Equivariant aspects of singular instanton Floer homology

Abstract

We associate several invariants to a knot in an integer homology 3-sphere using SU p2q singular instanton gauge theory. There is a space of framed singular connections for such a knot, equipped with a circle action and an equivariant Chern-Simons functional, and our constructions are morally derived from the associated equivariant Morse chain complexes. In particular, we construct a triad of groups analogous to the knot Floer homology package in Heegaard Floer homology, several Frøyshov-type invariants which are concordance invariants, and more. The behavior of our constructions under connected sums are determined. We recover most of Kronheimer and Mrowka's singular instanton homology constructions from our invariants. Finally, the ADHM description of the moduli space of instantons on the 4-sphere can be used to give a concrete characterization of the moduli spaces involved in the invariants of spherical knots, and we demonstrate this point in several examples.

Instanton Floer homology, sutures, and Heegaard diagrams

“…The second way is to study a set of explicit generators of the instanton knot homology and its variances for some special families of knots, and the number of generators bounds the dimension of homology. This idea has been exploited by Hedden, Herald, and Kirk [21] and Daemi and Scaduto [11]. Our Theorem 1.2 and Theorem 1.6 then offers a totally new way to obtain an upper bound of dim ℂ 𝐾𝐻𝐼, and the following corollary indicates that this bound is sharp for many examples.…”

“…The instanton homology of closed 3-manifolds and knots in 3-manifolds was introduced by Floer [12,13], which became a powerful tool in the study of 3-dimensional topology. Some related constructions were made by Kronheimer and Mrowka [34][35][36], the first author [42], and Daemi and Scaduto [11]. Apart from instanton Floer homology, there are three Floer homologies of closed 3-manifolds, knots, and balanced sutured manifolds: Heegaard Floer homology by Ozsváth and Szabó [52,53], Rasmussen [55], and Juhász [26], monopole Floer homology by Kronheimer and Mrowka [32,34], and embedded contact homology (𝐸𝐶𝐻) by Hutchings [25], Colin, Ghiggini, Honda, and Hutchings [10].…”

Involutions, knots, and Floer K-theory

“…Applying equivariant Floer theory to the double branched covers, several knot (concordance) invariants are defined in Heegaard Floer homology, such as [2,16,24,25,27,28,38], and Seiberg-Witten Floer theory [8]. In a certain orbifold setting, several versions of knot instanton Floer homology are developed ( [13,15,46]). We also provide a knot concordance invariant from our Floer K-theory.…”

An adjunction inequality for the Bauer-Furuta type invariants, with applications to sliceness and 4-manifold topology

“…Endo [11] proved that there are infinitely many topologically slice knots which are linearly independent in the knot concordance group by using Furuta's result [20]. Obstructions to smoothly slicing a knot in D 4 have been obtained via Heegaard Floer theory [1,7,24,25,28,51,56,[58][59][60], Khovanov homology [63], and gauge theory [5,42,44]. We study the the existence of slice disks in general 4-manifolds with S 3 -boundary.…”

“…This gives a contradiction. Now we will prove Theorem 1.14 which is giving some constrains on the topology of symplectic caps of S 3 and Σ (2,3,5).…”