Geometry and Topology Seminar

Shumakovich conjectured that the (unreduced) Khovanov homology of any nontrivial knot has 2-torsions. Inspired by the spectral sequence from Khovanov homology to singular instanton homology constructed by Kronheimer-Mrowka, we study the 2-torsions in unreduced variant of singular instanton homology for knots. When K is a fibered knot, we aim to show that the singular instanton homology has 2-torsions by comparing the homology groups with complex coefficients and Z/2 coefficients. Also, we aim to prove that the framed instanton homology of the closed 3-manifold obtained from S^3 by 1/2 surgery along any knot of genus>1 has 2-torsions. This is a joint work in progress with Deeparaj Bhat and Zhenkun Li.