Showing papers in "Journal of Knot Theory and Its Ramifications in 1996"
181 citations
TL;DR: In this article, the relationship between two-dimensional topological quantum field theories and Frobenius algebras is formulated as an equivalence of categories, and a classification of the indecomposable Frobenian alges as either annihilator alges or field extensions is provided.
Abstract: We characterize Frobenius algebras A as algebras having a comultiplication which is a map of A-modules. This characterization allows a simple demonstration of the compatibility of Frobenius algebra structure with direct sums. We then classify the indecomposable Frobenius algebras as being either “annihilator algebras” — algebras whose socle is a principal ideal — or field extensions. The relationship between two-dimensional topological quantum field theories and Frobenius algebras is then formulated as an equivalence of categories. The proof hinges on our new characterization of Frobenius algebras. These results together provide a classification of the indecomposable two-dimensional topological quantum field theories.
158 citations
TL;DR: In this article, the two variable polynomial invariant of knots defined using representations of the fundamental group of the knot complement into the Newton polygon of the polynomials are discussed.
Abstract: This paper reviews the two variable polynomial invariant of knots defined using representations of the fundamental group of the knot complement into . The slopes of the sides of the Newton polygon of this polynomial are boundary slopes of incompressible surfaces in the knot complement. The polynomial also contains information about which surgeries are cyclic, and about the shape of the cusp when the knot is hyperbolic. We prove that at least some mutants have the same polynomial, and that most untwisted doubles have non-trivial polynomial. We include several open questions.
90 citations
TL;DR: In this article, the problem of distinguishing mutant knots using invariants of their satellites was considered, and it was shown that the Homfly polynomial of the 3-parallel (and hence the related quantum invariants) will distinguish some mutant pairs.
Abstract: We consider the problem of distinguishing mutant knots using invariants of their satellites. We show, by explicit calculation, that the Homfly polynomial of the 3-parallel (and hence the related quantum invariants) will distinguish some mutant pairs. Having established a condition on the colouring module which forces a quantum invariant to agree on mutants, we explain several features of the difference between the Homfly polynomials of satellites constructed from mutants using more general patterns. We illustrate this by our calculations; from these we isolate some simple quantum invariants, and a framed Vassiliev invariant of type 11, which distinguish certain mutants, including the Conway and Kinoshita-Teresaka pair.
81 citations
TL;DR: The notion of finite type invariants for integral homology 3-spheres was introduced by Ohtsuki as mentioned in this paper, motivated by the notion of fixed-type knots invariants.
Abstract: Recently Ohtsuki [Oh2], motivated by the notion of finite type knot invariants, introduced the notion of finite type invariants for oriented, integral homology 3-spheres. In the present paper we propose another definition of finite type invariants of integral homology 3-spheres and give equivalent reformulations of our notion. We show that our invariants form a filtered commutative algebra. We compare the two induced filtrations on the vector space on the set of integral homology 3-spheres. As an observation, we discover a new set of restrictions that finite type invariants in the sense of Ohtsuki satisfy and give a set of axioms that characterize the Casson invariant. Finally, we pose a set of questions relating the finite type 3-manifold invariants with the (Vassiliev) knot invariants.
72 citations
47 citations
TL;DR: It is shown that the differences between canonical genus and free genus, and differences between free genus and usual genus of a knot can be arbitrarily large.
Abstract: We show that the differences between canonical genus and free genus, and differences between free genus and usual genus of a knot can be arbitrarily large.
43 citations
TL;DR: In this article, a category N of labeled trivalent graphs (nets) or ribbon graphs is extended by new generators called fusing, braiding, twist and switch with relations which can be called Moore-Seiberg relations.
Abstract: A category N of labeled (oriented) trivalent graphs (nets) or ribbon graphs is extended by new generators called fusing, braiding, twist and switch with relations which can be called Moore-Seiberg relations A functor to N is constructed from the category Surf of oriented surfaces with labeled boundary and their homeomorphisms Given a (eventually non-semisimple) k-linear abelian ribbon braided category with some finiteness conditions we construct a functor from a central extension of N with the set of labels ObC to k-vector spaces Composing the functors we get a modular functor from a central extension of Surfto k-vector spaces
39 citations
TL;DR: A surface braid is a generalization of classical braids, which is related to classical and 2-dimensional knot theory as discussed by the authors, and it is described by a diagram on a 2-disk called a chart.
Abstract: A surface braid is a generalization of classical braids, which is related to classical and 2-dimensional knot theory. It is described by a diagram on a 2-disk called a chart. We prove that surface braids are in one-to-one correspondence to such diagrams modulo some elementary moves. It helps us to handle surface braids. As an application we calculate the Grothendieck group of the semi-group of surface braids. A theorem on symmetric equivalence for the braid group is also given.
