cp's OEIS Frontend

A054993 Number of "long curves", i.e., topological types of smooth embeddings of the oriented real line into the oriented plane that coincide with the standard immersion x -> (x,0) in the neighborhood of -infinity and +infinity.

%I A054993 #45 Jan 22 2025 00:08:09
%S A054993 1,2,8,42,260,1796,13396,105706,870772,7420836,65004584,582521748,
%T A054993 5320936416,49402687392,465189744448,4434492302426,42731740126228,
%U A054993 415736458808868,4079436831493480,40338413922226212,401652846850965808,4024556509468827432,40558226664529024000,410887438338905738908,4182776248940752113344,42770152711524569532616,439143340987014152920384,4526179842103708969039296
%N A054993 Number of "long curves", i.e., topological types of smooth embeddings of the oriented real line into the oriented plane that coincide with the standard immersion x -> (x,0) in the neighborhood of -infinity and +infinity.
%C A054993 Also the number of knot diagrams with n crossings and two outgoing strings.
%D A054993 V. I. Arnold, Topological Invariants of Plane Curves and Caustics, American Math. Soc., 1994.
%D A054993 S. M. Gusein-Zade, On the enumeration of curves from infinity to infinity, in: Singularities and Bifurcations, Adv. Sov. Math., v. 21 (1994), pp. 189-198.
%H A054993 Steven R. Finch, <a href="/A002863/a002863_4.pdf">Knots, links and tangles</a>, August 8, 2003. [Cached copy, with permission of the author]
%H A054993 S. M. Gusein-Zade and F. S. Duzhin, <a href="http://dx.doi.org/10.4213/rm48">On the number of topological types of plane curves</a>; (Russian) Uspekhi Mat. Nauk 53 (1998), no. 3(321), 197-198. <a href="http://dx.doi.org/10.1070/RM1998v053n03ABEH000048">English translation</a>: Russian Mathematical Surveys 53 (1998) 626-627. <a href="http://www.pdmi.ras.ru/~arnsem/dataprog/">Related program and data</a>.
%H A054993 J. L. Jacobsen and P. Zinn-Justin, <a href="http://arXiv.org/abs/math-ph/0102015">A Transfer Matrix approach to the Enumeration of Knots</a>
%H A054993 J. L. Jacobsen and P. Zinn-Justin, <a href="http://arXiv.org/abs/math-ph/0104009">A Transfer Matrix approach to the Enumeration of Colored Links</a>, J. Knot Theory, 10 (2001), 1233-1267.
%H A054993 Christoph Lamm, <a href="https://arxiv.org/abs/2410.06601">The enumeration of doubly symmetric diagrams for strongly positive amphicheiral knots</a>, arXiv:2410.06601 [math.GT], 2024. See p. 14.
%H A054993 P. Zinn-Justin and J.-B. Zuber, <a href="https://arxiv.org/abs/1006.1812">Knot theory and matrix integrals</a>, arXiv:1006.1812 [math-ph], 2010.
%H A054993 <a href="/index/K#knots">Index entries for sequences related to knots</a>
%Y A054993 Cf. A008980, A008981, A008982, A008983, A008984, A008985.
%Y A054993 A151374 enumerates the long curves having Gauss diagrams without intersections, cf. A118814.
%Y A054993 Cf. A067647, A067648.
%Y A054993 A column of the triangles in A067640 and A062038.
%K A054993 nonn,nice
%O A054993 0,2
%A A054993 _Sergei Duzhin_, Nov 11 2000
%E A054993 Extended to n = 22 by J. L. Jacobsen and _Paul Zinn-Justin_, Jan 30 2002
%E A054993 More terms from _Paul Zinn-Justin_, Dec 13 2016