Flag algebras and the stable coefficients of the Jones polynomial

We study the structure of the stable coefficients of the Jones polynomial of an alternating link. We start by identifying the first four stable coefficients with polynomial invariants of a (reduced) Tait graph of the link projection. This leads us to introduce a free polynomial algebra of invariants of graphs whose elements give invariants of alternating links which strictly refine the first four stable coefficients. We conjecture that all stable coefficients are elements of this algebra, and give experimental evidence for the fifth and the sixth stable coefficient. We illustrate our results in tables of all alternating links with at most 10 crossings and all irreducible planar graphs with at most 6 vertices.     

Lagrangian Subspaces,Delta-Matroids,and Four-Term Relations

Finite-order invariants (Vassiliev invariants) of knots are expressed in terms of weight systems, that is, functions on chord diagrams (embedded graphs with a single vertex) satisfying the four-term relations. Weight systems have graph analogues, the so-called 4-invariants of graphs, i.e., functions on graphs that satisfy the four-term relations for graphs. Each 4-invariant determines a weight system.The notion of a weight system is naturally generalized to the case of embedded graphs with an arbitrary number of vertices. Such embedded graphs correspond to links; to each component of a link there corresponds a vertex of an embedded graph. Recently, two approaches have been suggested to extend the notion of 4-invariants of graphs to the case of combinatorial structures corresponding to embedded graphs with an arbitrary number of vertices. The first approach is due to V. Kleptsyn and E. Smirnov, who considered functions on Lagrangian subspaces in a 2n-dimensional space over F2 endowed with a standard symplectic form and introduced four-term relations for them. The second approach, due to V. Zhukov and S. Lando, gives four-term relations for functions on binary delta-matroids.In this paper, these two approaches are proved to be equivalent.     

Invariants of knots,surfaces in R<Superscript>3</Superscript>, and foliations

We give a survey of some known results related to combinatorial and geometric properties of finite-order invariants of knots in a three-dimensional space. We study the relationship between Vassiliev invariants and some classical numerical invariants of knots and point out the role of surfaces in the investigation of these invariants. We also consider combinatorial and geometric properties of essential tori in standard position in closed braid complements by using the braid foliation technique developed by Birman, Menasco, and other authors. We study the reductions of link diagrams in the context of finding the braid index of links. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 9, pp. 1239–1252, September, 2007.     

A Celtic Framework for Knots and Links

We describe a variant of a method used by modern graphic artists to design what are traditionally called Celtic knots, which are part of a larger family of designs called “mirror curves.” It is easily proved that every such design specifies an alternating projection of a link. We use medial graphs and graph minors to prove, conversely, that every alternating projection of a link is topologically equivalent to some Celtic link, specifiable by this method. We view Celtic representations of knots as a framework for organizing the study of knots, rather like knot mosaics or braid representations. The formalism of Celtic design suggests some new geometric invariants of links and some new recursively specifiable sequences of links. It also leads us to explore new variations of problems regarding such sequences, including calculating formulae for infinite sequences of knot polynomials. This involves a confluence of ideas from knot theory, topological graph theory, and the theory of orthogonal graph drawings.     

Moduli spaces for fundamental groups and link invariants derived from the lower central series

We consider an algebraic parametrization for the set of (Mal'cev completed) fundamental groups of the spaces with fixed first two Betti numbers, having in mind applications in low-dimensional topology and especially in link theory. The factor set of (restricted) isomorphism types of these groups acquires the structure of a ‘moduli space’, giving rise to invariants which, in the case of links, detect the isotopy type. We indicate two methods of computation for these invariants. We also prove a rigidity result for the associated graded Lie algebra of the fundamental group. A lot of examples are given.     

Strongly regular graphs and spin models for the Kauffman polynomial

We study spin models for invariants of links as defined by Jones [22]. We consider the two algebras generated by the weight matrices of such models under ordinary or Hadamard product and establish an isomorphism between them. When these algebras coincide they form the Bose-Mesner algebra of a formally self-dual association scheme. We study the special case of strongly regular graphs, which is associated to a particularly interesting link invariant, the Kauffman polynomial [27]. This leads to a classification of spin models for the Kauffman polynomial in terms of formally self-dual strongly regular graphs with strongly regular subconstituents [7]. In particular we obtain a new model based on the Higman-Sims graph [17].     

Finite type 3-manifold invariants and the structure of the Torelli group. I

Using the recently developed theory of finite type invariants of integral homology 3-spheres we study the structure of the Torelli group of a closed surface. Explicitly, we construct (a) natural cocycles of the Torelli group (with coefficients in a space of trivalent graphs) and cohomology classes of the abelianized Torelli group; (b) group homomorphisms that detect (rationally) the nontriviality of the lower central series of the Torelli group. Our results are motivated by the appearance of trivalent graphs in topology and in representation theory and the dual role played by the Casson invariant in the theory of finite type invariants of integral homology 3-spheres and in Morita's study [Mo2, Mo3] of the structure of the Torelli group. Our results generalize those of S. Morita [Mo2, Mo3] and complement the recent calculation, due to R. Hain [Ha2], of the I-adic completion of the rational group ring of the Torelli group. We also give analogous results for two other subgroups of the mapping class group. Oblatum 19-IX-1996 & 13-V-1997     

Modified 6j-symbols and 3-manifold invariants

We show that the renormalized quantum invariants of links and graphs in the 3-sphere, derived from tensor categories in Geer et al. (2009) [14], lead to modified 6j-symbols and to new state sum 3-manifold invariants. We give examples of categories such that the associated standard Turaev–Viro 3-manifold invariants vanish but the secondary invariants may be non-zero. The categories in these examples are pivotal categories which are neither ribbon nor semi-simple and have an infinite number of simple objects.     

