Signature (Knot)

The signature $s(K)$ of a Knot $K$ can be defined using the Skein Relationship

$$s({\rm unknot})=0$$

$$s(K_+)-s(K_-)\in\{0,2\},$$

and

$$4\vert s(K) \leftrightarrow \nabla(K)(2i)>0,$$

where $\nabla(K)$ is the Alexander-Conway Polynomial and $\nabla(K)(2i)$ is an Odd Number.

Many Unknotting Numbers can be determined using a knot's signature.

See also Skein Relationship, Unknotting Number

References

Gordon, C. McA.; 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.'' http://www.informatik.hu-berlin.de/~stoimeno/ptab/sig10.html.