Signature Function and Nullity

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 nullity at ω is the nullity of the matrix.

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

Numeric estimates for the values of the points of discontinuity of the signature function appear in the table and in the graphs. More precise values can be computed, as they are the roots of the Alexander polynomial.

Each signature function is given as a vector as in the example:

11n_75   {{0.2300534562, {0, 1, 2}, 1}, {0.3333333333, {2, 2, 2}, 2}}

This indicates that the Alexander polynomial has two roots on the unit circle, numbers A and B with t values roughly .23 and .33. At A the signture jumps from 0 to 2, and at A it equals 1. The nullity at A is 1. Near B the signature has constant value 2; the nullity at B is 2.

The following example is one for which the averaged signature function is trivial but the signature function is not trivial. This is an example of a slice knot with nontrivial signature function. The signature function obstructs this knot from being doubly slice.

12n_56   {{0.3333333333, {0, 1, 0}, 1}}

References

[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.