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.     

A framization of the Hecke algebra of type B

We introduce a framization of the Hecke algebra of type B. For this framization, we construct a faithful tensorial representation and two linear bases. We also construct a Markov trace on such an algebra, and from this trace we derive isotopy invariants for framed and classical knots and links in the solid torus.     

The Mod 2 Kauffman Bracket Skein Module of Thickened Torus

Framed links in thickened torus are studied. We define the mod 2 Kauff-man bracket skein module of thickened torus and give an expression of a framed link in this module. From this expression we propose a new ambient isotopic invariant of framed links.     

Scheduling independent jobs for torus connected networks with/without link contention

In this paper, we investigate the problem of how to schedule n independent jobs on an m × m torus based network. We develop a model to to quantify the effect of contention for communication links on the dilation of job execution time when multiple jobs share communication links; we then design an efficient algorithm to schedule a set of n independent jobs with different torus size requirements on a given torus with an objective to minimize the total schedule length. We also develop a feasibility algorithm for pre-emptively scheduling a given set of jobs on a torus of given size with a given deadline. We provide analysis for both the algorithms.     

On the Kauffman skein modules

 Let k be a subring of the field of rational functions in α, s which contains α ^±1 ,s ^±1 . Let M be a compact oriented 3-manifold, and let K(M) denote the Kauffman skein module of M over k. Then K(M) is the k-module freely generated by isotopy classes of framed links in M modulo the Kauffman skein relations. In the case of , the field of rational functions in α, s, we give a basis for the Kauffman skein module of the solid torus and a basis for the relative Kauffman skein module of the solid torus with two points on the boundary. We then show that K(S ¹ × S ² is freely generated by the empty link, i.e., . Received: 20 October 2001 / Revised version: 20 March 2002     

Spaces of knots in the solid torus,knots in the thickened torus,and links in the 3-sphere

We recursively determine the homotopy type of the space of any irreducible framed link in the 3-sphere, modulo rotations. This leads us to the homotopy type of the space of any knot in the solid torus, thus answering a question posed by Arnold. We similarly study spaces of unframed links in the 3-sphere, modulo rotations, and spaces of knots in the thickened torus. The subgroup of meridional rotations splits as a direct factor of the fundamental group of the space of any framed link except the unknot. Its generators can be viewed as generalizations of the Gramain loop in the space of long knots. Taking the quotient by certain such rotations relates the spaces we study. All of our results generalize previous work of Hatcher and Budney. We provide many examples and explicitly describe generators of fundamental groups.

     

Non-simple links with tunnel number one

We determine all non-simple links which admit an unknotting tunnel, i.e. links which contain an essential annulus or torus in its exterior and have tunnel number one.

     

加厚环面中标架环链的约化多项式

研究加厚环面中的标架环链.给出标架环链在Kauffman尖括号拆接代数中的表达式.利用Gro\"{o}bner基理论,我们从上述表达式中得到标架环链的约化多项式,该多项式是标架环链的同痕不变量且可计算.     

Invariants of B-Type Links Via an Extension of the Kauffman Bracket

We construct invariants of solid torus links ( -type invariants) generalizing the Kauffman bracket. These invariants give us expressions of statistical sums in some special cases. Bibliography: 29 titles.     

Tight contact structures on solid tori

In this paper we study properties of tight contact structures on solid tori. In particular we discuss ways of distinguishing two solid tori with tight contact structures. We also give examples of unusual tight contact structures on solid tori.

We prove the existence of a -valued and a -valued invariant of a closed solid torus. We call them the self-linking number and the rotation number respectively. We then extend these definitions to the case of an open solid torus. We show that these invariants exhibit certain monotonicity properties with respect to inclusion. We also prove a number of results which give sufficient conditions for two solid tori to be contactomorphic.

At the same time we discuss various ways of constructing a tight contact structure on a solid torus. We then produce examples of solid tori with tight contact structures and calculate self-linking and rotation numbers for these tori. These examples show that the invariants we defined do not give a complete classification of tight contact structure on open solid tori.

