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We present a framework for defining the congruent invariants of knots using equivariant singular instanton Floer theory with Chern-Simons filters. In recent years, it has been demonstrated that many of the matching invariants defined using instantons can be recovered from the framework. This relationship allows us to compute the Kronheimer and Mrowka $s^\sharp$ invariants and the partial ideal invariants, such as the two-bridge knot. In particular, in answer to Gong’s question, we prove the semi-additiveness of $s^\sharp$. We also introduce invariants formally similar to the Heegaard Floer $\tau$ invariants of Oszv\’ath and Szab\’o and the $\varepsilon$ invariants of Hom. We provide evidence for an exact relationship between these latter two invariants of his and the s^\sharp invariants.

Some new topology applications following our technique are: First, we generate a broad class of patterns with the property that satellite maps induced over concordance groups have infinite rank, giving a partial answer to the conjecture of Hedden and Pinz\”on-Caicedo. Second, generate infinitely many 2-bridge knots $K$. Although this is a kink in the algebraic coincidence group, it has the property that the sets of positive $1/n$-surgeries on $K$ are linearly independent sets in homology coboldism. group. Finally, for knots that are quasi-normal and not slices, we prove that matching from knots allows irreducible $SU(2)$ representations in the fundamental group of matching complements.

Although many of the papers focus on constructions using singular instanton theory with traceless meridional holonomy conditions, we have also developed a similar framework for congruent invariants for arbitrary holonomy parameters, Several applications are provided with this setup.

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