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A173466
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a(n) is the number of prime knots with n crossings such that the empirical unknotting numbers cannot be decided minimals using their signatures.
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3
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0, 0, 0, 1, 0, 2, 2, 13, 27, 114, 370, 1614
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OFFSET
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1,6
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COMMENTS
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Prime knots considered in the sequence are those satisfying (1/2)*abs(sigma(K)) < u(K), where sigma(K) denotes the signature of the knot K and u(K) the unknotting number.
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REFERENCES
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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.
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LINKS
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FORMULA
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EXAMPLE
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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).
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| 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)
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CROSSREFS
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KEYWORD
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nonn,more,hard
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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