The Alexander-polynomial conjecture for hyperbolic L-space knots

Let KK be a hyperbolic L-space knot, let gg be its Seifert genus, and let nn be its braid index. Let ΔK(t)\Delta_K(t) denote its Alexander polynomial. The hyperbolic L-space knot Alexander-polynomial conjecture. Then

ΔK(t)=1t+tn++t2gn1t2g1+t2g.\Delta_K(t) = 1-t+t^n + \ldots + t^{2g-n-1}- t^{2g-1}+ t^{2g}.

The hyperbolicity assumption is essential: the source notes that the assertion fails for the (2,3)(2,3)-cable of T(2,3)T(2,3), which is an L-space knot. The conjecture is otherwise presented as an open prediction.

Sources & referencesView supporting material

Primary source

Siddhi Krishna and Hugh Morton, “Twist positivity, L-space knots, and concordance”, arXiv:2211.17109 (2025).

Additional references

2 papers in this index state this conjecture (2008–2022). The statement above is taken from the most recent of them; the others are arXiv:0807.4891.

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