Thomas A. Gittings - cp's OEIS frontend

A094029 Number of n-crossing links with alternating braids of 3 strands.

1, 1, 4, 5, 13, 18, 38, 57, 115, 183, 354, 604, 1153, 2047, 3904, 7145, 13637, 25471, 48722, 92193

Offset: 4

Thomas A. Gittings, Apr 22 2004

Series is sum of three series for 1, 2 and 3 component links with alternating braids of 3 strands.

			4 crossing knot (1 component link) can be represented as 3-strand braid AbAb that is alternating (overcrossings and undercrossing alternate).

K. Murasugi and B. J. Kurpita, A Study of Braids, Kluwer Academic Publishers (1999), pp. 127-166

T. A. Gittings, Minimum braids: a complete invariant of knots and links.
V. F. R. Jones, Hecke algebra representations of braid groups and link polynomials, Ann. of Math., 126 (1987), pp. 335-388.

Cf. A094030, A094031, A094032.

A094030 Number of n-crossing knots with alternating braids of 3 strands.

Original entry on oeis.org

1, 0, 2, 0, 8, 0, 23, 0, 71, 0, 225, 0, 746, 0, 2555, 0, 8999, 0, 32297, 0

Offset: 4

Thomas A. Gittings, Apr 22 2004

			4 crossing knot (1 component link) can be represented as 3-strand braid AbAb that is alternating (overcrossings and undercrossing alternate).

K. Murasugi and B. J. Kurpita, A Study of Braids, Kluwer Academic Publishers (1999), pp. 127-166

T. A. Gittings, Minimum braids: a complete invariant of knots and links.
V. F. R. Jones, Hecke algebra representations of braid groups and link polynomials, Ann. of Math., 126 (1987), pp. 335-388.

Cf. A094029, A094031, A094032.

A094032 Number of n-crossing 3 component links with alternating braids of 3 strands.

Original entry on oeis.org

0, 0, 2, 0, 5, 0, 15, 0, 44, 0, 129, 0, 407, 0, 1349, 0, 4638, 0, 16425, 0

Offset: 4

Thomas A. Gittings, Apr 22 2004

K. Murasugi and B. J. Kurpita, A Study of Braids, Kluwer Academic Publishers (1999), pp. 127-166

T. A. Gittings, Minimum braids: a complete invariant of knots and links.
V. F. R. Jones, Hecke algebra representations of braid groups and link polynomials, Ann. of Math., 126 (1987), pp. 335-388.

Cf. A094029, A094030, A094031.

A094031 Number of n-crossing 2 component links with alternating braids of 3 strands.

Original entry on oeis.org

0, 1, 0, 5, 0, 18, 0, 57, 0, 183, 0, 604, 0, 2047, 0, 7145, 0, 25471, 0, 92193

Offset: 4

Thomas A. Gittings, Apr 22 2004

K. Murasugi and B. J. Kurpita, A Study of Braids, Kluwer Academic Publishers (1999), pp. 127-166

T. A. Gittings, Minimum braids: a complete invariant of knots and links.
V. F. R. Jones, Hecke algebra representations of braid groups and link polynomials, Ann. of Math., 126 (1987), pp. 335-388.

Cf. A094029, A094030, A094032.

A090597 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)).

Original entry on oeis.org

0, 1, 1, 3, 3, 8, 12, 27, 45, 96, 176, 363, 693, 1408, 2752, 5547, 10965, 22016, 43776, 87723, 174933, 350208, 699392, 1399467, 2796885, 5595136, 11186176, 22375083, 44741973, 89489408

Offset: 3

Thomas A. Gittings, Dec 11 2003

Arises from a conjecture about sequence of rational links with n crossings.
Conjecture derived from: s(n) = k(n) + l(n): definition of sum of rational knots (k) and links (l) s(n) = 6s(n-2) -8s(n-4): see A005418 (Jablan's observation) d(n) = d(n-2) + 2d(n-4): see A001045 (modified Jacobsthal sequence) l(n) = k(n-1) + d(n): conjecture.
a(n) is the number of rational (2-component) links. - Slavik Jablan, Dec 26 2003
Also yields the number of meanders, reduced by symmetry, on an n X 3 rectangle (see A200893). - Jon Wild, Nov 25 2011

Reinhard Zumkeller, 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 TL_n).
Index entries for linear recurrences with constant coefficients, signature (1,3,-1,0,-2,-4).

This is the difference between A005418 and A018240.
Cf. A018240 = sequence of rational knots, A005418 = number of rational knots and links, A001045 = Jacobsthal sequence, A329908, A336398.
Cf. A200893, and see the third column of the triangle read by rows there.

a090597 n = a090597_list !! (n-3)
a090597_list = [0,1,1,3,3,8,12] ++ zipWith (-)
   (drop 4 $ zipWith (-) (map (* 5) zs) (drop 2 a090597_list))
   (zipWith (+) (drop 2 $ map (* 2) zs) (map (* 8) zs))
   where zs = zipWith (+) a090597_list $ tail a090597_list
-- Reinhard Zumkeller, Nov 24 2011

f[x_] := (x-x^3-2x^4-3x^5) / (1-x-3x^2+x^3+2x^5+4x^6); CoefficientList[ Series[ f[x], {x, 0, 29}], x] (* Jean-François Alcover, Dec 06 2011 *)
J[n_] := (2^n - (-1)^n)/3; Table[(J[n - 3] + J[(n - If[OddQ[n], 3, 0])/2])/2 , {n, 3, 31}] (* David Scambler, Dec 13 2011 *)
LinearRecurrence[{1,3,-1,0,-2,-4},{0,1,1,3,3,8},30] (* Harvey P. Dale, Nov 12 2013 *)

a(n) = +a(n-1) +3*a(n-2) -a(n-3) -2*a(n-5) -4*a(n-6). - R. J. Mathar, Nov 23 2011
G.f.: -x^4*(-1+x^2+3*x^4+2*x^3) / ( (2*x-1)*(1+x)*(2*x^2-1)*(1+x^2) ). - R. J. Mathar, Nov 23 2011
a(n) = (J(n-3) + J((n-3)/2))/2 if n is odd; (J(n-3) + J(n/2))/2 if n is even, where J is the Jacobsthal number A001045. - David Scambler, Dec 12 2011

cp's OEIS Frontend

User: Thomas A. Gittings

A094029 Number of n-crossing links with alternating braids of 3 strands.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

A094030 Number of n-crossing knots with alternating braids of 3 strands.

Views

Author

Keywords

Examples

References

Links

Crossrefs

A094032 Number of n-crossing 3 component links with alternating braids of 3 strands.

Views

Author

Keywords

References

Links

Crossrefs

A094031 Number of n-crossing 2 component links with alternating braids of 3 strands.

Views

Author

Keywords

References

Links

Crossrefs

A090597 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)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Mathematica

Formula