37 citations
TL;DR: In this paper, the authors give a solution to the planarity problem for unsigned Gauss words, along the lines of Gauss's original condition, and give a necessary and sufficient condition for the signed Gauss word.
Abstract: C.F. Gauss gave a necessary condition for a word to be the intersection sequence of a closed normal planar curve and he gave an example which showed that his condition was not sufficient. Since then several authors have given algorithmic solutions to this problem. In a previous paper, along the lines of Gauss’s original condition, we gave a necessary and sufficient condition for the planarity of “signed” Gauss words. In this present paper we give a solution to the planarity problem for unsigned Gauss words.
35 citations
TL;DR: In this paper, a planar diagrammatic representation of a 3-manifold via o-graphs is presented, which can be used to obtain the Turaev-Viro invariant.
Abstract: In this paper we discuss the beautiful idea of Justin Roberts [7] (see also [8]) to re-obtain the Turaev-Viro invariants [11] via skein theory, and re-prove elementarily the Turaev-Walker theorem [9], [10], [13]. We do this by exploiting the presentation of 3-manifolds introduced in [1], [4]. Our presentation supports in a very natural way a formal implementation of Roberts’ idea. More specifically, what we show is how to explicitly extract from an o-graph (the object by which we represent a manifold, see below), one of the framed links in S3 which Roberts uses in the construction of his invariant, and a planar diagrammatic representation of such a link. This implies that the proofs of invariance and equality with the Turaev-Viro invariant can be carried out in a completely “algebraic” way, in terms of a planar diagrammatic calculus which does not require any interpretation of 3-dimensional figures. In particular, when proving the “term-by-term” equality of the expansion of the Roberts invariant with the state sum which gives the Turaev-Viro invariant, we simultaneously apply several times the “fusion rule” (which is formally defined, strictly speaking, only in diagrammatic terms), showing that the “braiding and twisting” which a priori may exist on tetrahedra is globally dispensable. In our point of view the success of this formal “algebraic” approach witnesses a certain efficiency of our presentation of 3-manifolds via o-graphs. In this work we will widely use recoupling theory which was very clearly exposed in [2], and therefore we will avoid recalling notations. Actually, for the purpose of stating and proving our results we will need to slightly extend the class of trivalent ribbon diagrams on which the bracket can be computed. We also address the reader to the references quoted in [2], in particular for the fundamental contributions of Lickorish to this area. In our approach it is more natural to consider invariants of compact 3-manifolds with non-empty boundary. The case of closed 3-manifolds is included by introducing a correction factor corresponding to boundary spheres, as explained in §2. Our main result is actually an extension to manifolds with boundary of the Turaev-Walker theorem: we show that the Turaev-Viro invariant of such a manifold coincides (up to a factor which depends on the Euler characteristic) with the Reshetikhin-Turaev-Witten invariant of the manifold mirrored in its boundary.
TL;DR: In this paper, the consequences of the possibility that Vassiliev invariants do not detect knot invertibility as well as the fact that quantum Lie group invariants are known not to do so are discussed.
Abstract: We discuss the consequences of the possibility that Vassiliev invariants do not detect knot invertibility as well as the fact that quantum Lie group invariants are known not to do so. On the other hand, finite group invariants, such as the set of homomorphisms from the knot group to M11, can detect knot invertibility. For many natural classes of knot invariants, including Vassiliev invariants and quantum Lie group invariants, we can conclude that the invariants either distinguish all oriented knots, or there exist prime, unoriented knots which they do not distinguish.
