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 describe the generalized Casimir operators and their actions on the positive
representations Pλ of the modular double of split real quantum groups Uqq (gR).
We introduce the notion of virtual highest and lowest weights, and show that the central
characters admit positive values for all parameters λ. We show that their image defines
a semi-algebraic region bounded by real points of the discriminant variety independent
of q, and we discuss explicit examples in the lower rank cases.
Let Sg be a closed surface of genus g and Mg be the moduli space of Sg endowed with
the Weil–Petersson metric. In this article we investigate the Weil–Petersson curvatures
of Mg for large genus g. First, we study the asymptotic behavior of the extremal
Weil–Petersson holomorphic sectional curvatures at certain thick surfaces in Mg as
g → ∞. Then we prove two curvature properties on the whole space Mg as g → ∞
in a probabilistic way.
This article is a continuation of our work on the classification of holomorphic framed vertex
operator algebras of central charge 24. We show that a holomorphic framed VOA of central charge 24
is uniquely determined by the Lie algebra structure of its weight one subspace. As a consequence, we
completely classify all holomorphic framed vertex operator algebras of central charge 24 and show
that there exist exactly 56 such vertex operator algebras, up to isomorphism.