Isotopy invariants for smooth tori in 4-manifolds

In this paper we define invariants under smooth isotopy for certain two-dimensional knots using some refined Cerf theory. One of the invariants is the knot type of some classical knot generalizing the string number of closed braids. The other invariant is a generalization of the unique invariant of degree 1 for classical knots in 3-manifolds. Possibly, these invariants can be used to distinguish smooth embeddings of tori in some 4-manifolds but which are equivalent as topological embeddings.     

Space curves with positive torsion

Summary Space curves may be classified under various kinds of deformation. The following six kinds of deformation have been of special interest; namely first, second and third order homotopy and isotopy. (We say the deformation is k-th order if the first k derivatives remain independent during the deformation.) The first order homotopy classification of space curves may be accomplished using well-known methods of Whitney; there is only one class. The second and third order homotopy classification was done by Feldman[1] and Little[6], respectively. The first order isotopy classification of space curves is knot theory; a subject of its own. The second order isotopy classification has been done by W. F. Pohl (unpublished). Thus, aside from knot theory, the only remaining problem is the third order isotopy problem. In this paper we give a partial answer. Our result is partial because we must restrict the class of curves with which we are dealing; namely to curves with a ? twist ?. But it may well be that every curve does have a twist, in which case our restricted class of curves would be all curves and the classification would be complete. In addition we construct a curve of positive torsion with any preassigned self-linking number in any preassigned knot class; a question raised by W. F. Pohl. Entrata in Redazione il 4 settembre 1976.     

Double braidings, twists and tangle invariants

A tortile (or ribbon) category defines invariants of ribbon (framed) links and tangles. We observe that these invariants, when restricted to links, string links, and more general tangles which we call turbans, do not actually depend on the braiding of the tortile category. Besides duality, the only pertinent data for such tangles are the double braiding and twist. We introduce the general notions of twine, which is meant to play the rôle of the double braiding (in the absence of a braiding), and the corresponding notion of twist. We show that the category of (ribbon) pure braids is the free category with a twine (a twist). We show that a category with duals and a self-dual twist defines invariants of stringlinks. We introduce the notion of turban category, so that the category of turban tangles is the free turban category. Lastly we give a few examples and a tannaka dictionary for twines and twists.     

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.     

Modelling supercoiled DNA knots and catenanes by means of a new regular isotopy invariant

This paper extends the knot polynomial classification of DNA knots and catenanes, by incorporating a measure of supercoiling. Cozzarelli, Millett, and White have used the Jones polynomial and the generalised 2-variable polynomial to describe the products of iterated Tn3 resolvase recombination and phage integrase mediated recombination. A new polynomial invariant, , is introduced; based on the regular isotopy invariants of Kauffman. The polynomial is an invariant of framed links and involves the Whitney degree of the link. This is useful because it not only allows a regular isotopy classification, but also distinguishes between plectonemic and solenoidal supercoils. The enzymes above require plectonemic supercoils for the synaptic substrate, so we put the polynomial to use to investigate the supercoiling of the recombination products.     

Maximal Thurston-Bennequin numbers of alternating links

We show that the upper bound of the maximal Thurston-Bennequin number for an oriented alternating link given by the Kauffman polynomial is sharp. As an application, we confirm a question of Ferrand. We also give a formula of the maximal Thurston-Bennequin number for all two-bridge links. Finally, we introduce knot concordance invariants derived from the Thurston-Bennequin number and the Maslov number of a Legendrian knot.     

Polynomial invariants and quandles of twisted links

Bourgoin defined the notion of a twisted link which corresponds to a stable equivalence class of links in oriented thickenings. It is a generalization of a virtual link. Some invariants of virtual links are extended for twisted links including the knot group and the Jones polynomial. In this paper, we generalize a multivariable polynomial invariant of a virtual link to a twisted link. We also introduce a quandle of a twisted link.     

Volume growth in the component of the Dehn–Seidel twist

We consider an entropy-type invariant which measures the polynomial volume growth of submanifolds under the iterates of a map, and we show that this invariant is at least 1 for every diffeomorphism in the symplectic isotopy class of the Dehn–Seidel twist. Received: April 2004 Revision: June 2004 Accepted: October 2004     

Diffeomorphic vs Isotopic Links in Lens Spaces

Links in lens spaces may be defined to be equivalent by ambient isotopy or by diffeomorphism of pairs. In the first case, for all the combinatorial representations of links, there is a set of Reidemeister-type moves on diagrams connecting isotopy equivalent links. In this paper, we provide a set of moves on disk, band and grid diagrams that connects diffeo-equivalent links: there are up to four isotopy equivalent links in each diffeo-equivalence class. Moreover, we investigate how the diffeo-equivalence relates to the lift of the link in the 3-sphere: in the particular case of oriented primitive-homologous knots, the lift completely determines the knot class in L(p, q) up to diffeo-equivalence, and thus only four possible knots up to isotopy equivalence can have the same lift.     