TL;DR: In this paper, Yetter showed that for any space X and a triangulation, T, a G-coloring of T is a map : T (1)! G such that given any 2 T (2), (e 1)'' 1 (e 2)'' 2 (e 3)'' 3 = 1, whenever @ = e'' 1 1 1 e'' 2 2 2 e'' 3 3, for i = 1 denoting in the rst expression noninversion or inversion in the group G and in the second preservation or reversal of
Abstract: In 20], Yetter makes the following deenition: Fix a nite group, G. For any space X and a triangulation, T, a G-coloring of T is a map : T (1) ! G such that given any 2 T (2) , (e 1) \" 1 (e 2) \" 2 (e 3) \" 3 = 1, whenever @ = e \" 1 1 e \" 2 2 e \" 3 3 , for \" i = 1 denoting in the rst expression non-inversion or inversion in the group G and in the second preservation or reversal of orientation. We denote the set of all G-colorings of T by G (T). Yetter then deenes Z G (X; T) to be the vector space having G (T) as basis. Restricting to the case where X is a surface, he shows that if T 0 is a triangulation obtained from T by iterated subdivision of edges, then there is a well deened map res T 0 ;T : G (T 0) ! G (T) which induces a map res T 0 ;T on the corresponding vector spaces. The Z G (X; T)s and res T 0 ;T deene a diagram of vector spaces and he takes Z G (X) to be the colimit of this diagram. It is then shown that 1. this deenes a (2 + 1)-dimensional topological quantum eld theory in the sense of Atiyah 1]; 2. the vector space Z G (X) is isomorphic to the vector space whose basis is the set of conjugacy classes of representations from (X) to G (i.e. of natural isomorphism classes of functors from (X) to G, regarded as a groupoid with one object). Yetter then extends the construction to take coeecients in a crossed G-set and in a second paper, 21], shows how to adapt the method to handling coeecients in an algebraic model of a homotopy 2-type. In both cases the theory gives a TQFT and there are hints at an interpretation in terms analogous to 2. above. Here we will provide alternative proofs of some of Yetter's results. This gives an interpretation in terms of simplicial bre bundles and of 2-descent data or non-abelian cocycles.
TL;DR: In this paper, invariants up to order six for arbitrary torus knots {n, m, with n and m coprime integers, with degree coincide with their order are computed.
Abstract: Vassiliev. invariants up to order six for arbitrary torus knots {n, m}, with n and m coprime integers, are computed. These invariants are polynomials in n and m whose degree coincide with their order. Furthermore, they turn out to be integer-valued in a normalization previously proposed by the authors.
TL;DR: In this paper, the generalized Casson invariant of lens spaces has been shown to hold for 3-manifolds, and necessary and sufficient conditions for lens spaces to have the same CSWJ invariants are given.
Abstract: Starting from evaluating all Gaussian sums, we calculate {τr, r≥2} (in the 4r-th cyclotomic fields) and {ξr, r odd≥3} (in the r-th cyclotomic fields) for all lens spaces L(p, q). We prove that they are all algebraic integers and show that ξr determines the Dedekind sum s(q, p), and hence determines the generalized Casson invariant of lens space. We conjecture these two properties hold for more general 3-manifold, and some evidences are discussed. Though the formulae are not simple, we are able to give necessary and sufficient conditions for lens spaces to have the same CSWJ invariants. Examples of lens spaces with the same invariants but different topological types are given. Some applications in number theory are also included.
TL;DR: In this paper, it was shown that there exists a link consisting of two components, each of which is individually unknotted, such that the link can be split with a single crossing change, however any such crossing change must knot one of the components.
Abstract: We will prove that there exists a link consisting of two components, each of which is individually unknotted, such that the link can be split with a single crossing change, however any such crossing change must knot one of the components.
TL;DR: For genus one knots, the Casson-Gordon invariant τ is expressed in terms of the classical signatures as discussed by the authors, generalizing an earlier result of P. Gilmer, and it is shown that pretzel knots K(3, 5, 7, 17) are not concordant to their reverses.
Abstract: For genus one knots the Casson-Gordon invariant τ is expressed in terms of the classical signatures, generalizing an earlier result of P. Gilmer. As an application it is shown that the pretzel knots K(3, –5, 7) and K(3, –5, 17) are not concordant to their reverses.
TL;DR: In this paper, the authors give a counterexample to the following conjecture of Kauffman [2] : if K is an amphicheiral alternating knot, then there exists a reduced alternating knot-diagram D of K, such that G(D) is isomorphic to G* (D), where D is a checkerboard graph of D and G*(D), its dual.
Abstract: We give a counterexample to the following conjecture of Kauffman [2]: Conjecture Let K be an amphicheiral alternating knot. Then there exists a reduced alternating knot-diagram D of K, such that G(D) is isomorphic to G* (D), where G(D) is a checkerboard-graph of D and G*(D) its dual.
TL;DR: In the class of links with one unknotting tunnel, the 2-bridge links are characterized by the property of consisting of two trivial components as discussed by the authors, and the tunnels of 2-bridges are classified.
Abstract: In the class of links with one unknotting tunnel the 2-bridge links are characterized by the property of consisting of two trivial components. The tunnels of 2-bridge links are classified.
TL;DR: In this paper, the mapping class group representation of Reshetikhin and Turaev is analyzed in terms of topological invariants, and it is shown that in genus 2, the representation splits into two summands, and the first summand factors through the action on the first homology of the surface with Z/3Z coefficients.
