The SL3-skein algebra of a closed surface Σg is a quantization of the SL3 character variety of Σg, where g denotes the genus of the surface. This algebra admits a basis consisting of non-elliptic web diagrams in Σg after Sikora–Westbury [22]. In this paper, we introduce explicit SL3 version of Dehn–Thurston coordinates for non-elliptic web diagrams on Σg, yielding a parametrization by a submonoid of Zd. The idea is to tropicalize the Fenchel–Nielsen coordinates for the SL3-Hitchin component [23]. Here d=16g−16 for g≥2 and d=4 in the torus case g=1, coinciding with the dimension of the corresponding character variety.
In the present paper, we first revisit the theory of distinguished varieties and develop some new results. These results are used to characterize the essential spectrum and the essential normality of distinguished quotient modules of Aα2(Dd), where α≥−1 is an integer. It is shown that the essential spectrum of a distinguished quotient module is identified with the topological boundary of the variety associated with the corresponding submodule. The essential normality of radical distinguished quotient modules is proven under the assumption on the regularity of the associated variety along ∂Dd. The non-triviality of K-homology elements of these distinguished quotient modules are verified.
This note corrects an error in the proof of Proposition 13 in [1] and simultaneously establishes a more general result. We prove that if M is a compact connected oriented 4-manifold with connected boundary ∂M, and if an unbounded number of disjoint copies of M embed topologically and locally flatly in the interior of a compact 4-manifold N, then TorH1(∂M;Z) is a direct double, i.e., TorH1(∂M;Z)≅A⊕A, with the linking pairing vanishing identically on the first summand, i.e., the linking pairing is split metabolic. This partially generalizes Hantzsche's theorem stating that the linking pairing for a closed 3-manifold that embeds in S4 is hyperbolic.
In 2020, Albers et al. [2] introduced a variational framework for Hamiltonian delay systems on closed symplectic manifolds in which the delay is confined to the nonlinear term. Their construction, however, is not applicable to the standard Hamiltonian delay systems on manifolds, or even in Euclidean space, given byx′(t)=J∇H(x(t−τ)).We address this gap through three complementary contributions. First, we devise a new variational principle for the above system in Euclidean space, embedding the delay in the kinetic (quadratic) term and thereby bypassing the limitations of the aforementioned framework. Second, assuming the Hamiltonian is asymptotically linear at both the origin and infinity, we combine critical point theory with an S1-index theory to obtain lower bounds on the number of periodic solutions; a fully worked example illustrates the computation. Third, we propose a Gaussian process with a delay kernel as a numerical method for finding periodic solutions of Hamiltonian delay systems. Experiments show that a composite kernel combining periodic and delay components reliably reconstructs periodic orbits from sparse data.
In this paper, we classify all simple quasi-finite weight modules over the Lie superalgebra of differential operators on the super circle S1|1. We obtain that any simple quasi-finite weight module over the differential operator superalgebra is a highest weight module, a lowest weight module, or a module of the intermediate series. As a by-product, we find a widely used approach to give a simple proof for the classification of simple cuspidal modules over the Lie algebra of differential operators on the circle S1, which is the main result of [33].
