Seifert hypersurfaces of 2-knots and Chern-Simons functional

Abstract

We introduce a real-valued functional csK on the SU (2)-representation space of the knot group for any oriented 2-knot K. In order to define the functionals, we use a generalization for homology S 1 × S 3 's of the Chern-Simons functionals introduced in [36]. We calculate our functionals for ribbon 2-knots and the twisted spun 2-knots of torus knots, Montesinos knots and 2-bridge knots. We show several properties of the images of the functionals including a connected sum formula and relationship to the Chern-Simons functionals of Seifert hypersurfaces of K. As a corollary, we show that every oriented 2-knot having an oriented homology 3-sphere of a certain class as its Seifert hypersurface admits an SU (2)-irreducible representation of a knot group. Moreover, we also relate the existence of embeddings from a homology 3-sphere into a negative definite 4-manifold to SU (2)-representations of their fundamental groups. For example, we prove that every closed definite 4-manifold containing Σ(2, 3, 5, 7) as a submanifold has an uncountable family of SU (2)-representations of its fundamental group. This implies that every 2-knot having Σ(2, 3, 5, 7) as a Seifert hypersurface has an uncountable family of SU (2)-representations of its knot group. The proofs of these results use several techniques from instanton Floer theory.

On meridian-traceless SU(2)-representations of link groups

“…In fact, most of the results listed below Question 1.1 were proved using instanton Floer theory. Although we only focus on the 3-dimensional case in this paper, we want to point out that instanton Floer theory has also been used to study SU(2)-representations of the fundamental groups of 4-manifolds (see [DLVVW19,Tan19,Dae20,DS20]).…”