The unknotting number and classical invariants, I

Abstract

Abstract. Given a knot K we introduce a new invariant coming from the Blanchfield pairing and we show that it gives a lower bound on the unknotting number of K. This lower bound subsumes the lower bounds given by the Levine-Tristram signatures, by the Nakanishi index and it also subsumes the Lickorish obstruction to the unknotting number being equal to one. Our approach in particular allows us to show for 25 knots with up to 12 crossings that their unknotting number is at least three, most of which are very difficult to treat otherwise.

The Unknotting Number and Classical Invariants Ii

“…Our main theorem is now the following result, first announced in [3], which says that n ‫ޒ‬ (K) is in fact determined by μ(K) and η(K). …”

An Explicit Computation of the Blanchfield Pairing for Arbitrary Links

“…For instance, applications of Bl(K) in knot concordance include a characterization of algebraic sliceness [27] and a crucial role in the obstruction theory underlying the solvable filtration of [15], see also [7,12,22,30,32]. Furthermore, Bl(K) has also served to compute unknotting numbers [3,4,5] and in the study of finite type invariants [33]. Finally, the Blanchfield pairing can be computed using Seifert matrices [21,27,31], is known to determine the Levine-Tristram signatures [5] and more generally the S-equivalence class of the knot [37].…”

“…Furthermore, Bl(K) has also served to compute unknotting numbers [3,4,5] and in the study of finite type invariants [33]. Finally, the Blanchfield pairing can be computed using Seifert matrices [21,27,31], is known to determine the Levine-Tristram signatures [5] and more generally the S-equivalence class of the knot [37].…”

“…In the case of links, the Blanchfield pairing generalizes to a hermitian pairing Bl(L) on the torsion submodule of the Alexander module of L. Although Bl(L) is still used to investigate concordance [8,11,13,19,28,36], unlinking numbers and splitting numbers [6], several questions remain: is there a natural definition of algebraic concordance for links and can it be expressed in terms of the Blanchfield pairing? Can one compute unlinking numbers and splitting numbers by generalizing the methods of [3,4,5] to links? Does the Blanchfield pairing determine the multivariable signature of [10]?…”

On the algebraic unknotting number

“…It is furthermore well-known that the 'classical' lower bounds on the unknotting number, i.e. the lower bounds which can be described in terms of the Seifert matrix of a knot, like the Nakanishi index [Na81], the Levine-Tristram signatures [Mus65, Le69,Tr69,Ta79,BF12], the Lickorish obstruction [Li85,CL86], the Murakami obstruction [Muk90] and the Jabuka obstruction [Ja09] give in fact lower bounds on the algebraic unknotting number.…”