Conjecture on band number for ribbon surfaces of fixed genus

For a knot KK, let its ribbon genus be the minimum genus of a ribbon surface bounding it, and let b(K)b(K) denote its band number. Fixed-genus band number conjecture. For every genus gg, there exists a knot KK with ribbon genus gg for which b(K)b(K) is lower than the number of bands in any ribbon surface of genus gg. This predicts that, even after fixing the ribbon genus, band number need not be realized by a ribbon surface of that genus. It is presented as a broader analogue of the conjectured distinction between band number and ribbon fusion number, and remains unresolved in the source.

Sources & referencesView supporting material

Primary source

Clayton McDonald, “Band Number and the Double Slice Genus”, arXiv:1901.07625 (2019).

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