Counting alternating knots by genus
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- Stoimenow, A. & Vdovina, A. Math. Ann. (2005) 333: 1. doi:10.1007/s00208-005-0659-x
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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.