Abstract
We relate certain abelian invariants of a knot, namely the Alexander polynomial, the Blanchfield form, and the Arf invariant, to intersection data of a Whitney tower in the 4-ball bounded by the knot. We also give a new 3-dimensional algorithm for computing these invariants.
Highlights
We show that intersection data in Whitney towers determines abelian invariants of knots, the Blanchfield form, the Alexander polynomial, and the Arf invariant
In the following theorem we show that the matrix presents the Blanchfield pairing up to Witt equivalence, and determines the Alexander polynomial up to norms and units
Theorem 1.6 (Freedman, Kirby, Matsumoto, Quinn) The Arf invariant Arf(K ) vanishes if and only if K is the boundary of framed Whitney tower of order two in D4
Summary
We show that intersection data in Whitney towers determines abelian invariants of knots, the Blanchfield form, the Alexander polynomial, and the Arf invariant. Our main result algorithmically computes the Blanchfield form and the Alexander polynomial of a knot using intersection data from an order two twisted Whitney tower in the 4-disc bounded by the knot. This relates two incarnations of the Arf invariant of a knot using. A 4-dimensional argument—one characterizing the Arf invariant in terms of Whitney towers, the other in terms of the Alexander polynomial
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