition. Let M and N be m-dimensional compact manifoldswith-boundaries. Let M ∪∂ N be a boundary-union of M and N. Then we have0≤ν(M∪∂N)≤ max{υ(M), ν(N)}. We define υ(X) as follows: First, define an invariant of (n - 1)-closed manifolds. Take the maximum of the invariant of all connected-components of the boundary of each handle-body of an ordered-handle-decomposition with a fixed base A, where we impose the condition that the base A is a (not necessarily connected) closed manifold. Take the minimum of the maximum for all ordered-handle-decompositions with the base A. It is our another invariant υ(X,A). Take the maximum of the minimum,υ(X, ), for all basis to satisfy the above condition. It is υ(X). See the body of the paper for the precise definition. [ABSTRACT FROM AUTHOR]

In this paper, we consider generalizations of the Alexander polynomial and signature of 2-bridge knots by considering the Gordon-Litherland bilinear forms associated with essential state surfaces of the 2-bridge knots. We show that the resulting invariants are well-defined and explore properties of these invariants. Finally, we realize the boundary slopes of essential surfaces as differences of signatures of the knot. [ABSTRACT FROM AUTHOR]

Geometric interpretations of some virtual knot invariants are given in terms of invariants of links in S³. Alexander polynomials of almost classical knots are shown to be specializations of the multi- variable Alexander polynomial of certain two-component boundary links of the form J ⊔K with J a fibered knot. The index of a crossing, a common ingredient in the construction of virtual knot invariants, is related to the Milnor triple linking number of certain three-component links J⊔K1⊔K2 with J a connected sum of trefoils or figure-eights. Our main technical tool is virtual covers. This technique, due to Manturov and the first author, associates a virtual knot u to a link J ⊔ K, where J is fibered and lk(J;K) = 0. Here we extend virtual covers to all multi-component links L = J⊔ K, with K a knot. It is shown that an unknotted component J0 can be added to L so that J0⊔J is fibered and K has algebraic intersection number zero with a fiber of J0 ⊔ J. This is called fiber stabilization. It provides an avenue for studying all links with virtual knots. [ABSTRACT FROM AUTHOR]

We use twisted Alexander polynomials to show that certain algebraically slice 2-bridge knots are not topologically slice, even though all prime power Casson–Gordon signatures vanish. We also provide some computations indicating the efficacy of Casson–Gordon signatures in obstructing the smooth sliceness of 2-bridge knots. [ABSTRACT FROM PUBLISHER]

We show that if the connected sum of two knots with coprime Alexander polynomials is doubly slice, then the Ozsváth-Szabó correction terms, as smooth double sliceness obstructions, vanish for both knots. Recently, Meier gave smoothly slice knots that are topologically doubly slice, but not smoothly doubly slice. As an application, we give a new example of such a knot that is distinct from Meier's knots modulo doubly slice knots. [ABSTRACT FROM AUTHOR]

The Tait conjecture states that alternating reduced diagrams of links in have the minimal number of crossings. It has been proved in 1987 by Thistlethwaite, Kauffman and Murasugi studying the Jones polynomial. In [A. Carrega, The Tait conjecture in , J. Knot Theory Ramifications 25(11) (2016) 1650063], the author proved an analogous result for alternating links in giving a complete answer to this problem. In this paper, we extend the result to alternating links in the connected sum of copies of . In and , the appropriate version of the statement is true for -homologically trivial links, and the proof also uses the Jones polynomial. Unfortunately, in the general case, the method provides just a partial result and we are not able to say if the appropriate statement is true. [ABSTRACT FROM AUTHOR]

In this paper, the properties of the Kauffman bracket skein module (KBSM) of are investigated. Links in lens spaces are represented both through band and disk diagrams. The possibility to transform between the diagrams enables us to compute the KBSM on an interesting class of examples consisting of inequivalent links with equivalent lifts in the -sphere. The computations show that the KBSM is an essential invariant, that is, it may take different values on links with equivalent lifts. We also show how the invariant is related to the Kauffman bracket of the lift in the -sphere. [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]

The Turaev genus and dealternating number of a link are two invariants that measure how far away a link is from alternating. We determine the Turaev genus of a torus knot with five or fewer strands either exactly or up to an error of at most one. We also determine the dealternating number of a torus knot with five or fewer strands up to an error of at most two. We also give bounds on the Turaev genus and dealternating number of torus links with five or fewer strands and on some infinite families of torus links on six strands. [ABSTRACT FROM AUTHOR]

The aim of this paper is to define certain algebraic structures coming from generalized Reidemeister moves of singular knot theory. We give examples, show that the set of colorings by these algebraic structures is an invariant of singular links. As an application, we distinguish several singular knots and links. [ABSTRACT FROM AUTHOR]

For two knots and , is said to be -adjacent to , if admits a knot diagram containing crossings such that crossing changes at any non-empty subset of them yield a knot diagram of . Using a surgical view of Alexander invariants, we will characterize the Alexander polynomials of knots which are -adjacent to the trivial knot. [ABSTRACT FROM AUTHOR]

