Knot traces and concordance

Abstract

We give a method for constructing many pairs of distinct knots K0 and K1 such that the two 4‐manifolds obtained by attaching a 2‐handle to B4 along Ki with framing zero are diffeomorphic. We use the d‐invariants of Heegaard Floer homology to obstruct the smooth concordance of some of these K0 and K1, thereby disproving a conjecture of Abe. As a consequence, we obtain a proof that there exist patterns P in solid tori such that P(K) is not always concordant to P(U)#K and yet whose action on the smooth concordance group is invertible.

Stein traces and characterizing slopes

“…In [Bra80], Brakes introduced dualizable patterns for smooth knots as a method to create pairs of knots with diffeomorphic zero surgeries. In [MP18], smooth dualizable patters are extensively studied, and a criteria for a pattern to be dualizable is given. In addition, the relation to the smooth annulus twist is discussed in [MP18],…”

“…This answers the above question in the affirmative. Our first construction is the contact annulus twist, a contact version of the smooth annulus twist developed in the series of articles [AJLO15, AJOT13,MP18,Oso06]. This is discussed in Section 3 and will lead to constructions of infinite families of Legendrian knots in (S 3 , ξ st ) that have the same Legendrian surgeries and, in some cases, the same Stein traces.…”

“…In Section 3.5 we will also develop other methods to create pairs of Legendrian knots that share the same Legendrian surgery or Stein traces. In particular, we will generalize dualizable patterns [Bra80,MP18] and RGB links [Pic19,Tag20b] to the contact and symplectic framework.…”

Getting a handle on the Conway knot

“…Allison Miller and Lisa Piccirillo [MP18] proved something even stronger: they showed that there exist knots K and K with the same trace such that K and K are not even concordant. This disproved a conjecture of Abe [Abe16].…”

Knots and 4-manifolds

“…The smaller D 4 inside X K is standard (it is a ball neighbourhood of the centre of D 4 ⊂ X K ) so its complement in S 4 is a ball in which we see a slice disk. For details see Kirby-Melvin or Miller-Piccirillo [32,44].…”