Heegaard Floer homology and its generalization

2016 Fiscal Year Final Research Report

• Project/Area Number: 26887010
• Research Category: Grant-in-Aid for Research Activity Start-up
• Allocation Type: Single-year Grants
• Research Field: Geometry
• Research Institution: The University of Tokyo (2015-2016); Tohoku University (2014)
• Principal Investigator: BAO Yuanyuan 東京大学, 大学院数理科学研究科, 助教 (00710823)
• Research Collaborator: WU Zhongtao
• Project Period (FY): 2014-08-29 – 2016-03-31
• Keywords: Heegaard Floer / graph / Alexander polynomial / MOY relation

Outline of Final Research Achievements

In the past two years, I studied the Heegaard Floer homology for an embedded bipartite graph in a closed 3-manifold. The Euler characteristic of the homology is the Alexander polynomial, which is a classical invariant in knot theory. During this academic year, my coworker and I found that this polynomial satisfies some relations similar with MOY relations for sl(n) quantum polynomial, and we showed that these relations, in turn, provide a characterization of the Alexander polynomial for a graph. One of the important questions in Heegaard Floer theory is how to understand the theory from the quantum topological viewpoint. In the future, we will study the quantum topological meaning of the Alexander polynomial and then that of its categorification, the Heegaard Floer homology.

Free Research Field

Low-dimensional Topology