In the previous note, Okada and the author gave an approach to give a characterization of Alexander polynomials for knots which are transformed by a single crossing change into a given knot whose Alexander polynomial is monic. In this note, we give a characterization in the case of , and show that the Gordian distance of and is two. We also give a characterization in the cases of more three knots. [ABSTRACT FROM AUTHOR]

We consider a local move, denoted by , on knot diagrams or virtual knot diagrams.If two (virtual) knots and are transformed into each other by a finite sequence of moves, the distance between and is the minimum number of times of moves needed to transform into . By , we denote the set of all (virtual) knots which can be transformed into a (virtual) knot by moves. A geodesic graph for is the graph which satisfies the following: The vertex set consists of (virtual) knots in and for any two vertices and , the distance on the graph from to coincides with the distance between and . When we consider virtual knots and a crossing change as a local move , we show that the -dimensional lattice graph for any given natural number and any tree are geodesic graphs for . [ABSTRACT FROM AUTHOR]

This paper extends the work by Mednykh and Rasskazov presented in [On the structure of the canonical fundamental set for the 2-bridge link orbifolds, Universität Bielefeld, Sonderforschungsbereich 343, Discrete Structuren in der Mathematik, Preprint (1988), pp. 98-062, ]. By using their approach, we derive the Riley-Mednykh polynomial for a family of -bridge knot orbifolds. As a result, we obtain explicit formulae for the volumes and Chern-Simons invariants of orbifolds and cone-manifolds on the knot with Conway's notation . [ABSTRACT FROM AUTHOR]

A virtual link diagram is even if the virtual crossings divide each component into an even number of arcs. The set of even virtual link diagrams is closed under classical and virtual Reidemeister moves, and it contains the set of classical link diagrams. For an even virtual link diagram, we define a certain linking invariant which is similar to the linking number. In contrast to the usual linking number, our linking invariant is not preserved under the forbidden moves. In particular, for two fused isotopic even virtual link diagrams, the difference between the linking invariants of them gives a lower bound of the minimal number of forbidden moves needed to deform one into the other. Moreover, we give an example which shows that the lower bound is best possible. [ABSTRACT FROM AUTHOR]

A conjecture of Aganagic and Vafa relates the open Gromov-Witten theory of X = OP1 ( 1, 1) to the augmentation polynomial of Legendrian contact homology. We describe how to use this conjecture to compute genus zero, one boundary component open Gromov-Witten invariants for Lagrangian submanifolds LK ⊂ X obtained from the conormal bundles of knots K ⊂ S³. This computation is then performed for two non-toric examples (the figure-eight and three-twist knots). For (r, s) torus knots, the open Gromov-Witten invariants can also be computed using Atiyah-Bott localization. Using this result for the unknot and the (3, 2) torus knot, we show that the augmentation polynomial can be derived from these open Gromov-Witten invariants. [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 present a new paradigm for three-dimensional chaos, and specifically for the Lorenz equations. The main difficulty in these equations and for a generic flow in dimension 3 is the existence of singularities. We show how to use knot theory as a way to remove the singularities. Specifically, we claim: (i) for certain parameters, the Lorenz system has an invariant onedimensional curve, which is a trefoil knot. The knot is a union of invariant manifolds of the singular points. (ii) The flow is topologically equivalent to an Anosov flow on the complement of this curve, and moreover to a geodesic flow. (iii) When varying the parameters, the system exhibits topological phase transitions, i.e. for special parameter values, it will be topologically equivalent to an Anosov flow on a knot complement. Different knots appear for different parameter values and each knot controls the dynamics at nearby parameters. [ABSTRACT FROM AUTHOR]

Taking advantage of a numerical invariant, we visualize a distribution of rational homology 3-spheres on a plane via the Casson-Walker-Lescop (CWL) invariant and observe several aspects of the distribution. In particular, we study the characteristics of the distribution of lens spaces as a fundamental family of rational homology 3-spheres with a way to yield a family of estimation for the Dedekind sum. The CWL invariant captures the finiteness of lens space surgeries along knots. According to the finiteness, for example, the CWL invariant determines possible lens spaces as the results of integral surgeries along a knot with . [ABSTRACT FROM AUTHOR]

Checkerboard framings are an extension of checkerboard colorings for virtual links. From checkerboard framings, we obtain an independent invariant of virtual links: the cut point number. Checkerboard framings and cut points can be used as a tool to extend other classical invariants to virtual links. [ABSTRACT FROM AUTHOR]

Virtual knot theory is a generalization of knot theory which is based on Gauss chord diagrams and link diagrams on closed oriented surfaces. A twisted knot is a generalization of a virtual knot, which corresponds to a link diagram on a possibly non-orientable surface. In this paper, we discuss an invariant of twisted links which is obtained from the JKSS invariant of virtual links by use of double coverings. We also discuss some properties of double covering diagrams. [ABSTRACT FROM AUTHOR]

We revisit the issue of the existence of infinitely many distinct prime knots with the same Alexander invariant. We present infinitely many distinct families, each family made up of infinitely many distinct knots. Within each family, the Alexander invariant is the same. Unlike, other examples in the literature, ours are elementary and based on a sub-collection of pretzel knots with three tassels. [ABSTRACT FROM AUTHOR]