A Jones polynomial for braid-like isotopies of oriented links and its categorification

A braid-like isotopy for links in 3-space is an isotopy which uses only those Reidemeister moves which occur in isotopies of braids. We define a refined Jones polynomial and its corresponding Khovanov homology which are, in general, only invariant under braid-like isotopies.Comment: 19 pages, many figure

Track association performance of the best hypotheses search method

Uncontrolled space objects in the geostationary orbit domain are hazardous threats for active satellites. Catalogs need to be build up, in order to protect this precious domain. The Swiss ZimSMART telescope, located in Zimmerwald, regularly scans the geostationary ring in order to provide a homogenous coverage. This surveying technique typically yields short measurement arcs, called tracklets. Each tracklet provides information about the line-of-sight and the rates of change but typically not about the full state of the observed object. Computationally intensive multi-hypothesis filter methods have been developed to associate tracklets with each other. An effective implementation to this approach is presented that uses an optimization algorithm to reduce the number of initial hypotheses. The method is tested with a set of real measurements of the aforementioned telescope

Polynomial invariants which can distinguish the orientations of knots

This paper contains the first knot polynomials which can distinguish the orientations of classical knots and which make no excplicit use of the knot group. But they make extensive use of the meridian and of the longitude in a geometric way. Let $M$ be the topological moduli space of long knots up to regular isotopy, and for any natural number $n > 1$ let $M_n$ be the moduli space of all n-cables $nK$ of framed long knots $K$ which are twisted by a given string link $T$ to close to a knot in the solid torus, with a marked point on the knot at infinity. First we construct integer valued combinatorial 1-cocycles for $M_n$ by using Gauss diagram formulas for finite typ invariants. We observe then that our 1-cocycles allow to fix certain crossings of $nK$ as local parameters of the 1-cocycles. Finally, we transform the local parameter into an unordered set of global parameters by following the crossings in the isotopy. We evaluate now the 1-cocycles on a canonical loop in $M_n$. The outcome are polynomial valued invariants of $K$, where the variables are indexed by finite type invariants and by regular isotopy types of string links $T$.Comment: 36 pp, 19 figure