9,565 research outputs found
A Turaev surface approach to Khovanov homology
We introduce Khovanov homology for ribbon graphs and show that the Khovanov
homology of a certain ribbon graph embedded on the Turaev surface of a link is
isomorphic to the Khovanov homology of the link (after a grading shift). We
also present a spanning quasi-tree model for the Khovanov homology of a ribbon
graph.Comment: 30 pages, 18 figures, added sections on virtual links and
Reidemeister move
Milnor Invariants and Twisted Whitney Towers
This paper describes the relationship between the first non-vanishing Milnor
invariants of a classical link and the intersection invariant of a twisted
Whitney tower. This is a certain 2-complex in the 4-ball, built from immersed
disks bounded by the given link in the 3-sphere together with finitely many
`layers' of Whitney disks.
The intersection invariant is a higher-order generalization of the
intersection number between two immersed disks in the 4-ball, well known to
give the linking number of the link on the boundary, which measures
intersections among the Whitney disks and the disks bounding the given link,
together with information that measures the twists (framing obstructions) of
the Whitney disks.
This interpretation of Milnor invariants as higher-order intersection
invariants plays a key role in the classifications of both the framed and
twisted Whitney tower filtrations on link concordance (as sketched in this
paper). Here we show how to realize the higher-order Arf invariants, which also
play a role in the classifications, and derive new geometric characterizations
of links with vanishing Milnor invariants of length less than or equal to 2k.Comment: Typo corrected in statement of Theorem 16; no change to proof needed.
Otherwise, this revision conforms with the version published in the Journal
of Topology. 36 pages, 23 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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