The vanishing conjecture for equal-rank Lie superalgebra invariants of slice virtual knots

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Let KK be a slice virtual knot, meaning that it is virtually concordant to the unknot, and let mm be an integer with m≥1m\geq 1. The invariant f~K m∣m(q,w)\widetilde{f}^{\,m|m}_K(q,w) is defined from the generalized Uq(gl(m∣m))U_q(\mathfrak{gl}(m|m)) polynomial.

Vanishing conjecture. For all m≥1m\geq 1, if KK is a slice virtual knot, then

f~K m∣m(q,w)=0.\widetilde{f}^{\,m|m}_K(q,w)=0.

The case m=1m=1 is the generalized Alexander polynomial, which is a virtual slice obstruction, and direct calculations give vanishing for the listed slice virtual knots when 1≤m≤51\leq m\leq 5. The conjecture proposes that this vanishing holds for every equal-rank case and every slice virtual knot.

References

Primary source

Micah Chrisman and Anup Poudel, “Lie superalgebra invariants and almost classical knots”, arXiv:2405.07375 (2025).

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