The purpose of this paper is to interpret polynomial invariants of strongly invertible
links in terms of Khovanov homology theory. To a divide, that is a proper generic
immersion of a finite number of copies of the unit interval and circles in a
–disc,
one can associate a strongly invertible link in the
–sphere.
This can be generalized to signed divides: divides with
or
sign assignment to each crossing point. Conversely, to any link
that is strongly invertible
for an involution
,
one can associate a signed divide. Two strongly invertible links that are isotopic through
an isotopy respecting the involution are called strongly equivalent. Such isotopies give
rise to moves on divides. In a previous paper [Topology 47 (2008) 316-350], the
author finds an exhaustive list of moves that preserves strong equivalence, together
with a polynomial invariant for these moves, giving therefore an invariant for strong
equivalence of the associated strongly invertible links. We prove in this paper that
this polynomial can be seen as the graded Euler characteristic of a graded complex of
–vector
spaces. Homology of such complexes is invariant for the moves on divides and so is
invariant through strong equivalence of strongly invertible links.
Keywords
strongly invertible links, divides, Morse signed divides,
Khovanov homology
