A173466
a(n) is the number of prime knots with n crossings such that the empirical unknotting numbers cannot be decided minimals using their signatures.
Original entry on oeis.org
0, 0, 0, 1, 0, 2, 2, 13, 27, 114, 370, 1614
Offset: 1
From _Franck Maminirina Ramaharo_, Aug 14 2018: (Start)
Let K denote a prime knot in Alexander-Briggs notation, s(K) = (1/2)*abs(sigma(K)) and u(K) = unknotting number of K. The following table gives some of the first prime knots with the property s(K) != u(K).
==============================================================
| K | 4_1 | 6_1 | 6_3 | 7_4 | 7_7 | 8_1 | 8_3 | 8_4 | 8_6 |
-------+-----+-----+-----+-----+-----+-----+-----+-----+-----+
| s(K) | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 1 | 1 |
-------+-----+-----+-----+-----+-----+-----+-----+-----+-----+
| u(K) | 1 | 1 | 1 | 2 | 1 | 1 | 2 | 2 | 2 |
==============================================================
(End)
- Peter R. Cromwell, Knots and Links, Cambridge University Press, November 15, 2004, pp. 151-154.
- W. B. R. Lickorish, An introduction to Knot Theory, Springer, 1997, Table 8.1, p. 85.
- S. A. Bleiler, A note on unknotting number, Math. Proc. Camb. Phil. Soc. Vol. 96 (1984).
- J. C. Cha and C. Livingston, KnotInfo: Table of Knot Invariants.
- T. Kanenobu and S. Matsumura, Lower bound of the unknotting number of prime knots with up to 12 crossings, Journal of Knot Theory and Its Ramifications Vol. 24 (2015).
- K. Murasugi, On a certain numerical invariant of link types, Trans. Am. Math. Soc. Vol. 117 (1965), 387-422.
- Eric Weisstein's World of Mathematics, Knot Signature.
- Eric Weisstein's World of Mathematics, Unknotting Number.
- Index entries for sequences related to knots
A318050
Triangle read by rows: T(n,k) is the number of prime knots with n crossings whose unknotting numbers are k.
Original entry on oeis.org
0, 1, 0, 1, 0, 1, 1, 0, 3, 0, 3, 3, 1, 0, 9, 11, 1, 0, 17, 22, 9, 1
Offset: 3
Triangle begins:
n\k| 0 1 2 3 4
---+-------------------
3 | 0 1
4 | 0 1
5 | 0 1 1
6 | 0 3
7 | 0 3 3 1
8 | 0 9 11 1
9 | 0 17 22 9 1
- P. R. Cromwell, Knots and Links, Cambridge University Press, 2004, pp. 151-154.
- S. A. Bleiler, A note on unknotting number, Math. Proc. Camb. Phil. Soc. Vol. 96 (1984).
- M. Borodzik and S. Friedl, The unknotting number and classical invariants, I, Algebraic and Geometric Topology Vol. 15 (2015), 85-135.
- J. C. Cha and C. Livingston, KnotInfo: Table of Knot Invariants.
- S. Jablan and L. Radovic, Unknotting numbers of alternating knot and link families, Publications de l'Institut Mathématiques, Nouvelle série, tome 95 (2014), 87-99.
- K. Murasugi, On a certain numerical invariant of link types, Trans. Am. Math. Soc. Vol. 117 (1965), 387-422.
- Eric Weisstein's World of Mathematics, Unknotting Number.
- Index entries for sequences related to knots
A318052
Number of prime knots with n crossings whose unknotting numbers are given by their signatures.
Original entry on oeis.org
0, 0, 1, 0, 2, 1, 5, 8, 22, 51, 182, 562
Offset: 1
Let K denote a prime knot in Alexander-Briggs notation, and let sigma(K) and u(K) denote the signature and the unknotting number of the knot K, respectively. The following table gives some of the first prime knots with the property u(K) = (1/2)*abs(sigma(K)).
==================================================================
| K | 3_1 | 5_1 | 5_2 | 6_2 | 7_1 | 7_2 | 7_5 | 7_6 | 8_2 |
-----------+-----+-----+-----+-----+-----+-----+-----+-----+-----+
| sigma(K) | -2 | -4 | -2 | -2 | -6 | -2 | -4 | -2 | -4 |
-----------+-----+-----+-----+-----+-----+-----+-----+-----+-----+
| u(K) | 1 | 2 | 1 | 1 | 3 | 1 | 2 | 1 | 2 |
==================================================================
- P. R. Cromwell, Knots and Links, Cambridge University Press, 2004, pp. 151-154.
- S. A. Bleiler, A note on unknotting number, Math. Proc. Camb. Phil. Soc. Vol. 96 (1984).
- J. C. Cha and C. Livingston, KnotInfo: Table of Knot Invariants.
- T. Kanenobu and S. Matsumura, Lower bound of the unknotting number of prime knots with up to 12 crossings, Journal of Knot Theory and Its Ramifications Vol. 24 (2015).
- K. Murasugi, On a certain numerical invariant of link types, Trans. Am. Math. Soc. Vol.117 (1965), 387-422.
- Eric Weisstein's World of Mathematics, Knot Signature.
- Eric Weisstein's World of Mathematics, Unknotting Number.
- Index entries for sequences related to knots
Cf.
A002863,
A078477,
A089797,
A089891,
A089892,
A172293,
A172184,
A172441,
A172444,
A172486,
A173466,
A318050,
A318051.
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