We construct an enhanced version of knot contact homology, and
show that we can deduce from it the group ring of the knot group together
with the peripheral subgroup. In particular, it completely determines a knot up
to smooth isotopy. The enhancement consists of the (fully noncommutative)
Legendrian contact homology associated to the union of the conormal torus
of the knot and a disjoint cotangent fiber sphere, along with a product on a
filtered part of this homology. As a corollary, we obtain a new, holomorphiccurve
proof of a result of the third author that the Legendrian isotopy class of
the conormal torus is a complete knot invariant.
We introduce symplectic Calabi–Yau caps to obtain new obstructions to exact fillings. In
particular, they imply that any exact filling of the standard contact structure on the unit
cotangent bundle of a hyperbolic surface has vanishing first Chern class and has the same integral
homology and intersection form as its disk cotangent bundle. This gives evidence to a conjecture
that all of its exact fillings are diffeomorphic to the disk cotangent bundle. As a result, we also
obtain the first infinite family of Stein fillable contact 3-manifolds with uniform bounds on the
Betti numbers of its exact fillings but admitting minimal strong fillings of arbitrarily large b2.
Moreover, we introduce the notion of symplectic uniruled/adjunction caps and uniruled/
adjunction contact structures to present a unified picture to the existing finiteness results on the
topological invariants of exact/strong fillings of a contact 3-manifold. As a byproduct, we find
new classes of contact 3-manifolds with the finiteness properties and extend Wand’s obstruction
of planar contact 3-manifolds to uniruled/adjunction contact structures with complexity zero.
We prove a version of equivariant split generation of Fukaya category when a symplectic manifold admits a free action of a finite group <i>G</i>. Combining this with some generalizations of Seidel's algebraic frameworks from , we obtain new cases of homological mirror symmetry for some symplectic tori with non-split symplectic forms, which we call <i>special isogenous tori</i>. This extends the work of AbouzaidSmith . We also show that derived Fukaya categories are complete invariants of special isogenous tori.