Signature Function

The Tristram-Levine signature function of a knot, σ(K), is equal to σ((1- ω)V + (1- ω)*Vt), where V is a Seifert Matrix for the knot and ω is the unit complex number, exp(π i t).

The signature function provides bounds for the smooth 4-genus of a knot, for instance, slice knots have signature zero.

The numeric values of the jump points shown in the graphs are rough numerical estimates. Better estimates will be forthcoming. In the meantime, more precise values can be computed as they are the roots of the Alexander polynomial.


[1] Levine, J., "Invariants of knot cobordism," Invent. Math. 8 (1969), 98-110.

[2] Tristram, A. "Some cobordism invariants for links," Proc. Camb. Phil. Soc. 66 (1969), 251-264.