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Search: MSC category 20F36 ( Braid groups; Artin groups )

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1. CJM Online first

Ashraf, Samia; Azam, Haniya; Berceanu, Barbu
Representation stability of power sets and square free polynomials
The symmetric group $\mathcal{S}_n$ acts on the power set $\mathcal{P}(n)$ and also on the set of square free polynomials in $n$ variables. These two related representations are analyzed from the stability point of view. An application is given for the action of the symmetric group on the cohomology of the pure braid group.

Keywords:symmetric group modules, square free polynomials, representation stability, Arnold algebra
Categories:20C30, 13A50, 20F36, 55R80

2. CJM 2007 (vol 59 pp. 418)

Stoimenow, A.
On Cabled Knots and Vassiliev Invariants (Not) Contained in Knot Polynomials
It is known that the Brandt--Lickorish--Millett--Ho polynomial $Q$ contains Casson's knot invariant. Whether there are (essentially) other Vassiliev knot invariants obtainable from $Q$ is an open problem. We show that this is not so up to degree $9$. We also give the (apparently) first examples of knots not distinguished by 2-cable HOMFLY polynomials which are not mutants. Our calculations provide evidence of a negative answer to the question whether Vassiliev knot invariants of degree $d \le 10$ are determined by the HOMFLY and Kauffman polynomials and their 2-cables, and for the existence of algebras of such Vassiliev invariants not isomorphic to the algebras of their weight systems.

Categories:57M25, 57M27, 20F36, 57M50

3. CJM 2003 (vol 55 pp. 822)

Kim, Djun Maximilian; Rolfsen, Dale
An Ordering for Groups of Pure Braids and Fibre-Type Hyperplane Arrangements
We define a total ordering of the pure braid groups which is invariant under multiplication on both sides. This ordering is natural in several respects. Moreover, it well-orders the pure braids which are positive in the sense of Garside. The ordering is defined using a combination of Artin's combing technique and the Magnus expansion of free groups, and is explicit and algorithmic. By contrast, the full braid groups (on 3 or more strings) can be ordered in such a way as to be invariant on one side or the other, but not both simultaneously. Finally, we remark that the same type of ordering can be applied to the fundamental groups of certain complex hyperplane arrangements, a direct generalization of the pure braid groups.


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