Correction Terms and the Nonorientable Slice Genus

Abstract

Abstract. By considering negative surgeries on a knot K in S 3 , we derive a lower bound to the non-orientable slice genus γ 4 (K) in terms of the signature σ(K) and the concordance invariants V i (K), which strengthens a previous bound given by Batson, and which coincides with Ozsváth-Stipsicz-Szabó's bound in terms of their υ invariant for L-space knots and quasi-alternating knots. A curious feature of our bound is superadditivity, implying, for instance, that the bound on the stable non-orientable genus is sometimes better than the one on γ 4 (K).

On the nonorientable four-ball genus of torus knots

“…There are now several lower bounds on γ 4 that look similar to that given in Lemma 2.5. First, Golla and Marengon [GM18] showed that γ 4 (K) ≥ σ(K) 2 − min m≥0 m + 2V m ( K) for any knot K in S 3 , where K denotes the mirror of K and {V i } i are the concordance invariants explored in [NW15] (and originally defined in [Ras04]). This bound agrees with the bound in Lemma 2.5 for alternating knots and L-space knots, in particular for torus knots.…”

Upsilon-like concordance invariants from 𝔰𝔩n knot cohomology

“…Related to υ(K) is an invariant ϕ(K) defined by Golla and Marengon [GM16] in terms of earlier invariants defined by Rasmussen [Ras04] and studied by Ni and Wu [NW15]. Again, this invariant ϕ(K) gives rise to a lower bound on the smooth non-orientable 4-ball genus.…”

Nonorientable link cobordisms and torsion order in Floer homologies

“…Since 1 .SO.3// D Z=2Z, the rotation by 4 is isotopic to the identity. Thus, F 4 is isotopic to the identity cobordism, and (8) .…”

“…On the other hand, the study of nonorientable surfaces and knot cobordisms in I S 3 has received increasing attention in the last decade -see Batson [3], Ozsváth, Stipsicz and Szabó [25], Golla and Marengon [8] and Fan [7] -and there are now several bounds to the nonorientable slice genus of a knot. However, if a knot bounds a nonorientable surface of a given "genus", it is not clear how complicated the embedding must be.…”