AI 中文总结
该研究针对Khovanov等人引入的薄平曲面范畴TFS，证明了连通可视tf-配边的范式定理与关系充分性定理，给出了TFS的完整生成元-关系表示。
AI 中文摘要
Khovanov、Qi和Rozansky引入的薄平曲面范畴TFS是一个严格对称幺半范畴，其态射为带拐角的紧定向曲面，这些曲面源自带形中浸入图的邻域。我们证明两个主要结果：一是范式定理，每个连通可视tf-配边在TFS中等于由其拓扑类型确定的典范复合；二是充分性定理，我们给出TFS范畴中一组关系，它们是充分的，即任何表示同一tf-配边的生成元的两个复合都可通过该组关系的有限序列关联。这些结果共同给出了TFS的完整生成元-关系表示。
英文摘要
The category TFS of thin flat surfaces, introduced by Khovanov, Qi, and Rozansky, is a strict symmetric monoidal category whose morphisms are compact oriented surfaces with corners arising as neighbourhoods of immersed graphs in a strip. We prove two main results. First, a normal form theorem: every connected viewable tf-cobordism is equal, in TFS, to a canonical composite determined by its topological type. Second, a sufficiency theorem: we establish a list of relations in the TFS category which are sufficient, meaning that any two composites of generators representing the same tf-cobordism are related by a finite sequence of the listed relations. Together, these results give a complete generators-and-relations presentation of TFS.
Comments28 pages, 42 figures
