Knots in Hellas '98

The following sections are included:

• Preface

• Contents

We use the Topological Quantum Field Theory derived from the skein theory of the Kauffman bracket to compute TQFT-invariants at infinity for Whitehead's manifold.

We show that certain "grid" local moves are unknotting operations. For that we use and extend ideas involved in [A2]. We find basic estimates for the knot invariants these unknotting operations define working as in [A2].

The following sections are included:

• References

Nakanishi conjectured in 1981 that any link is 3-equivalent to a trivial link.

J. Przytycki proved that some classes of links, including 11 crossing links, are 3-equivalent to trivial links . Based on this result we prove that all closed 4-braids are 3-equivalent to trivial links. Then we show that all links with no more than 12 crossings are 3-equivalent to trivial links. For closed 5-braids only 7 of them (up to conjugacy class) have more than 12 crossings. We reduce five of them by hand. The remaining two are 3-equivalent. Hence there is only one unsettled case which is a 5-braid with 20 crossings as shown in Figure 15. We prove also similar results for t₃, -moves.

We show that the singular braid monoid SB₃ is linear by using a generalization of the classical Burau representation to the singular braid monoid. Furthermore, we use the HNN-structure of the group SG₃ in which SB₃ embeds to give a second - group theoretically motivated - solution to the word problem in SB₃. As an application of our approach we compute the homology of SG₃.

We study the new formulas of the first author for the degree-3-Vassiliev invariants for knots in the 3-sphere and solid torus and present some results obtained by them. We show that a knot with Jones polynomial consisting of exactly two monomials must have at least 20 crossings.

Gluck surgery is the operation of cutting out S² × D², a tubular neighborhood of a 2-knot and pasting it back in a 4-manifold. It may be expected to make a fake pair of 4-manifolds, which means a pair that are homotopy equivalent but non-diffeomorphic. We will give an alternative proof of a theorem: Gluck surgery along a banded 2-knot is independent of the bands, which was proved in P. Melvin's thesis and prove that: if a 2-knot is obtained by ribbon moves from another 2-knot, then the Gluck surgeries along them are diffeomorphic. Our method, which we call "framed links in 4-manifolds", is a (4,5)-dimensional version of the usual ((3,4)-dimensional) framed links.

We review the definition of a general planar algebra V = ∪V_k. We show how to construct a general planar algebra from a bipartite graph by creating a specific model using statistical mechanical sums defined by labelled tangles. These planar algebras support a partition function for a closed tangle which is spherically invariant and defines a positive definite inner product on each V_k. We then describe how any planar algebra is naturally a cylic module in the sense of Connes and do some computations.

A rack is a non-empty set with a binary operation such that right multiplication is an automorphism. New racks closely related to the braid groups and the mapping class groups will be introduced using "cords" on surfaces, and their structure will be investigated. Emphasis will be laid on the racks of cords on the plane and the sphere. In particular, their presentations and the center of the associated group of the rack of cords on the sphere will be given.

We show that the values at t = 1 of the first r coefficient polynomials of the HOMFLY polynomial of an r-component link L are determined by the linking numbers. This generalizes a formula of Hoste and one of Lickorish and Millett. The formula was proved before, using a different method, by J. Przytycki and related to homotopy skein modules [15].

We give a new method of distinguishing elements of skein homologies. Using this method we show that each skein homology module is at least as big as the Kauffman bracket skein module of a given 3-manifold.

This paper is a survey of the theory of virtual knots. It is a modified version of the lectures given by the author in the "Knots in Hellas" meeting in Delphi, Greece.

In this paper, by applying the results in [4], we shall compute lower bounds for the unknotting numbers of the knots obtained from certain links, for example, by band surgery.

We focus an interest on the torsion linking of a surface-knot. It is a knot invariant independent of the surface-knot group and its peripheral subgroup. It is identified with the torsion linking of any associated closed 4-manifold with infinite cyclic first homology. In the case of such a 4-manifold with an exact leaf, the linking of the leaf is identified with an orthogonal sum of it and a hyperbolic linking.

