  The Morse-Novikov number MN(L) of a smooth link L in the three-dimensional
sphere is by definition the minimal possible number of critical points of a
regular circle-valued Morse function on the link complement (the term regular
means that the Morse function must have nice behaviour in a tubular
neighbourhood of L). Novikov homology provides lower bounds for MN(L). In the
present paper we introduce the notion of twisted Novikov homology, which allows
to obtain better lower bounds for MN(L) than the usual Novikov homology. Our
twisted Novikov homology is a module over the Novikov ring Z((t)) but it
contains the information coming from the non abelian homological algebra of the
group ring of the fundamental group of the link complement. Using this
technique we prove that the Morse-Novikov number of the knot nC (the connected
sum of n copies of the Conway knot) is not less than 2n/5 for every positive
integer n. We prove also that MN(nC) is not greater than 2n. The same estimates
hold for the Morse-Novikov numbers of the connected sum of n copies of the
Kinoshita-Terasaka knot.