At the end, we construct a family of tight contact structure on a solid torus such that the induced contact structure on a finite-sheeted cover of that solid torus is no longer tight. This answers negatively a question asked by Eliashberg in 1990. We also give an example of tight contact structure on an open solid torus which cannot be contactly embedded into a sphere with the standard contact structure, another example of unexpected behavior.

     

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.     

Dynamical systems of translation groups on a torus

The criterion for linearizability is obtained by a property of denominator of rational numbers that are given by continued fractions. We can regard this criterion as a condition given by behaviors of orbits of dynamical systems of translation groups on a one-dimensional real torus. By using continued fractions, we obtain a criterion that is an analogous phenomenon in several complex variables that is determined by behaviors of dynamical systems of translation groups on a one-dimensional complex torus.     

Generalizations of the Wada representations and virtual link groups

We extend the Wada representations of the classical braid group to the virtual and welded braid groups. Using the resulting representations, we construct the groups of virtual links and prove that they are link invariants. We give some examples of calculating the groups of torus (virtual) links.     

Skeins,SU(N) three-manifold invariants and TQFT

The skein theory associated to the HOMFLY polynomial invariant of oriented knots and links in the three-sphere is explored in order to provide the background results necessary for the creation of a Topological Quantum Field Theory. A simple local duality result in the skein theory is proved. It allows vector space dimensions in the theory to be correlated with the structure constants in a skein algebra associated to the solid torus.     

The short ruler on the torus

ABSTRACT

We can shorten any path that links two given points by applying short ruler transforms iteratively. In this article we take a closer look at a short ruler process on the torus. The torus is a compact Riemannian manifold and at least a subsequence of the process converges to a geodesic between the two points. However, on compact Riemann manifolds there might exist different limit geodesics (with the same length). On the torus, the geodesics with the same length are isolated and the limit geodesic is unique.     

On the equivariant formality of Kähler manifolds with torus group actions

We consider simply connected compact Kähler manifolds which have a holomorphic action of a torus group. We use the existing equivariant models for rational homotopy to show that these spaces satisfy an equivariant formality condition over the complex numbers.     

Torus rational fibrations

We study rational fibrations where the fibre is an r-dimensional torus and the base is a formal space. We make use of the Eilenberg–Moore Spectral Sequence to prove the Toral Rank Conjecture in some cases.     

Skein modules and the noncommutative torus

We prove that the Kauffman bracket skein algebra of the cylinder over a torus is a canonical subalgebra of the noncommutative torus. The proof is based on Chebyshev polynomials. As an application, we describe the structure of the Kauffman bracket skein module of a solid torus as a module over the algebra of the cylinder over a torus, and recover a result of Hoste and Przytycki about the skein module of a lens space. We establish simple formulas for Jones-Wenzl idempotents in the skein algebra of a cylinder over a torus, and give a straightforward computation of the -th colored Kauffman bracket of a torus knot, evaluated in the plane or in an annulus.

     

Rational smoothness and fixed points of torus actions

We obtain a criterion for rational smoothness of an algebraic variety with a torus action, with applications to orbit closures in flag varieties, and to closures of double classes in regular group completions. Dedicated to the memory of Claude Chevalley     

Dimension and enumeration of primitive ideals in quantum algebras

In this paper, we study the primitive ideals of quantum algebras supporting a rational torus action. We first prove a quantum analogue of a Theorem of Dixmier; namely, we show that the Gelfand-Kirillov dimension of primitive factors of various quantum algebras is always even. Next we give a combinatorial criterion for a prime ideal that is invariant under the torus action to be primitive. We use this criterion to obtain a formula for the number of primitive ideals in the algebra of 2×n quantum matrices that are invariant under the action of the torus. Roughly speaking, this can be thought of as giving an enumeration of the points that are invariant under the induced action of the torus in the “variety of 2×n quantum matrices”. The first author thanks NSERC for its generous support. This research was supported by a Marie Curie Intra-European Fellowship within the 6th European Community Framework Programme held at the University of Edinburgh, by a Marie Curie European Reintegration Grant within the 7th European Community Framework Programme and by Leverhulme Research Interchange Grant F/00158/X.