Knot Signature

The signature  of a knot  can be defined using the skein relationship

(1)

(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.