On the secondary Upsilon invariant

Abstract

In this paper we construct an infinite family of knots with vanishing Upsilon invariant Υ, although their secondary Upsilon invariants Υ 2 show that they are linearly independent in the smooth knot concordance group. We also prove a conjecture in a paper by Allen.

A further note on the concordance invariants epsilon and upsilon

“…A very simple alternative proof of the main theorem and a conjecture in [All20] will also be provided in the course of the above computation of the ϕ-invariant. (The conjecture was first proved by Xu [Xu18].) On the contrary, [DHST19, Proposition 1.10] gives a knot Floer complex, a knot with which would have vanishing ϕ-invariant but nonvanishing Υ-invariant.…”

“…From this formula, one immediately knows that any knot of the form T q,kq+p #−T p,q #−kT q,q+1 has vanishing Υ invariant, where −K means the mirror image of the knot K with reversed orientation (representing the inverse element of K in C) and kK means the connected sum of k copies of K. By a theorem of Litherland [Lit79] that torus knots are linearly independent in the concordance group, it is easy to give subgroups of C isomorphic to Z ∞ included in the kernel of the Υ homomorphism, but its elements may also have vanishing ε invariant [Xu18,Remark 4.11]. In Proposition 3.4, we will show a bound for some of the knots of the form T q,kq+p # − T p,q # − kT q,q+1 with respect to the total order given by the ε invariant, which enables us to prove Theorem 1.1.…”