Andrew Lobb
The Kanenobu knots and Khovanov-Rozansky homology.
Mon, 05/23/2011 - 15:01 — SomNambulAuthors: Andrew Lobb
Subjects: Geometric Topology
AbstractKanenobu has given infinite families of knots with the same HOMFLY
polynomials. We show that these knots also have the same sl(n) and HOMFLY
homologies, thus giving the first example of an infinite family of knots
undistinguishable by these invariants. This is a consequence of a structure
theorem about the homologies of knots obtained by twisting up the ribbon of a
ribbon knot with one ribbon.2-strand twisting and knots with identical quantum knot homologies.
Wed, 03/09/2011 - 16:01 — SomNambulAuthors: Andrew Lobb
Subjects: Geometric Topology
AbstractGiven a knot, we ask how its Khovanov and Khovanov-Rozansky homologies change
under the operation of introducing twists in a pair of strands. We obtain long
exact sequences in homology and further algebraic structure which is then used
to derive topological and computational results. In particular, we show that
the suite of Khovanov-Rozansky invariants does not form a complete invariant.
Specifically, for any natural number m we show that there exist m distinct
knots with identical sl(n) homologies for all n simultaneously and hence also
with identical HOMFLY homologies.Computable bounds for Rasmussen's concordance invariant.
Thu, 08/20/2009 - 23:06 — SomNambulAuthors: Andrew Lobb
Subjects: Geometric Topology
AbstractGiven a diagram D of a knot K, we give easily computable bounds for
Rasmussen's concordance invariant s(K). The bounds are not independent of the
diagram chosen, but we show that for knots satisfying a given condition the
bounds are tight. This improves on previously known Bennequin-type bounds.
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