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.

S. 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.

P. Zinn-Justin and J.-B. Zuber. Knot theory and matrix integrals. In The Oxford Handbook of Random Matrix Theory. 2011. Eds Akemann, Baik and Di Francesco. arXiv.

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

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