  Heegaard Floer theory produces chain complexes associated to knots. Viewed as
modules over polynomial rings, such complexes yield torsion invariants that
offer constraints on cobordisms between knots. For instance, Juhasz, Miller and
Zemke used torsion invariants to bound the number of local maxima and minima in
cobordisms between pairs of knots. Gong and Marengon defined a related torsion
invariant and used it to study nonorientable knot cobordisms.
  In this paper we define a one parameter family of Heegaard Floer torsion
invariants that yields a piecewise linear function defined on the interval
[0,2]. We call this the Upsilon torsion function; it is closely related to the
Heegaard Floer Upsilon function defined by Ozsvath, Stipsicz and Szabo. In a
natural way, this Upsilon torsion function interpolates between the
Juhasz-Miller-Zemke invariant and the Gong-Marengon invariant. In addition to
bounding the number of local maxima and minima in knot cobordisms, the Upsilon
torsion function provides new obstructions related to the Gordian distance
between knots.
