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

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.

Sources & referencesView supporting material

Primary source

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

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