An unoriented analogue of slice-torus invariant

A slice-torus invariant is an $\mathbb{R}$-valued homomorphism on the knot concordance group whose value gives a lower bound for the 4-genus such that the equality holds for any positive torus knot. Such invariants have been discovered in many of knot homology theories, while it is known that any slice-torus invariant does not factor through the topological concordance group. In this paper, we introduce the notion of “unoriented slice-torus invariant”, which can be regarded as the same as slice-torus invariant except for the condition about the orientability of surfaces. Then we show that the Ozsváth-Stipsicz-Szabó $\upsilon$-invariant, the Ballinger $t$-invariant and the Daemi-Scaduto $h_{\mathbb{Z}}$-invariant (shifted by a half of the knot signature) are unoriented slice-torus invariants. As an application, we give a new method for computing the above invariants, which is analogous to Livingston’s method for computing slice-torus invariants. Moreover, we use the method to prove that any unoriented slice-torus invariant does not factor through the topological concordance group.