2024/04/05 by Kouki Sato, Sato, Kouki
Chemistry · Mathematics · #57K10 #57K18 #Chemistry and Stereochemistry Studies #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology
paper · pdf · doi:10.48550/arxiv.2404.04056
openalex publication_date 2024/04/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A slice-torus invariant is an 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 υ-invariant, the Ballinger t-invariant and the Daemi-Scaduto hℤ-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.