Addendum to “Groups of ribbon knots”

The inductive step that was described in the proof of Theorem 3.2 of Ng (Topology 37(2) (1998) 441) is clarified.     

Knots of genus one or on the number of alternating knots of given genus

We prove that any non-hyperbolic genus one knot except the trefoil does not have a minimal canonical Seifert surface and that there are only polynomially many in the crossing number positive knots of given genus or given unknotting number.

     

On Legendrian knots and polynomial invariants

It is proved in this note that the analogues of the Bennequin inequality which provide an upper bound for the Bennequin invariant of a Legendrian knot in the standard contact three dimensional space in terms of the least degree in the framing variable of the HOMFLY and the Kauffman polynomials are not sharp. Furthermore, the relationships between these restrictions on the range of the Bennequin invariant are investigated, which leads to a new simple proof of the inequality involving the Kauffman polynomial.

     

Universal Formulae for SU Casson Invariants of Knots

An Casson invariant of a knot is an integer which can be thought of as an algebraic-topological count of the number of characters of representations of the knot group which take a longitude into a given conjugacy class. For fibered knots, these invariants can be characterized as Lefschetz numbers which, for generic conjugacy classes, can be computed using a recursive algorithm of Atiyah and Bott, as adapted by Frohman. Using a new idea to solve the Atiyah-Bott recursion (as simplified by Zagier), we derive universal formulae which explicitly compute the invariants for all . Our technique is based on our discovery that the generating functions associated to the relevant Lefschetz numbers (and polynomials) satisfy certain integral equations.

     

Vassiliev invariants of type one for links in the solid torus

In Bataineh (2003) [2] we studied the type one invariants for knots in the solid torus. In this research we study the type one invariants for n-component links in the solid torus by generalizing Aicardi's invariant for knots in the solid torus to n-component links in the solid torus. We show that the generalized Aicardi's invariant is the universal type one invariant, and we show that the generalized Aicardi's invariant restricted to n-component links in the solid torus with zero winding number for each component is equal to an invariant we define using the universal cover of the solid torus. We also define and study a geometric invariant for n-component links in the solid torus. We give a lower bound on this invariant using the type one invariants, which are easy to calculate, which helps in computing this geometric invariant, which is usually hard to calculate.     

Finite-Type Invariants of Cubic Complexes

The paper is for a general audience and may serve as a preliminary introduction to the theory of finite-type invariants.     

一些特殊变换群的联合不变量和联合微分不变量

The theory of moving frames developed by Peter J Olver and M Fels has importaut applications to geometry,classical invariant theory.We will use this theory to classify joint invariants and joint differential invariants of some transformation groups.     

Legendrian Knots in Overtwisted Contact Structures on S3

We study the problem of classifying Legendrian knots in overtwisted contact structures on S ³. The question is whether topologically isotopic Legendrian knots have to be Legendrian isotopic if they have equal values of the well-known invariants rot and tb. We give positive answer in the case that there is an overtwisted disc intersecting none of the knots and we construct an example of a knot intersecting each overtwisted disc (this provides a counterexample to the conjecture of Eliashberg). Our proof needs some results on the structure of the group of contactomorphisms of S ³. We divide the subgroup Cont₊(S ³, ) of coorientation-preserving contactomorphisms for an overtwisted contact distribution into two classes.     

For a fixed Turaev shadow Jones-Vassiliev invariants depend polynomially on the gleams

We use Turaev's technique of shadows and gleams to parametrize the set of all knots in S ³ with the same Hopf projection. We show that the Vassiliev invariants arising from the Jones polynomial J _t (K) are polynomials in the gleams, i.e., for , the n-th order Vassiliev invariant u _n , defined by , is a polynomial of degree 2n in the gleams. Received: April 30, 1996     

The Invariants of the Clifford Groups

The automorphism group of the Barnes-Wall lattice L _m in dimension 2 ^m (m ; 3) is a subgroup of index 2 in a certain Clifford group of structure 2 ₊ ^1+2m . O ⁺(2m,2). This group and its complex analogue of structure .Sp(2m, 2) have arisen in recent years in connection with the construction of orthogonal spreads, Kerdock sets, packings in Grassmannian spaces, quantum codes, Siegel modular forms and spherical designs. In this paper we give a simpler proof of Runge@apos;s 1996 result that the space of invariants for of degree 2k is spanned by the complete weight enumerators of the codes , where C ranges over all binary self-dual codes of length 2k; these are a basis if m k - 1. We also give new constructions for L _m and : let M be the -lattice with Gram matrix . Then L _m is the rational part of M ^m, and = Aut(M^m). Also, if C is a binary self-dual code not generated by vectors of weight 2, then is precisely the automorphism group of the complete weight enumerator of . There are analogues of all these results for the complex group , with doubly-even self-dual code instead of self-dual code.     