Delta move and polynomial invariants of links

We investigate how a self-delta move, which is a delta move on the same component, influences the HOMFLY polynomial of a link. Then we reveal some relationships among finite type invariants, which are coming from the derivatives of the Jones polynomials and the first HOMFLY coefficient polynomials, of the four links involving in a self-delta move.     

Knot graphs

We consider the equivalence classes of graphs induced by the unsigned versions of the Reidemeister moves on knot diagrams. Any graph that is reducible by some finite sequence of these moves, to a graph with no edges, is called a knot graph. We show that the class of knot graphs strictly contains the set of delta‐wye graphs. We prove that the dimension of the intersection of the cycle and cocycle spaces is an effective numerical invariant of these classes. © 2000 John Wiley & Sons, Inc. J Graph Theory 34: 100–111, 2000     

Signature invariants of covering links

We apply the theory of signature invariants of links in rational homology spheres to covering links of homology boundary links. From patterns and Seifert matrices of homology boundary links, we derive an explicit formula to compute signature invariants of their covering links. Using the formula, we produce fused boundary links that are positive mutants of ribbon links but are not concordant to boundary links. We also show that for any finite collection of patterns, there are homology boundary links that are not concordant to any homology boundary links admitting a pattern in the collection.

     

Finite type invariants of nanowords and nanophrases

Homotopy classes of nanowords and nanophrases are combinatorial generalizations of virtual knots and links. Goussarov, Polyak and Viro defined finite type invariants for virtual knots and links via semi-virtual crossings. We extend their definition to nanowords and nanophrases. We study finite type invariants of low degrees. In particular, we show that the linking matrix and T invariant defined by Fukunaga are finite type of degree 1 and degree 2 respectively. We also give a finite type invariant of degree 4 for open homotopy of Gauss words.     

The Alexander polynomial and finite type 3-manifold invariants

Using elementary counting methods, we calculate a universal perturbative invariant (also known as the LMO invariant) of a 3-manifold M, satisfying , in terms of the Alexander polynomial of M. We show that +1 surgery on a knot in the 3-sphere induces an injective map from finite type invariants of integral homology 3-spheres to finite type invariants of knots. We also show that weight systems of degree 2m on knots, obtained by applying finite type 3m invariants of integral homology 3-spheres, lie in the algebra of Alexander-Conway weight systems, thus answering the questions raised in [Ga]. Received: 27 April 1998 / in final form: 8 August 1999     

Goussarov-Habiro theory for string links and the Milnor-Johnson correspondence

We study the Goussarov-Habiro finite type invariants theory for framed string links in homology balls. Their degree 1 invariants are computed: they are given by Milnor's triple linking numbers, the mod 2 reduction of the Sato-Levine invariant, Arf and Rochlin's μ invariant. These invariants are seen to be naturally related to invariants of homology cylinders through the Milnor-Johnson correspondence: in particular, an analogue of the Birman-Craggs homomorphism for string links is computed. The relation with Vassiliev theory is studied.     

Strong band sum and dichromatic invariants

Strong band sum is a natural construction from links to dichromatic links. We compute Hoste and Kidwell's dichromatic link invariant of a strong band sum in terms of monochromatic invariants of the data (original link, band). It turns out that the two-variable Conway polynomial of a strong fusion only depends on the monochromatic Conway polynomial of the original link. In particular, it does not depend on the band. Cochran's series of concordance invariants is discussed in this framework. partially supported by NATO via DAAD     

<Emphasis Type="Italic">S</Emphasis>-graphs and braid index of links

We study the relationship between reduction operations on link diagrams and S-graphs associated with them. We are motivated by the problem of computing the braid index of a link and some well known conjectures concerning the braid index of a link and the writhe of its diagrams. Possible counterexamples are discussed in terms of both S-graphs and link diagrams. We also indicate the relation of S-graphs to singular links regarded up to an appropriate equivalence relation.     

Bar complex, configuration spaces and finite type invariants for braids

We show that the bar complex of the configuration space of ordered distinct points in the complex plane is acyclic. The 0-dimensional cohomology of this bar complex is identified with the space of finite type invariants for braids. We construct a universal holonomy homomorphism from the braid group to the space of horizontal chord diagrams over Q, which provides finite type invariants for braids with values in Q.     

Finite type invariants for knotoids

We extend the theory of Vassiliev (or finite type) invariants for knots to knotoids using two different approaches. Firstly, we take closures on knotoids to obtain knots and we use the Vassiliev invariants for knots, proving that these are knotoid isotopy invariant. Secondly, we define finite type invariants directly on knotoids, by extending knotoid invariants to singular knotoid invariants via the Vassiliev skein relation. Then, for spherical knotoids we show that there are non-trivial type-1 invariants, in contrast with classical knot theory where type-1 invariants vanish. We give a complete theory of type-1 invariants for spherical knotoids, by classifying linear chord diagrams of order one, and we present examples arising from the affine index polynomial and the extended bracket polynomial.     

Facet defining inequalities among graph invariants: The system GraPHedron

We present a new computer system, called GraPHedron, which uses a polyhedral approach to help the user to discover optimal conjectures in graph theory. We define what should be optimal conjectures and propose a formal framework allowing to identify them. Here, graphs with n nodes are viewed as points in the Euclidian space, whose coordinates are the values of a set of graph invariants. To the convex hull of these points corresponds a finite set of linear inequalities. These inequalities computed for a few values of n can be possibly generalized automatically or interactively. They serve as conjectures which can be considered as optimal by geometrical arguments.We describe how the system works, and all optimal relations between the diameter and the number of edges of connected graphs are given, as an illustration. Other applications and results are mentioned, and the forms of the conjectures that can be currently obtained with GraPHedron are characterized.