Knot Signature
المؤلف:
Gordon, C. M.; Litherland, R. A.; and Murasugi, K.
المصدر:
"Signatures of Covering Links." Canad. J. Math. 33
الجزء والصفحة:
...
14-6-2021
5042
Knot Signature
The signature 
of a knot 
can be defined using the skein relationship
 |
(1)
|
![s(K_+)-s(K_-) in <span style=]() {0,2}, " src="https://mathworld.wolfram.com/images/equations/KnotSignature/NumberedEquation2.gif" style="height:15px; width:134px" /> |
(2)
|
and
 |
(3)
|
where 
is the Conway polynomial and 
is an odd number.
Many unknotting numbers can be determined using a knot's signature.
Knot signatures are implemented in the Wolfram Language as KnotData[knot, "Signature"]. The following table summarizes knot signatures for knots on 10 of fewer crossings.
REFERENCES:
Gordon, C. M.; Litherland, R. A.; and Murasugi, K. "Signatures of Covering Links." Canad. J. Math. 33, 381-394, 1981.
Murasugi, K. "On the Signature of Links." Topology 9, 283-298, 1970.
Murasugi, K. "Signatures and Alexander Polynomials of Two-Bridge Knots." C. R. Math. Rep. Acad. Sci. Canada 5, 133-136, 1983.
Murasugi, K. "On the Signature of a Graph." C. R. Math. Rep. Acad. Sci. Canada 10, 107-111, 1988.
Murasugi, K. "On Invariants of Graphs with Applications to Knot Theory." Trans. Amer. Math. Soc. 314, 1-49, 1989.
Rolfsen, D. Knots and Links. Wilmington, DE: Publish or Perish Press, 1976.
Stoimenow, A. "Signatures." https://www.ms.u-tokyo.ac.jp/~stoimeno/ptab/sig10.html.
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