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.

1, 2, 8, 42, 260, 1796, 13396, 105706, 870772, 7420836, 65004584, 582521748, 5320936416, 49402687392, 465189744448, 4434492302426, 42731740126228, 415736458808868, 4079436831493480, 40338413922226212, 401652846850965808, 4024556509468827432, 40558226664529024000, 410887438338905738908, 4182776248940752113344, 42770152711524569532616, 439143340987014152920384, 4526179842103708969039296

Offset

0,2

Comments

Also the number of knot diagrams with n crossings and two outgoing strings.

References

V. I. Arnold, Topological Invariants of Plane Curves and Caustics, American Math. Soc., 1994.

S. M. Gusein-Zade, Adv. Sov. Math., v. 21 (1994), pp. 189-198.

Links

Table of n, a(n) for n=0..27.

Steven R. Finch, Knots, links and tangles, August 8, 2003. [Cached copy, with permission of the author]

S. M. Gusein-Zade and F. S. Duzhin, On the number of topological types of plane curves; (Russian) Uspekhi Mat. Nauk 53 (1998), no. 3(321), 197-198. English translation: Russian Mathematical Surveys 53 (1998) 626-627. Related program and data.

J. L. Jacobsen and P. Zinn-Justin, A Transfer Matrix approach to the Enumeration of Knots

J. L. Jacobsen and P. Zinn-Justin, A Transfer Matrix approach to the Enumeration of Colored Links, J. Knot Theory, 10 (2001), 1233-1267.

Christoph Lamm, The enumeration of doubly symmetric diagrams for strongly positive amphicheiral knots, arXiv:2410.06601 [math.GT], 2024. See p. 14.

P. Zinn-Justin and J.-B. Zuber, Knot theory and matrix integrals, arXiv:1006.1812 [math-ph], 2010.

Index entries for sequences related to knots

Crossrefs

Cf. A008980, A008981, A008982, A008983, A008984, A008985.

Cf. A067647, A067648.

A column of the triangles in A067640 and A062038.

Sequence in context: A107588 A013999 A130649 * A188912 A229285 A339460

Adjacent sequences: A054990 A054991 A054992 * A054994 A054995 A054996

Keyword

nonn,nice,changed

Author

Sergei Duzhin, Nov 11 2000

Extensions

Extended to n = 22 by J. L. Jacobsen and Paul Zinn-Justin, Jan 30 2002

More terms from Paul Zinn-Justin, Dec 13 2016

Status

approved