Surgery obstructions from Khovanov homology

Abstract

For a 3-manifold with torus boundary admitting an appropriate involution, we show that Khovanov homology provides obstructions to certain exceptional Dehn fillings. For example, given a strongly invertible knot in S 3 , we give obstructions to lens space surgeries, as well as obstructions to surgeries with finite fundamental group. These obstructions are based on homological width in Khovanov homology, and in the case of finite fundamental group depend on a calculation of the homological width for a family of Montesinos links.

Khovanov homology and the symmetry group of a knot

“…There is a third natural grading to consider: Setting δ = u − q records diagonals of slope 1 in the (u, q)-plane and gives rise to a 1 2 Z-grading on Kh(L). This may be relaxed to a relative Z-grading (compare [47]). It is an absolute Z-grading for knots and we have that…”

“…Precisely, if T n is the representative obtained from T by adding n half-twists, then we have T (n) = T n (0) (see Figure 3). Rational tangle attachments other than these horizontal twists will not, in general, preserve the suture despite the fact that the equivalence class of the underlying (unsutured) tangle is preserved (see [47], for example, for details on this more standard notion of tangle equivalence).…”

“…Other examples include Ng's bound on the Thurston-Bennequin number [28], Plamenevskaya's invariant of transverse knots [35], and obstructions to finite fillings on strongly invertible knots due to the author [47, 48].There are two features common to each of these applications. First, the quantity extracted from Khovanov homology is an integer (a particular grading [28,35,37], a count of a collection of gradings [47], or a dimension count [21]); and second, the quantity measured is not one that can be extracted from the Jones polynomial -additional structure in Khovanov homology is essential in each case. The latter points to a clear advantage of Date: April 5, 2017.…”

Contact structures and reducible surgeries

“…(Berge knots are strongly invertible by a result of Osborne [57], cf. [78], so they are already known to satisfy the cabling conjecture [23].) Proposition 3.6.…”

A Survey on the Turaev Genus of Knots

“…Lowrance [22] and Watson [36] have proved that the width of Khovanov homology remains unchanged after replacing a crossing in a link diagram with an alternating rational tangle, provided the crossing satisfies certain conditions. Using Corollary 4.6, this generates many families of knots with unbounded Turaev genus.…”