Search results for "Mathematics - Geometric Topology"
showing 10 items of 55 documents
HOMFLY-PT skein module of singular links in the three-sphere
2012
For a ring R, we denote by [Formula: see text] the free R-module spanned by the isotopy classes of singular links in đ3. Given two invertible elements x, t â R, the HOMFLY-PT skein module of singular links in đ3 (relative to the triple (R, t, x)) is the quotient of [Formula: see text] by local relations, called skein relations, that involve t and x. We compute the HOMFLY-PT skein module of singular links for any R such that (t-1 - t + x) and (t-1 - t - x) are invertible. In particular, we deduce the Conway skein module of singular links.
Compressed Drinfeld associators
2004
Drinfeld associator is a key tool in computing the Kontsevich integral of knots. A Drinfeld associator is a series in two non-commuting variables, satisfying highly complicated algebraic equations - hexagon and pentagon. The logarithm of a Drinfeld associator lives in the Lie algbera L generated by the symbols a,b,c modulo [a,b]=[b,c]=[c,a]. The main result is a description of compressed associators that satisfy the compressed pentagon and hexagon in the quotient L/[[L,L],[L,L]]. The key ingredient is an explicit form of Campbell-Baker-Hausdorff formula in the case when all commutators commute.
Teichmuller Space and Related Topics : Proceedings of the workshop on Geometry, January 20, 2011, JOSAI UNIVERSITY
2012
The theory of geometric structures on a surface with nonempty boundary can be developed by using a decomposit,ion of such a surface into hexagons, in the same way as the theory of geometric structures on a surface without boundary is developed using the decomposition of such a surface into pairs of pants. The basic elements of the theory for surfaces with boundary include the study of measured foliations and of hyperbolic structures on hexagons. It turns out that there is an interesting space of measured foliations on a hexagon, which is equipped with a piecewise-Iinear structure (in fact, a natural cell-decomposition), and this space is a natural boundary for the space of hyperbolic structâŚ
Birman's conjecture for singular braids on closed surfaces
2003
Let M be a closed oriented surface of genus gâĽ1, let Bn(M) be the braid group of M on n strings, and let SBn(M) be the corresponding singular braid monoid. Our purpose in this paper is to prove that the desingularization map Ρ : SBn(M)ââ¤[Bn(M)], introduced in the definition of the Vassiliev invariants (for braids on surfaces), is injective.
On cyclic branched coverings of prime knots
2007
We prove that a prime knot K is not determined by its p-fold cyclic branched cover for at most two odd primes p. Moreover, we show that for a given odd prime p, the p-fold cyclic branched cover of a prime knot K is the p-fold cyclic branched cover of at most one more knot K' non equivalent to K. To prove the main theorem, a result concerning the symmetries of knots is also obtained. This latter result can be interpreted as a characterisation of the trivial knot.
On codimension two embeddings up to link-homotopy
2017
We consider knotted annuli in 4-space, called 2-string-links, which are knotted surfaces in codimension two that are naturally related, via closure operations, to both 2-links and 2-torus links. We classify 2-string-links up to link-homotopy by means of a 4-dimensional version of Milnor invariants. The key to our proof is that any 2-string link is link-homotopic to a ribbon one; this allows to use the homotopy classification obtained in the ribbon case by P. Bellingeri and the authors. Along the way, we give a Roseman-type result for immersed surfaces in 4-space. We also discuss the case of ribbon k-string links, for $k\geq 3$.
Hyperbolic isometries versus symmetries of links
2009
We prove that every finite group is the orientation-preserving isometry group of the complement of a hyperbolic link in the 3-sphere.
A note on the Lawrence-Krammer-Bigelow representation
2002
A very popular problem on braid groups has recently been solved by Bigelow and Krammer, namely, they have found a faithful linear representation for the braid group B_n. In their papers, Bigelow and Krammer suggested that their representation is the monodromy representation of a certain fibration. Our goal in this paper is to understand this monodromy representation using standard tools from the theory of hyperplane arrangements. In particular, we prove that the representation of Bigelow and Krammer is a sub-representation of the monodromy representation which we consider, but that it cannot be the whole representation.
On the classification of mapping class actions on Thurston's asymmetric metric
2011
AbstractWe study the action of the elements of the mapping class group of a surface of finite type on the TeichmĂźller space of that surface equipped with Thurston's asymmetric metric. We classify such actions as elliptic, parabolic, hyperbolic and pseudo-hyperbolic, depending on whether the translation distance of such an element is zero or positive and whether the value of this translation distance is attained or not, and we relate these four types to Thurston's classification of mapping class elements. The study is parallel to the one made by Bers in the setting of TeichmĂźller space equipped with TeichmĂźller's metric, and to the one made by Daskalopoulos and Wentworth in the setting of TeâŚ
From braid groups to mapping class groups
2005
This paper is a survey of some properties of the braid groups and related groups that lead to questions on mapping class groups.