The normalized flat Jones–Krushkal polynomial conjecture for almost classical flat knots

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Let α\alpha be an almost classical flat knot, and let Jˉα(z)\bar J_\alpha(z) denote its normalized flat Jones–Krushkal polynomial, defined using a minimal diagram of α\alpha.

Normalized flat Jones–Krushkal polynomial conjecture. For any almost classical flat knot α\alpha, the normalized flat Jones–Krushkal polynomial satisfies

Jˉα(−1)=1.\bar J_\alpha(-1)=1.

This was observed empirically for every almost classical flat knot up to 1010 crossings. Whether the identity holds for all almost classical flat knots remains open.

References

Primary source

Jie Chen, “FlatKnotInfo: the first 1.24 million flat knots”, arXiv:2410.00216 (2024).

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