Coarse equality between 2g4 and the classical signature for all torus knots

Establish that for every integer k ≥ 1 there exists a constant Ck such that for all integers N coprime to k, the positive torus knot T(k,N) satisfies |2 g4(T(k,N)) − σ(T(k,N))| ≤ Ck, i.e., that the topological 4-genus and the classical knot signature of T(k,N) agree up to a bounded additive error depending only on k.

Background

The paper studies the locally flat cobordism distance between torus knots T(m,m+1) and T(k,N), showing it is essentially determined by signature-type invariants when k ∈ {2,3,4,6} (even k use the classical signature, while for k=3 one uses the Levine–Tristram signature at ω = e^{2πi/3}).

A key input for these cases is the known coarse equality 2g4(T(k,N)) ≈ σ(T(k,N)) up to a constant error for k = 2, 3, 4, 6. The authors point out that extending this coarse equality to all k would generalize their main results to all even k (and, analogously, to all odd k with an appropriate choice of Levine–Tristram signature). This identifies the general coarse equality 2g4 ≈ σ for torus knots as a conjectural open direction.