Research & Projects

The sliceness of Conway knot used to be an open question for decades until Lisa Piccirillo gave a negative answer using trace embedding lemma, RBG link construction and Rassmussen’s s-invariant.  This work not only denied a longstanding possible counter example to Slice-Ribbon conjecture but also proposed tools worthing further development. In our paper, we first review theproof and then summerize background information and further application of the tools. We will mainly focus on Khovanov homology and s-invariant, RBG link construction and trace embedding lemma.  Besides backrgound and basic of these tools, we also include their further and combined applications, e.g. bounding projective slice genus.  Moreover,  we give a detailed verision of computation on Conway knot and compute some new examples in the appendix.

See the paper from the link below, if you are interested in it! 

https://drive.google.com/file/d/1yt_IrjFbadtI-mzbuSiRfYh6eLd98mvI/view?usp=drive_link

Heegaard Floer homology is a useful homology theory developed by Ozsv´ath and Szab´o for studying properties of three manifolds. It is in particular important for investigate three manifolds obtained from surgery on knots on links.  Our paper aims at summarize definition and basic properties of this homology theory. We also include useful computational tools, basic examples and its application to surgery investigation.

See the paper from the link below, if you are interested in it! 

https://drive.google.com/file/d/1USW0OjY4MSQOkkGUoCaSDzutpKUYXVH1/view?usp=drive_link

This article will lay out main properties of Heegaard Floer homology and Knot Floer homology, including the 3-manifold invariant correction term from Heegaard Floer theory, concordance invariants 𝜏,𝜈.  Our main purpose is to state and prove that knot surgery formula for Heeggard Floer homology of integral and rational homology spheres( a summary of  works of  Ozsv´ath and Szab´o) and apply it to prove cosmetic surgery conjecture( a restate of work of Yi Ni and Zongtao Wu.

See the paper from the link below, if you are interested in it! 

Apologize to foreign viewers that this is my thesis as undergraduate student, so it is written in Chinese.

https://drive.google.com/file/d/1st6gZGV1OirIbYbG3N5Zzd1TYkpG2_u9/view?usp=drive_link

In this paper, we define grid homology for singular links in lens spaces and use it to construct a resolution cube for knot Floer homology of regular links in lens spaces. The results will first be proven over Z/2Z and then extended to be over Z via a sign assignment.

This paper has been uploaded to arxiv, the link is shown below.

https://arxiv.org/abs/2501.03579

We show that for a real rational homology sphere equipped with a real structure, 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 Frøyshov-type invariants and prove two Smith-type inequalities. 

This paper has been uploaded to arxiv, the link is shown below.

[2510.03709] The equivalence between two real Seiberg-Witten homologies