Our aim in this paper is to introduce and study the properties of different kinds of singular braids. Our starting point being the singular braids with double points as introduced in [3] and [5], we define a much bigger class of singular braids which we call geometric singular braids. Geometric singular braids form a monoid and we give an explicit presentation for this monoid. In this paper, we consider only singular braids where the tangents of the strings around a singular point generate a plane. In general this need not be the case. Finally, geometric singular links are defined and it is shown that there is a one-to-one correspondence between geometric singular braids and geometric singular links up to Markov equivalence.

We examine spaces of connected tri-/univalent graphs subject to local relations which are motivated by the theory of Vassiliev invariants. It is shown that the behaviour of ladder-like subgraphs is strongly related to the parity of the number of rungs: there are similar relations for ladders of even and odd lengths, respectively. Moreover, we prove that - under certain conditions - an even number of rungs may be transferred from one ladder to another.

We consider braids on m + n strands, such that the first m strands are trivially fixed. We denote the set of all such braids by B_m,n. Via concatenation B_m,n acquires a group structure. The objective of this paper is to find a presentation for B_m,n using the structure of its corresponding pure braid subgroup, P_m,n, and the fact that it is a subgroup of the classical Artin group B_m+n. Then we give an irredundant presentation for B_m,n. The paper concludes by showing that these braid groups or appropriate cosets of them are related to knots in handlebodies, in knot complements and in c.c.o. 3–manifolds.

Denote by W(m, n) the hyperbolic cone-manifold whose underlying space is the 3-sphere and singular geodesics are formed by two components of the Whitehead link with cone angles 2π/m and 2π/n. The aim of the paper is to establish the Tangent and Sine Rules relating the complex lengthes of the singular geodesics and the cone angles of W(m, n). An explicit upper bound for the real length of the singular geodesic is also given.

Polygonal knots are embeddings of polygons in three space. For each n, the collection of embedded n-gons determines a subset of Euclidean space whose structure is the subject of this paper. Which knots can be constructed with a specified number of edges? What is the likelihood that a randomly chosen polygon of n-edges will be a knot of a specific topological type? At what point is a given topological type most likely as a function of the number of edges? Are the various orderings of knot types by means of "physical properties" comparable? These and related questions are discussed and supporting evidence, in many cases derived from Monte Carlo explorations, is provided.

This work is concerned with detecting when a closed braid and its axis are 'mutually braided' in the sense of Rudolph [7]. It deals with closed braids which are fibred links, the simplest case being closed braids which present the unknot. The geometric condition for mutual braiding refers to the existence of a close control on the way in which the whole family of fibre surfaces meet the family of discs spanning the braid axis. We show how such a braid can be presented naturally as a word in the 'band generators' of the braid group discussed by Birman, Ko and Lee [1] in their recent account of the band presentation of the braid groups. In this context we are able to convert the conditions for mutual braiding into the existence of a suitable sequence of band relations and other moves on the braid word, and thus derive a combinatorial method for deciding whether a braid is mutually braided.

We show an integrality of the quantum SU(2)-invariant associated with a non-trivial first cohomology class modulo two.

In this note, we will study on generalized unknotting operations for links, especially with the condition to restrict their deformations for the same component, and we will show their differences.

In this paper we address the problem of measuring structural complexity of generic tangles of vortex lines in a fluid domain, by using a combination of geometric and topological techniques. To this end new concepts based on the idea of structural 'tropicity' are introduced to determine 'tubeness', "sheetness" and 'bulkiness' of a vortex tangle and to evaluate the degree of topological entanglement. A number of cases are considered: from highly organised, coherent vortex regions, given by the embedding of vortex coils, knots and links on nested tori, to less organised vortical flows, such as tangles of chaotic vortex lines. Various measures of linking (and helicity) are presented as well as estimates of writhing and crossing numbers based on geometric and topological information. Moreover, by using the concept of signature preserving flow we extend the definition of classical stability to include wilder vortex dynamics that during evolution preserve structural complexity. The tools and the new concepts presented in this paper are useful for the classification and study of general flow fields and can be employed to develop computational techniques for measuring structural complexity.

We consider smooth knottings of compact n-dimensional manifolds (not necessarily orientable) in Rⁿ⁺² (or Sⁿ⁺²). We generalize to higher dimensions the classical notion of a knotting in general position with respect to projection. It is shown that any knotting is equivalent to one which is in general position with respect to projection. In higher dimensions, unlike classical knot theory, a generic projection is not necessarily an immersion. Thus we need to consider maps such that the set of non-immersion points is well behaved. The geometry of projections is examined and we use this information to show that any smooth knotting of an orientable n-manifold in Rⁿ⁺² is smoothly isotopic to one whose projection into Rⁿ⁺¹ is an immersion. As corollary we obtain an elementary geometric proof that an orientable n-dimensional submanifold of Rⁿ⁺² has trivial normal bundle.

