arXiv subjects

Shangjun Shi

Publications and source records attributed to Shangjun Shi.

3 recordsLinked to original sources

arXiv2026

Building Foliations from Heegaard Diagrams

Every closed orientable 3-manifold admits both a Heegaard splitting and a coorientable codimension-one foliation. We address the foliation realization problem: constructing a foliation directly from a Heegaard diagram of arbitrary genus. Using Gabai's sutured manifold theory, we introduce the notion of a meridional sutured handlebody and prove that every such handlebody decomposes, via basic sutured decomposition operations, into a Reeb component together with product disks. We then present a three-step construction: (i) building two meridional sutured handlebodies from a given Heegaard diagram, (ii) gluing them along compatible disk regions by reversing disk decompositions, and (iii) filling the remaining cavities with a meridional sutured handlebody and some Reeb components. The construction applies to Heegaard diagrams of any genus.

math.GT↗

arXiv2024

Kauffman bracket skein module of two families of Seifert manifolds

We compute the Kauffman bracket skein modules of Seifert manifolds $\Sigma_{0,1}((k_1,1),(k_2,1))$ and $\Sigma_{0,0}((k_1,1),(k_2,1),(k_3,1))$ by providing presentations of them. From the obtained presentations, we show that the Kauffman bracket skein modules of $\Sigma_{0,1}((k_1,1),(k_2,1))$ are free with infinitely many generators when $k_1,k_2\ge 1$ and that of $\Sigma_{0,0}((k_1,1),(k_2,1),(k_3,1))$ are finitely generated when $k_1,k_2,k_3 \ge 2$. We also show that the empty link in either case is not trivial.

math.GT↗

arXiv2024

On the Kauffman bracket skein module of $(S^1 \times S^2) \ \# \ (S^1 \times S^2)$

Determining the structure of the Kauffman bracket skein module of all $3$-manifolds over the ring of Laurent polynomials $\mathbb Z[A^{\pm 1}]$ is a big open problem in skein theory. Very little is known about the skein module of non-prime manifolds over this ring. In this paper, we compute the Kauffman bracket skein module of the $3$-manifold $(S^1 \times S^2) \ \# \ (S^1 \times S^2)$ over the ring $\mathbb Z[A^{\pm 1}]$. We do this by analysing the submodule of handle sliding relations, for which we provide a suitable basis. Along the way we compute the Kauffman bracket skein module of $(S^1 \times S^2) \ \# \ (S^1 \times D^2)$. We also show that the skein module of $(S^1 \times S^2) \ \# \ (S^1 \times S^2)$ does not split into the sum of free and torsion submodules. Furthermore, we illustrate two families of torsion elements in this skein module.

math.GT↗