Instanton knot homology and its canonical isomorphism class

Assume full AC. For a based smooth knot $K\subset S^3$ with the earring and admissible singular-bundle data above, let $C^{\mathrm{♮}}(K;\mathbb{Z})$ be the integral singular-instanton cochain complex: its generators are regular critical points of the perturbed Chern–Simons functional on the corresponding singular configuration space, and its degree-one differential counts oriented zero-dimensional moduli spaces of trajectories for the perturbed singular anti-self-dual equation. Here finite energy is the flow energy $2{\int }_{\mathbb{R}}\mathrm{\parallel }{\mathrm{\partial }}_{t}B{\mathrm{\parallel }}_2^{2}dt$ in temporal gauge, equivalently the perturbed topological energy of the relative trajectory class, rather than unrenormalized curvature energy over the infinite cylinder. The grading is relative modulo four, and the differential has degree one. Define $I^{\mathrm{♮}}(K;\mathbb{Z}):=H^{\ast }(C^{\mathrm{♮}}(K;\mathbb{Z}))$ using the analytic construction and independence of choices established in Analytic foundations for the singular instanton knot complex. Write $I^{\mathrm{♮}}(K;\mathbb{Q}):=I^{\mathrm{♮}}(K;\mathbb{Z}){\otimes }_{\mathbb{Z}}\mathbb{Q}$. The invariant class specified here is the isomorphism class of this group with its relative grading; the continuation identifications retain the declared overall sign ambiguity.

Based earring and singular-bundle data

Integral analytic construction and continuation invariance ↗

Kronheimer–Mrowka, Khovanov homology is an unknot-detector (2011), §§1.1–1.2, 2–4, and 8.7

Kronheimer–Mrowka, Knots, three-manifolds and instantons (2018), §5.1

Yang, An Overview of Floer Homologies (2025), §§5.5–5.7

Analytic foundations for the singular instanton knot complex ↗

The word “class” here refers to the invariant isomorphism class specified in the Definition, not to an additional distinguished element of $I^{\mathrm{♮}}(K)$. The original operative Definition specifies a group, not such a vector. In particular, the local unknot computation gives a rank-one group whose generator depends on its orientation sign. Kronheimer–Mrowka, Khovanov homology is an unknot-detector, §1.1, PDF pp. 1–2, explicitly records the framed-basepoint cobordism maps and their overall sign ambiguity. No later rank argument needs a selected element. This clarification makes no canonical-element existence claim.

Kronheimer–Mrowka, Khovanov homology is an unknot-detector (2011), §1.1, PDF pp. 1–2

Kronheimer–Mrowka, Knots, three-manifolds and instantons (2018), §5.1