arXiv subjects

Yonghan Xiao

Publications and source records attributed to Yonghan Xiao.

4 recordsLinked to original sources

arXiv2026

Real link Floer homology

In this paper, we define real link Floer homology for strongly invertible and doubly periodic links in closed real $3$-manifolds with connected fixed sets, which generalizes real Heegaard Floer homology and real sutured Heegaard Floer homology. We give a combinatorial description of the theory in $S^3$ via real grid diagrams and use it to investigate structural properties of the theory as well as properties of strongly invertible knots. A computer implementation computing real grid homology of knots was written by Zhenkun Li. An appendix including real grid homology of 50+ small knots is made jointly by Zhenkun Li and the author, from which we observe several interesting phenomenon.

math.GT↗

arXiv2026

Real sutured Heegaard Floer homology

We develop a theory of real sutured manifolds and a real Heegaard Floer theory for these manifolds. We develop a notion of real nice diagrams, and prove that our invariant is combinatorially computable. Our theory shares many structural properties with Juhász's sutured Floer homology, as does the topological theory of real sutured manifolds with Gabai's original sutured manifold theory. We also show that our invariant has several new structural properties differentiating it from sutured Floer homology.

math.GT↗

arXiv2026

The equivalence between two real Seiberg-Witten Floer homologies

We show that for a real rational homology sphere $Y$ equipped with a real $\mathrm{spin^c}$ structure $\mathfrak{s}$, the real monopole Floer homology defined by Li and the real Seiberg-Witten Floer homology defined by Konno, Miyazawa and Taniguchi are isomorphic. As corollaries, we identify some Froyshov-type invariants and prove two Smith-type inequalities.

math.GT↗

arXiv2025

Grid homology for singular links in lens space and a resolution cube

In this paper, we define grid homologies for singular links in lens spaces and use them to construct a resolution cube for knot Floer homology of regular links in lens spaces. The results will first be proved over $\mathbb{Z}/2\mathbb{Z}$ and then over $\mathbb{Z}$ with the help of sign assignments. We will also identify the signed grid homology and classical knot Floer homology over $\mathbb{Z}$ for regular links in lens spaces, illustrating the fact that our resolution cube is genuinely one for knot Floer homology. The main advancement in the paper is that we give a complete description of singular knot theory in lens spaces which was only defined in $S^3$ previously and we construct a signed combinatorial resolution cube for knot Floer homology in lens spaces which may be powerful in relating $HFK^\circ$ to other link homology theories.

math.GT↗