Conjecture on band number for ribbon surfaces of fixed genus
Conjecture on band number for ribbon surfaces of fixed genus
For a knot , let its ribbon genus be the minimum genus of a ribbon surface bounding it, and let denote its band number. Fixed-genus band number conjecture. For every genus , there exists a knot with ribbon genus for which is lower than the number of bands in any ribbon surface of genus . 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).
Progress summary
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