Instanton Floer homology, sutures, and Euler characteristics

Quantum topology

article

Zhenkun Li¹  Fan Ye² 

[1] Stanford University;University of Cambridge

关键词: Instanton;    Floer homology;    sutures;    Euler characteristics;   

DOI  :  10.4171/qt/182

学科分类：内科医学

来源: European Mathematical Society

【 摘 要 】

This is a companion paper to an earlier work of the authors. In this paper, we provide an axiomatic definition of Floer homology for balanced sutured manifolds and prove that the graded Euler characteristic χgr\chi_{\textup{gr}}χgr​ of this homology is fully determined by the axioms we proposed. As a result, we conclude that χgr(SHI(M,γ))=χgr(SFH(M,γ))\chi_{\textup{gr}}(\mathrm{SHI}(M,\gamma))=\chi_{\textup{gr}}(\mathrm{SFH}(M,\gamma))χgr​(SHI(M,γ))=χgr​(SFH(M,γ)) for any balanced sutured manifold (M,γ)(M,\gamma)(M,γ). In particular, for any link LLL in S3S^3S3, the Euler characteristic χgr(KHI(S3,L))\chi_{\textup{gr}}(\mathrm{KHI}(S^3,L))χgr​(KHI(S3,L)) recovers the multi-variable Alexander polynomial of LLL, which generalizes the knot case. Combined with the authors’ earlier work, we provide more examples of (1,1)(1,1)(1,1)-knots in lens spaces whose KHI\mathrm{KHI}KHI and HFK^\widehat{\mathrm{HFK}}HFK have the same dimension. Moreover, for a rationally null-homologous knot in a closed oriented 3-manifold YYY, we construct canonical Z2\mathbb{Z}_2Z2​-gradings on KHI(Y,K)\mathrm{KHI}(Y,K)KHI(Y,K), the decomposition of I♯(Y)I^\sharp(Y)I♯(Y) discussed in the previous paper, and the minus version of instanton knot homology KHI‾−(Y,K)\underline{\mathrm{KHI}}^-(Y,K)KHI​−(Y,K) introduced by Zhenkun Li.

【 授权许可】

CC BY