Abstract: The quantum group construction of Reshetikhin and Turaev provides representations of the mapping class group, indexed by an integer parameter r. This paper presents computations of these representations when r=6, and analyzes their relationship to other topological invariants. It is shown that in genus 2, the representation splits into two summands. The first summand factors through the mapping class group action on the first homology of the surface with Z/3Z coefficients, while the second summand can be analyzed via its restriction to the subgroup of the mapping class group which is normally generated by the sixth power of a Dehn twist on a nonseparating curve. This analysis reveals a connection to the homology intersection pairing on the surface, and also yields information about the kernel and image of the representation. It is also shown that the representation yields a family of 2-dimensional nonabelian representations of the Torelli group. This paper continues the program established by the author in [Wr] to relate the Reshetikhin-Turaev representations at specific roots of unity to classical invariants.
TL;DR: An invariant polynomial s(t) for framed knots in the solid torus is defined in this paper, where the coefficients are Vassiliev invariants of order one.
Abstract: An invariant polynomial s(t) is defined for framed knots in the solid torus. The coefficients are Vassiliev invariants of order one. An invariant polynomial A(t) of Legendrian curves is introduced and it is shown how to calculate it from their fronts. The coefficient of A(t) of the order n term is the restriction to the discriminant of the selftangencies with partial index n of the Arnold invariant J+ of wave fronts. The polynomial A(t) of a Legendrian curve is recovered from the polynomial s(t) of the Legendrian knot, provided with its natural contact framing.
TL;DR: In this article, an explicit formulation of a third order finite knot invariant is derived from the perturbative expression of the Wilson loop integral along a knotted line in Chern-Simons theory.
Abstract: An explicit formulation of a third order finite knot invariant is derived from the perturbative expression of the Wilson loop integral along a knotted line in Chern-Simons theory. This is achieved by an appropriate deformation of the knot line in three dimensional space. It is demonstrated that this formulation fulfils the axioms of the Vassiliev invariants. We use our formula in order to calculate the invariant for knots with up to nine crossings and for some torus knots.
TL;DR: In this paper, it was shown that all Lins-Mandel spaces S (b, l, t, c) are branched cyclic coverings of the 3-sphere.
Abstract: In this paper we show that all Lins-Mandel spaces S (b, l, t, c) are branched cyclic coverings of the 3-sphere. When the space is a 3-manifold, the branching set of the covering is a two-bridge knot or link of type (l, t) and otherwise is a graph with two vertices joined by three edges (a θ-graph). In the latter case the singular set of the space is always composed by two points with homeomorphic links. The first homology groups of the Lins-Mandel manifolds are computed when t=1 and when the branching set is a knot of genus one. Furthermore the family of spaces has been extended in order to contain all branched cyclic coverings of two-bridge knots or links.
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TL;DR: In this paper, a finite set of moves to such sequences of classical braid words such that two sequences represent the same knotting if and only if they are related by a sequence of moves in this set.
Abstract: Knotted surfaces in 4-space are described by sequences of classical braids. We provide a finite set of moves to such sequences of classical braid words such that two sequences represent the same knotting if and only if they are related by a sequence of moves in this set.
TL;DR: The weak unknotting number of a 2-knot K in a 4-sphere is the minimum number of elements g1, g2, g3, g4, g5, g6, g7, g8, g9, g10, g11, g12, g13, g14, g15, g16, gn of the knot group π K of K such that πK becomes the infinite cyclic group by adding the relations gix=xgi, where x is a meridian.
Abstract: The weak unknotting number of a 2-knot K in a 4-sphere is the minimum number of elements g1, g2,…, gn of the knot group πK of K such that πK becomes the infinite cyclic group by adding the relations gix=xgi, where x is a meridian. We give an example of the nonadditivity of the weak unknotting number under connected sum of 2-knots.
TL;DR: In this paper, it was shown that the set of all smooth, closed space curves having the same knot type and writhe is path-connected, i.e., it is smooth and closed space.
Abstract: This paper proves that the set of all smooth, closed space curves having the same knot type and writhe is path-connected.
TL;DR: In this paper, an explicit combinatorial formula for an invariant of (n − 1)-singular knots with weight system denoted by P is given. But this formula is restricted to regular knots and cannot be used to prove invariance under singular Reidemeister moves.
Abstract: Starting from a Weight-System denoted by P and defined on the n-Chord-Diagrams with values in an arbitrary Q–module, we give an explicit combinatorial formula for an invariant of (n–1)-singular knots which has P as its derivative. The formula is defined for regular knot projections. Its invariance under singular Reidemeister moves is then proved.