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Examples of finding norm-minimizing surfaces and Thurston polytope

Let $Y$ be a (compact, oriented, connected) $3$-manifold. Thurston introduced a norm on $H_2(Y, \partial Y)$, which is defined as follow: any class $x \in H_2(Y, \partial Y; \mathbb{Z})$ is ...
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Isomorphims on homologies induced by cylindrical structure

This question is related to my previous question Cylindrical structure and homology. Let $S$ be a oriented compact (topological) 2-manifold. We consider a cylinder $M=S\times I$ over $S$, here $I=[0, ...
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All Embeddings of a 3-D g-Handlebody Hg in S^4 is Trivial (i.e., any two embeddings are isotopic)

I think we can argue that Sg --the genus-g surface -- has only a trivial embedding in S4 , since Sg is topologically a wedge of g S1's, and there are no knotted S1's in S4 (meaning that any two ...