On the Melvin–Morton–Rozansky conjecture

We prove a conjecture stated by Melvin and Morton (and elucidated further by Rozansky) saying that the Alexander–Conway polynomial of a knot can be read from some of the coefficients of the Jones polynomials of cables of that knot (i.e., coefficients of the “colored” Jones polynomial). We first reduce the problem to the level of weight systems using a general principle, which may be of some independent interest, and which sometimes allows to deduce equality of Vassiliev invariants from the equality of their weight systems. We then prove the conjecture combinatorially on the level of weight systems. Finally, we prove a generalization of the Melvin–Morton–Rozansky (MMR) conjecture to knot invariants coming from arbitrary semi-simple Lie algebras. As side benefits we discuss a relation between the Conway polynomial and immanants and a curious formula for the weight system of the colored Jones polynomial. Oblatum 28-VII-1994 & 5-XI-1995 & 22-XI-1995     

Topological Bifurcations of Attracting 2-Tori of Quasiperiodically Driven Oscillators

We examine the solutions to a damped, quasiperiodic (QP) Mathieu equation with cubic nonlinearities. The system is suspended in a four-dimensional phase space ℝ² × T² in which there exist attracting, knotted 2-tori called torus braids. We develop a topological classification scheme in which a torus braid is characterized by closed braids that exist in two Poincare sections, ℝ² \times S¹ × {·} and ℝ² × {·} \times S¹. Based on the classification scheme, we develop numerical invariants that describe the linkedness of attractors and provide information about the global dynamics. Numerical simulations show that changes of a single parameter lead to a global bifurcation through which the attracting torus loses stability and locally "doubles," shedding a torus of different equivalence class. We call this a topological torus bifurcation of the doubling variety (TTBD). We provide a topological analysis of the doubling produced by TTBDs and we examine the qualitative dynamical changes that result. We also examine the effect of TTBDs on the spectrum of Lyapunov exponents and the time series power spectrum.     

Vassiliev invariants for braids on surfaces

We show that Vassiliev invariants separate braids on a closed oriented surface, and we exhibit a universal Vassiliev invariant for these braids in terms of chord diagrams labeled by elements of the fundamental group of the surface.

     

Chern–Simons Invariants of Torus Links

We compute the vacuum expectation values of torus knot operators in Chern–Simons theory, and we obtain explicit formulae for all classical gauge groups and for arbitrary representations. We reproduce a known formula for the HOMFLY invariants of torus knots and links, and we obtain an analogous formula for Kauffman invariants. We also derive a formula for cable knots. We use our results to test a recently proposed conjecture that relates HOMFLY and Kauffman invariants.     

Notes on the braid index of closed positive braids

The Morton–Franks–Williams inequality for a link gives a lower bound for the braid index in terms of the HOMFLY polynomial. Franks and Williams conjectured that for any closed positive braid the lower bound coincides with the braid index. In this paper, we show that the bound is achieved for a certain class of closed positive braids. We also give an infinite family of prime closed positive braids such that the lower bound does not coincide with their braid indices.     

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.     

The colored Jones polynomial and the A-polynomial of Knots

We study relationships between the colored Jones polynomial and the A-polynomial of a knot. The AJ conjecture (of Garoufalidis) that relates the colored Jones polynomial and the A-polynomial is established for a large class of two-bridge knots, including all twist knots. We formulate a weaker conjecture and prove that it holds for all two-bridge knots. Along the way we also calculate the Kauffman bracket skein module of the complements of two-bridge knots. Some properties of the colored Jones polynomial are established.     

Yang-Baxter invariants for line configurations

We study line configurations in 3-space by means of “line diagrams”, projections into a plane with an indication of over and under crossing at the vertices. If we orient such a diagram, we can associate a “contracted tensor”T with it in the same spirit as is done in Knot Theory. We give conditions to makeT independent of the orientation, and invariant under isotopy. The Yang-Baxter equation is one such condition. Afterwards we restrict ourselves to Yang-Baxter invariants with a topological state model, and give some new invariants for line isotopy.     

Some applications of Tristram-Levine signatures and relation to Vassiliev invariants

Tristram and Levine introduced a continuous family of signature invariants for knots. We show that any possible value of such an invariant is realized by a knot with given Vassiliev invariants of bounded degree. We also show that one can make a knot prime preserving Alexander polynomial and Vassiliev invariants of bounded degree. Finally, the Tristram-Levine signatures are applied to obtain a condition on (signed) unknotting number.     

On Invariants of Virtual Links

We propose a new method of generalizing classical link invariants for the case of virtual links. In particular, we have generalized the knot quandle, the knot fundamental group, the Alexander module, and the coloring invariants. The virtual Alexander module leads to a definition of VA-polynomial that has no analogue in the classical case (i.e. vanishes on classical links).