Commutativity conjecture for instanton knot Floer differentials

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Suppose K⊂S3K\subset S^3 is a quasi-alternating knot. Let KHI⁡‾(−S3,K)\underline{\operatorname{KHI}}(-S^3,K) be the instanton knot Floer homology group, and let d+d_+ and d−d_- be maps on it. The maps shift the grading associated to the Seifert surface by one. Commutativity conjecture for instanton knot Floer differentials. The maps satisfy

d+∘d−≐d−∘d+,d_+\circ d_-\doteq d_-\circ d_+,

where ≐\doteq means equality up to multiplication by a unit in C\mathbb{C}. This is proposed so that algebraic methods used for quasi-alternating knots can determine the differentials from the Alexander polynomial and signature; the source does not state that the conjecture has been proved.

References

Primary source

Zhenkun Li and Fan Ye, “SU(2) representations and a large surgery formula”, arXiv:2107.11005 (2026).

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