A122495 -id:A122495 - cp's OEIS frontend

A018240 Number of rational knots (or two-bridge knots) with n crossings (up to mirroring).

1, 1, 2, 3, 7, 12, 24, 45, 91, 176, 352, 693, 1387, 2752, 5504, 10965, 21931, 43776, 87552, 174933, 349867, 699392, 1398784, 2796885, 5593771, 11186176, 22372352, 44741973, 89483947, 178962432, 357924864, 715838805, 1431677611, 2863333376, 5726666752

Offset: 3

Alexander Stoimenow (stoimeno(AT)math.toronto.edu)

			The a(7)=7 rational knots with 7 crossings are 7, 52, 43, 322, 313, 2212, 21112. All the rational knots are listed in A122495.

S. Jablan and R. Sazdanović, LinKnot: Knot Theory by Computer, World Scientific Press, 2007.

Vincenzo Librandi, Table of n, a(n) for n = 3..1000
C. Ernst and D. W. Sumners, The Growth of the Number of Prime Knots, Math. Proc. Cambridge Philos. Soc. 102, 303-315, 1987 (see Theorem 5, formulas for TK_n).
Taizo Kanenobu and Toshio Sumi, Polynomial Invariants of 2-Bridge Knots through 22 Crossings, Math. Comp. 60 (1993), 771-778, S17 (see Table 2).
P.-V. Koseleff, D. Pecker, Conway polynomials of two-bridge links, arXiv:1011.5992 [math.GT], 2010-2012 (only version 1 contains tables).
P.-V. Koseleff, D. Pecker, On Alexander-Conway polynomials of two-bridge links, Journal of Symbolic Computation 68 (2015), 215-229.
A. Stoimenow, Generating functions, Fibonacci numbers and rational knots, Journal of Algebra, 310 (2007), 491-525.
Index entries for sequences related to knots
Index entries for linear recurrences with constant coefficients, signature (-1,5,5,-2,-2,-8,-8).

Cf. A122495, A005418, A002863, A086825, A089790.

Cf. A018240 = number of rational knots, A005418 = number of rational knots and links, A001045 = Jacobsthal sequence (the difference between the number of rational links and knots), A090597 = rational links with n crossings, A329908, A336398.

LinearRecurrence[{-1, 5, 5, -2, -2, -8, -8}, {1, 1, 2, 3, 7, 12, 24}, 50] (* Harvey P. Dale, Sep 03 2013 *)
CoefficientList[Series[(1 - 2 x^2 - x^3 - x^4)/((1 - 2 x) (1 + x) (1 - 2 x^2) (1 + x^2)), {x, 0, 50}], x] (* Vincenzo Librandi, Aug 07 2014 *)

Vec((1-2*x^2-x^3-x^4)*x^3/((1-2*x)*(1+x)*(1-2*x^2)*(1+x^2))+O(x^66)) \\ Joerg Arndt, Aug 07 2014

a(n) = - a(n-1) + 5*(a(n-2)+a(n-3)) - 2*(a(n-4)+a(n-5)) - 8*(a(n-6)+a(n-7)). [Originally contributed as a separate sequence entry by Thomas A. Gittings, Dec 11 2003; see Stoimenow, Corollary 5.1 for proof]

G.f.: (1-2*x^2-x^3-x^4)*x^3/((1-2*x)*(1+x)*(1-2*x^2)*(1+x^2)). - R. J. Mathar, Sep 08 2008

Edited by Andrey Zabolotskiy, Jun 18 2020

A173637 Conway notation for rational 2-component links.

Original entry on oeis.org

2, 4, 212, 6, 33, 222, 412, 3112, 232, 8, 53, 422, 323, 3122, 242, 21212, 211112, 612, 5112, 432, 414, 4113, 3312, 32112, 3132, 31113, 252, 22212, 221112

Offset: 1

Jonathan Vos Post, Nov 23 2010

nonn

The ordering of the list is based on increasing crossing numbers and inverse lexicographical order for the terms with the same crossing number.
This is to links what A122495 is to knots.
All these links are chiral.
Each term is actually an ordered set of positive integers, concatenated; as long as all integers are 1-digit, it's not a problem, but a(30) requires "digit" 10, so at that point the sequence becomes not fully well-defined. An irregular array of these numbers would be well-defined.
Number of the terms of this sequence with crossing number k plus number of the terms of A122495 with crossing number k equals A005418(k-2). - Andrey Zabolotskiy, May 23 2017

			a(1) = 2 because 2 is the Conway notation for the Hopf link.
a(2) = 4 because 4 is the Conway notation for the (2,4) torus link.

C. Cerf, Atlas of oriented knots and links, Topology Atlas 3 no.2 (1998).
Peter R. Cromwell, Knots and Links, Cambridge University Press, November 15, 2004, p.210.

J. H. Conway, An enumeration of knots and links and some of their algebraic properties, 1970. Computational Problems in Abstract Algebra (Proc. Conf., Oxford, 1967) pp. 329-358 Pergamon, Oxford.
C. Giller, A family of links and the Conway calculus, Trans. American Math Soc., 270 (1982) 75-109.
Index entries for sequences related to knots

Cf. A002863, A002864, A018240, A090597, A122495.

Sequence edited and more terms added by Andrey Zabolotskiy, May 23 2017

cp's OEIS Frontend

A018240 Number of rational knots (or two-bridge knots) with n crossings (up to mirroring).

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