Alexander Modules, Skein Modules and Mapping Class Groups
Open Access DepositedFrom Branched Covers to 4-Manifolds
Covering spaces and branched coverings have played a fundamental role in topology and knot theory since the nineteenth century. In this thesis, we investigate several aspects of quantum invariants and four-dimensional topology through constructions based on covering spaces. The results presented here develop a unified framework applying covering techniques to algebraic and geometric invariants, including Alexander invariants, Fox colorings, skein modules, and mapping class groups. Our results in this thesis focus on three main themes
the Alexander module, skein modules and the mapping class groups of $S^2$-fiber bundles over surfaces. In Chapters 1–3, we study the Alexander module from the perspective of the infinite cyclic cover and its relationship with the Fox module. Using the framework of pseudo-colorings and Fox modules, we prove the generalization of Kauffman–Harary conjecture followed by several families of examples\cite{2}, as well as for certain non-alternating examples. We also present new observations on the Alexander modules of the Turk’s head link family\cite{3}, which suggest a potential connection to extensions of Plan’s Theorem over Laurent polynomial rings. In Chapter 4, we introduce the Nakanishi–Montesinos 3-move conjecture and construct six explicit counterexamples\cite{4}. We also provide a complete classification of all links with up to 20 crossings under the 3-move equivalence. In Chapter 5, we study the cubic skein module and its applications to coloring problems\cite{5}. In particular, we investigate conjectural approaches related to 7-colorings and prove a conjecture for the family of 2-bridge links. We show that, in the cubic skein module, the difference between a 2-bridge link and its mirror image can always be expressed as a multiple of the difference of Hopf link and its mirror diagram. In Chapter 6, we apply a generalization of the Dax invariants for embedded surfaces in nontrivial $S^2$-bundles over surfaces to construct a surjective homomorphism from the smooth mapping class group onto $\mathbb{Z}^{\infty}$. Motivated by viewpoint of coverings, we study lifted diffeomorphisms in mapping class group($MCG$) to apply the generalized Dax invariants in the trivial bundle case. We show that the $\mathbb{Z}^{\infty}$ surjectivity property of nontrivial $S^2$-bundle is inherited from the trivial $S^2$-bundle\cite{6}. In the appendix A, we show the construction of a new Cabling link\cite{A1} that is Khovanov A-adequate but not A-adequate.
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