Fenn–Kauffman–Manturov bracket-polynomial conjecture for virtual knots

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Let KK be a virtual knot, and let ⟨K⟩\langle K\rangle denote its bracket polynomial. Two virtual knots are Z-equivalent when they are related by classical and virtual Reidemeister moves together with the Z-move.

Fenn–Kauffman–Manturov conjecture. If

⟨K⟩=1,\langle K\rangle=1,

then KK is Z-equivalent to the unknot.

This conjecture, attributed in the source to Fenn, Kauffman, and Manturov, asserts that the bracket polynomial detects the unknot up to Z-equivalence in the value-one case. The source does not give a resolution.

References

Primary source

Kumud Bhandari, H. A. Dye and Louis H. Kauffman, “Lower bounds on virtual crossing number and minimal surface genus”, arXiv:0904.1525 (2009).

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