Vassiliev invariants of type two for a link

We show that any type two Vassiliev invariant of a link can be expressed as a linear combination of the second coefficients of the Conway polynomials of its components and a quadratic expression of linking numbers.

     

Black and white local invariants of mappings from surfaces into three-space

Goryunov proved that the space of local invariants of Vassiliev type for generic maps from surfaces to three-space is three-dimensional. The basic invariants were the number of pinch points, the number of triple points and one linked to a Rokhlin type invariant. In this paper we show that, by colouring the complement of the image of the map with a chess board pattern, we can produce a six-dimensional space of local invariants. These are essentially black and white versions of the above. Simple examples show how these are more effective. This revised version was published online in August 2006 with corrections to the Cover Date.     

A geometric characterization of Vassiliev invariants

It is a well-known paradigm to consider Vassiliev invariants as polynomials on the set of knots. We prove the following characterization: a rational knot invariant is a Vassiliev invariant of degree if and only if it is a polynomial of degree on every geometric sequence of knots. Here a sequence with is called geometric if the knots coincide outside a ball , inside of which they satisfy for all and some pure braid . As an application we show that the torsion in the braid group over the sphere induces torsion at the level of Vassiliev invariants: there exist knots in that can be distinguished by -invariants of finite type but not by rational invariants of finite type. In order to obtain such torsion invariants we construct over a universal Vassiliev invariant of degree for knots in .

     

Topological Correlations in Trivial Knots: New Arguments in Favor of the Representation of a Crumpled Polymer Globule

We prove the representation of a fractal crumpled structure of a strongly collapsed unknotted polymer chain. In this representation, topological considerations result in the chain developing a densely packed system of folds, which are mutually segregated at all scales. We investigate topological correlations in randomly generated knots on rectangular lattices (strips) of fixed widths. We find the probability of the spontaneous formation of a trivial knot and the probability that each finite part of a trivial knot becomes a trivial knot itself after joining its ends in a natural way. The complexity of a knot is characterized by the highest degree of the Jones–Kauffman polynomial topological invariant. We show that the knot complexity is proportional to the strip length in the case of long strips. Simultaneously, the typical complexity of a quasi-knot, which is a part of a trivial knot, is substantially less. Our analysis shows that the latter complexity is proportional to the square root of the strip length. The results obtained clearly indicate that the topological state of any part of a trivial knot densely filling the lattice is also close to the trivial state.     

Seiberg–Witten Invariants of Nonsimple Type and Einstein Metrics

We study the behavior of the moduli space of solutions to theSeiberg–Witten equations under a conformal change in the metric of aKähler surface (M,g). If the canonical line bundle K _M is ofpositive degree, we prove there is only one (up to gauge) solution tothe equations associated to any conformal metric to g. We use this, toconstruct examples of four dimensional manifolds withSpin ^c -structures, whose moduli spaces of solutions to theSeiberg–Witten equations, represent a nontrivial bordism class ofpositive dimension, i.e. the Spin ^c -structures are not inducedby almost complex structures. As an application, we show the existenceof infinitely many nonhomeomorphic compact oriented 4-manifolds withfree fundamental group and predetermined Euler characteristic andsignature that do not carry Einstein metrics.     

Clasp-pass move and Vassiliev invariants of type three for knots

Recently it has been proved that if and only if two knots and have the same value for the Vassiliev invariant of type two, then can be deformed into by a finite sequence of clasp-pass moves. In this paper, we determine the difference of the values of the Vassiliev invariant of type three between two knots which can be deformed into each other by a clasp-pass move.

     

SL(2)-Solution of the Pentagon Equation and Invariants of Three-Dimensional Manifolds

We construct an algebraic complex corresponding to a triangulation of a three-manifold starting with a classical solution of the pentagon equation, constructed earlier by the author and Martyushev and related to the flat geometry, which is invariant under the group SL(2). If this complex is acyclic (which is confirmed by examples), we can use it to construct an invariant of the manifold.     

Invariants of Algebras and Finite Groups

Group theoretical constructions usually terminate in the problem to decide whether two groups are isomorphic. In the case of arbitrary finite groups the calculation of ordinary group characters is not sufficient to decide about it. R. Brauer posed the problem, to find suitable additional group invariants. Applying the theory of norm-type forms of associative algebras, specialized to group algebras, we found, that the 1-, 2-, and 3-characters in the nonmodular case (with some restrictions on the characteristic), especially over the field of complex numbers, are necessary and sufficient for the finite groups. This sharpens a recent result of E. Formanek and D. Sibley on group determinants. Detailed proofs will be given elsewhere. Here we give an overview on recent related results and add a remark concerning calculations in the modular group case.     

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.