Knot 4-genus and the rank of classes in W(ℚ(t))

Abstract

The Witt rank .w/ of a class w in the Witt group W.F/ of a field with involution F is the minimal rank of a representative of the class. In the case of the Witt group of hermitian forms over the rational function field Q.t/, we define an easily computed invariant r.w/ and prove that, modulo torsion in the Witt group, r determines ; more specifically, . 4w/D r.4w/ for all w2 W.Q.t//. The need to determine the Witt rank arises naturally in the study of the 4-genus of knots; we illustrate the application of our algebraic results to knot theoretic problems, providing examples for which r provides stronger bounds on the 4-genus of a knot than do classical signature bounds or Ozsvath‐Szabo and Rasmussen‐Khovanov bounds.