The paper introduces the concept of Kauffman bracket skein modules, which are 3-manifold invariants that generalize the skein theory of polynomial link invariants in S3 to arbitrary 3-manifolds. The authors focus on computing the Kauffman bracket skein module of small Seifert manifolds S2(k1, k2, k3), where ki are integers.
The key highlights and insights are:
The authors define elements T±(k, n) in the relative skein module S2,∞(S1 × I, {u, v}) and establish their recursive properties, which are important for the subsequent calculations.
The authors prove that small Seifert manifolds S2(k1, k2, k3) have special Heegaard diagrams, which are used to compute their skein modules.
The main result is a presentation of the Kauffman bracket skein module S2,∞(S2(k1, k2, k3)) as a quotient of the free module S2,∞(H2) by a submodule Jk1,k2,k3 generated by specific elements Rk,k′
l,m,n and R̃k,k′
l,m,n.
As an application, the authors re-calculate the skein module of lens spaces L(p, 1) using their main theorem.
The paper provides a comprehensive and detailed analysis of the Kauffman bracket skein module of small Seifert manifolds, extending previous results and demonstrating the power of the techniques developed.
إلى لغة أخرى
من محتوى المصدر
arxiv.org
الرؤى الأساسية المستخلصة من
by Minyi Liang,... في arxiv.org 09-30-2024
https://arxiv.org/pdf/2409.09438.pdfاستفسارات أعمق