The Witt rank ρ(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 ρ; more
specifically, ρ(4w) = r(4w) for all w ∈ 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 Ozsváth–Szabó and Rasmussen–Khovanov
bounds.
Keywords
knot genus, four genus, algebraic concordance, Witt group
