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Combinatorial Heegaard-Floer Homology:
Invariance and Invariants of Transverse Knots
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(Joint with Ciprian Manolescu, Peter Ozsvath, and Zoltan Szabo.  See
math.GT/0610559.)

This is a continuation of the talk on September 22, although the basic
definitions will be recalled.  We look at one variant of the homology
theory, HFK-, which is a module over Z[u], where u is an indeterminate
with non-zero grading.  We give a direct combinatorial proof that the
homology is an invariant of the knot and exhibit explicit chain maps
relating the complexes for different representations of the knot.

Using these chain maps, we define an invariant of transversal knots
(knots transverse to the standard contact structure on R^3), which
lives in HFK-.  Stabilizing the knot multiplies the transverse
invariant by u.  Since the transverse invariant is never in the u
torsion of the homology, this gives an upper bound on how many times a
transverse knot can be destabilized.  This bound is always at least as
good as the Thurston-Bennequin inequality, related to the smooth
4-ball genus.
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