Abstract
It is shown that the number of alternating knots of given genus g>1 grows as a polynomial of degree 6g−4 in the crossing number. The leading coefficient of the polynomial, which depends on the parity of the crossing number, is related to planar trivalent graphs with a Bieulerian path. The rate of growth of the number of such graphs is estimated.
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Stoimenow, A., Vdovina, A. Counting alternating knots by genus. Math. Ann. 333, 1–27 (2005). https://doi.org/10.1007/s00208-005-0659-x
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DOI: https://doi.org/10.1007/s00208-005-0659-x