Let F be an embedded Klein bottle in S⁴\{∞}. If the singular set Γ(F*) of the projection F* of F into R³ consists of at most three disjoint simple dosed curves, then F bounds a said Klein bottle in the 4-sphere S⁴, i.e., F can be moved to the standard Klein bottle.

We prove that the Kauffman bracket skein module of a 3-manifold M at a 4r-th root of unity divided by the Jones-Wenzl idempotent depends only on ∂M.

Virtual knot groups are characterized, and their properties are compared with those of classical knot groups. A coloring theory, generalizing the usual notion of Fox n-coloring, is introduced for virtual oriented links.

An new invariant of a knot, called the zeta-function, is introduced. It is defined by counting the non-diagonalizable representations of the fundamental group into SL₂(𝔽_q), up to conjugacy, and forming a formal power series, similar to a zeta-function from algebraic geometry. The structure of this invariant in the general case is given. The special case of (2, n) torus knots is studied, and basic topological properties are noted.

Delta finite-type invariants are defined analogously to finite-type invariants, using delta moves instead of crossing changes. We show that they are closely related to the lower central series of the commutator subgroup of the pure braid group.

Knot theory was boosted in its development when in 1860ties Kelvin proposed that knots made out of vortex lines of ether constitute elementary particles of matter. Tait and other associates of Kelvin started to catalogue various types of knots with a hope to create a periodic table of knots which would correspond to periodic tables of chemical elements. However, development of atomic physics, first with classical and then with quantum models of atoms failed to show a connection between knots and atoms. More recently, though, the superstring theory revived the idea that knots can be relevant to understanding elementary particles. Studying ideal geometric configuration of knots we noticed that their writhe is quantized into units of 4/7 and 10/7 whereby the total writhe of a knot is determined by a simple arithmetic sum of four elementary types of crossings which need to be "removed" to convert a standard minimal crossing diagram of a given knot into a diagram of a trivial knot without nugatory crossings. There are parallel and antiparallel crossings of positive and negative sign. Interestingly parallel and antiparallel crossings of the same sign show frequently superposition of states. This resembles the quantum properties of elementary particles and their known superposition principle. A model of physical knotted systems is discussed which can change energy states by adopting discrete shapes with associated discrete energies whereby the writhe difference between any two discrete shapes is n*constant.

We give examples showing that the Fiedler solid torus degree 3 Gauß sum invariants can be used to detect mutation of links.

The Turaev-Viro-Ocneanu invariant of three manifolds is a generalization of the Turaev-Viro invariant of three manifolds, and is derived from the paragroup of the subfactor. We investigate the E₆ Turaev-Viro-Ocneanu invariant derived from the E₆ subfactor. We give a formula for the invariant of the lens space L(p, 1). The value of the E₆ Turaev-Viro-Ocneanu invariant can be a complex number, in contrast with the fact that the value of the Turaev-Viro invariant must be a real number because of the relation "Turaev-Viro invariant" = |"Reshetikhin-Turaev invariant"|². Thus the E₆ Turaev-Viro-Ocneanu invariant can distinguish the lens space L(p,1) from the same one with reversed orientation L(p, p- 1) for p = 12k + 3,12k + 4,12k + 8,12k + 9 while the Turaev-Viro invariant can not.

Delta-unknotting operation is a local move, as shown in Figure 2. Murakami-Nakanishi (MN) showed this move to be an unknotting operation. We will show that for many periodic knots the Δ-unknotting number is greater than one.

This paper is part expository and part presentation of calculational results. The target space of the Kontsevich integral for knots is a space of diagrams; this space has various algebraic structures which are described here. These are utilized with Le's theorem on the behaviour of the Kontsevich integral under cabling and with the Melvin-Morton Theorem, to obtain, in the Kontsevich integral for torus knots, both an explicit expression up to degree five and the general coefficients of the wheel diagrams.

Please refer to full text.

The following sections are included:

• Organising Committee and List of Participants