Legendrian contact homology and nondestabilizability

Abstract

We provide the first example of a Legendrian knot with nonvanishing contact homology whose Thurston-Bennequin invariant is not maximal.

Legendrian contact homology in the boundary of a subcritical Weinstein 4-manifold

“…For the purposes of this result, it suffices to work over the ring Z/2 instead of Z[t, t −1 ] by setting t = 1 and reducing mod 2. We will show that the homology for the DGA over this ring has a certain quotient that has (somewhat remarkably) previously appeared in the literature on Legendrian contact homology in a rather different context [31].…”

“…Thus φ descends to a surjection from H(A, ∂) to A ′′ . But it was proven in [31] that A ′′ is nonzero.…”

The contact homology of Legendrian knots with maximal Thurston–Bennequin invariant

“…Conversely, there are conjecturally nondestabilizable knots of type m(10 139 ), 10 161 , and m(12n 242 ) with nonmaximal tb and vanishing contact homology [3,20]. On the other hand, it is an open question whether there is a Legendrian knot K for which tb(K) is maximal but the contact homology of K vanishes.…”

“…Finally, the proof that K 2 has nonvanishing contact homology uses an action of C(K 2 ) on an infinite-dimensional vector space, just as the nonvanishing examples in [20] did. In Section 3, we will show that this is necessary in the sense that C(K 2 ) does not have any finite-dimensional representations.…”

“…, x 22 from left to right and right cusps x 23 , x 24 , x 25 from top to bottom with differentials specified in Appendix B. In order to show that K 2 has nontrivial contact homology, it will suffice to show that the characteristic algebra C 2 = C(K 2 ) is nonvanishing [20]. We remark that the abelianization of C 2 does vanish, however, so that both K 1 and K 2 have the same abelianized characteristic algebra.…”