277 research outputs found
Commutators and squares in free groups
Let F_2 be the free group generated by x and y. In this article, we prove
that the commutator of x^m and y^n is a product of two squares if and only if
mn is even. We also show using topological methods that there are infinitely
many obstructions for an element in F_2 to be a product of two squares.Comment: Published by Algebraic and Geometric Topology at
http://www.maths.warwick.ac.uk/agt/AGTVol4/agt-4-27.abs.htm
A refinement of Rasmussen's s-invariant
In a previous paper we constructed a spectrum-level refinement of Khovanov
homology. This refinement induces stable cohomology operations on Khovanov
homology. In this paper we show that these cohomology operations commute with
cobordism maps on Khovanov homology. As a consequence we obtain a refinement of
Rasmussen's slice genus bound s for each stable cohomology operation. We show
that in the case of the Steenrod square Sq^2 our refinement is strictly
stronger than s.Comment: 26 pages, 2 figure
A perturbation of the geometric spectral sequence in Khovanov homology
We study the relationship between Bar-Natan's perturbation in Khovanov
homology and Szabo's geometric spectral sequence, and construct a link
invariant that generalizes both into a common theory. We study a few properties
of the new invariant, and introduce a family of s-invariants from the new
theory in the same spirit as Rasmussen's s-invariant
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