Narrow Results

We construct new invariant polynomials in two and multiple variables for virtual knots and links. They are defined as determinants of Alexander-like matrices whose determinants are virtual link invariants. These polynomials vanish on classical links. In some cases, they separate links that can not be separated by the Jones–Kauffman polynomial [Kau] and the polynomial proposed in [Ma3].

A classical theorem of R. H. Bing states that a closed connected 3-manifold M is homeomorphic to the 3-sphere if and only if every knot in M is contained in a 3-ball. We give a simple proof of this characterization based on the surgery presentation of 3-manifolds.

Given a Bianchi Group , and a Hyperbolic manifold M, where π₁(M) is of finite index in Γ_d, we show that all boundary slopes are realized as the boundary slope of an immersed totally geodesic surface and hence are virtually embedded boundary slopes.

There are some phenomena arising in the virtual knot theory which are not the case for classical knots. One of them deals with the "breaking" procedure of knots and obtaining long knots. Unlike the classical case, they might not be the same. The present work is devoted to construction of some invariants of long virtual links. Several explicit examples are given. For instance, we show how to prove the non-triviality of some knots obtained by breaking virtual unknot diagrams by very simple means.

We discuss the strong invariant of virtual links proposed in [23]. This invariant is obtained as a generalization of the Jones–Kauffman polynomial (generalized Kauffman's bracket) by adding to the sum some equivalence classes of curves in two-dimensional surfaces. Thus, the invariant is valued in the infinite-dimensional free module over Z[q,q^-1].

We prove that this invariant can be decomposed into finite type Vassiliev invariant of virtual links (in Kauffman's sense); thus we present new infinite series of Vassiliev invariants.

It is also proved that this invariant is strictly stronger than the Jones–Kauffman polynomial for virtual knots proposed by Kauffman. Some examples when the invariant can recognize virtual knots that can not be recognized by other invariants are given.

A method is given for economically constructing any algebraic knot or link K. This construction, which involves tree diagrams, gives a new upper bound for the edge number of K that is proven to be at most twice the crossing number of K. Furthermore, it realizes a minimal-crossing projection.

The first author defined a well-order in the set of links by embedding it into a canonical well-ordered set of (integral) lattice points. He also gave elementary transformations among lattice points to enumerate the prime links in terms of lattice points under this order. In this paper, we add some new elementary transformations and explain how to enumerate the prime links. We show a table of the first 443 prime links arising from the lattice points of lengths up to 10 under this order. Our argument enables us to find 7 omissions and 1 overlap in Conway's table of prime links of 10 crossings.

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.

In the present paper, we construct Khovanov homology theory with arbitrary coefficients for arbitrary virtual knots. We give a definition of the complex, which is homotopy equivalent to the initial Khovanov complex in the classical case; our definition works in the virtual case as well. The method used in this work allows us to construct a Khovanov homology theory for "twisted virtual knots" in the sense of Bourgoin and Viro [4, 27] (in particular, for knots in RP³). We also generalize some results of the Khovanov homology for virtual knots with arbitrary atoms (Wehrli and Kofman–Champanerkar spanning tree, minimality problems, Frobenius extensions) and orientable ones (Rasmussen's invariant).

Johnson and Livingston have characterized peripheral structures in homomorphs of knot groups. We extend their approach to the case of links. The main result is an algebraic characterization of all possible peripheral structures in certain homomorphic images of link groups.

This paper is an unpublished version of a talk that the author gave in Brazil in 1987 at a meeting of the Brazilian Mathematical Society, while visiting Sostenes Lins in Recife, Brazil. The paper is an introduction to state models for link invariants, and it gives a surface-genus interpretation of the extreme terms in the bracket polynomial.

Study of equivalence classes of links up to n-moves plays an important role in the theory of invariants based on the skein relation and, in particular, skein modules. In this paper, we consider Nakanishi's 4-move conjecture [12]. The modification of the conjecture to 2-component link (homotopically trivial links) is a question proposed by Kawauchi [10]. We define a new invariant of links which is preserved by 4-moves and analyze its potential strength. In particular, we show that our invariant allows us to obtain results of [8, 9, 13] concerning 4-moves.

We use Reidemeister torsion to study a twisted Alexander polynomial, as defined by Turaev, for links in the projective space. Using sign-refined torsion, we derive a skein relation for a normalized form of this polynomial.

In 2003, Naik and Stanford proved that two S-equivalent knots are related by a finite sequence of doubled-delta moves on their knot diagrams. We show that classical S-equivalence is not sufficient to extend their result to ordered links. We define a new algebraic relation on Seifert matrices, called strong S-equivalence, and prove that two oriented, ordered links L and L′ are related by a sequence of doubled-delta moves if and only if they are strongly S-equivalent. We also show that this is equivalent to the fact that L′ can be obtained from L through a sequence of Y-clasper surgeries, where each clasper leaf has total linking number zero with L.

It is shown that given any link-manifold, there is an algorithm to decide if the manifold contains an embedded, essential planar surface; if it does, the algorithm will construct one. Two similar results are obtained with added boundary conditions. Namely, given a link-manifold M, a component B of ∂M, and a slope γ on B, there is an algorithm to decide if there is an embedded punctured-disk in M with boundary γ and punctures in ∂M\B; and given a link-manifold M, a component B of ∂M, and a meridian slope μ on B, there is an algorithm to decide if there is an embedded punctured-disk with boundary a longitude on B and punctures in ∂M\B. In both cases, if there is one, the algorithm will construct one. The proofs introduce a number of new methods and differ from the classical proofs, using normal surfaces, as solutions may not be found among the fundamental solutions.

We introduce a complexity c(X) ∈ ℕ for pairs X = (M,L), where M is a closed orientable 3-manifold and L ⊂ M is a link. The definition employs simple spines, but for well-behaved X's, we show that c(X) equals the minimal number of tetrahedra in a triangulation of M containing L in its 1-skeleton. Slightly adapting Matveev's recent theory of roots for graphs, we carefully analyze the behaviour of c under connected sum away from and along the link. We show in particular that c is almost always additive, describing in detail the circumstances under which it is not. To do so we introduce a certain (0, 2)-root for a pair X, we show that it is well-defined, and we prove that X has the same complexity as its (0, 2)-root. We then consider, for links in the 3-sphere, the relations of c with the crossing number and with the hyperbolic volume of the exterior, establishing various upper and lower bounds. We also specialize our analysis to certain infinite families of links, providing rather accurate asymptotic estimates.

We introduce a new numerical invariant of knots and links from the descending diagrams. It is considered to live between the unknotting number and the bridge number. Some fundamental results and an incomplete table of the invariant for knots with 8-crossings or less are given.

In this paper, we construct new links invariants from a type I basic Lie superalgebra 𝔤. The construction uses the existence of an unexpected replacement of the vanishing quantum dimension of typical module, by non-trivial "fake quantum dimensions". Using this, we get a multivariable link invariant associated to any one parameter family of irreducible 𝔤-modules.

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.

We give some relationships of the Jones and Q polynomials between two links which are related by a band surgery. Then we consider two applications: The first one is to an evaluation of the ribbon-fusion number, the least fusion number of a ribbon knot. The second one is to DNA knot theory, helping us to understand the action of the Xer site-specific recombination at psi site.