Equality of Involutive and Arf Knot Invariants

Definitions and mathematical statement

Use the standard convention that the smooth knot concordance group $\mathcal C$ consists of smooth concordance classes of oriented knots in $S^3$, with connected sum as addition and the unknot class as zero. Two knots are smoothly concordant if they cobound a smoothly embedded annulus in $S^3\times[0,1]$. A knot $K$ represents torsion if \[ \exists n\in{\mathbb{Z}}_{\geq 1}\quad n[K]=0\quad\text{in }\mathcal C. \] This includes the zero class. Equivalently, some positive connected sum of copies of $K$ bounds a smoothly embedded disk in $B^4$.

The Arf invariant $\operatorname{Arf}(K)\in{\mathbb{Z}}/2{\mathbb{Z}}$ is the Arf invariant of the quadratic refinement on $H_1(F;{\mathbb{Z}}/2{\mathbb{Z}})$ associated to a Seifert surface $F$ for $K$. This refinement is given by $q_F(x)=\operatorname{lk}(x,x^+)\bmod 2$, where $x^+$ is a positive normal push-off of a representative cycle. For a symplectic basis $a_1,b_1,\ldots,a_g,b_g$, its Arf invariant is $\sum_{i=1}^g q_F(a_i)q_F(b_i)\in{\mathbb{Z}}/2{\mathbb{Z}}$.

Let $A$ denote the involutive knot invariant referred to in the supplied source record. Its construction, codomain, and normalization are not supplied, so this notation remains source-dependent. The intended conjecture is \[ \forall K\subset S^3,\qquad \left(\exists n\in{\mathbb{Z}}_{\geq1}:n[K]=0\text{ in }\mathcal C\right) \Longrightarrow A(K)=\operatorname{Arf}(K). \] The equality is to be understood using the source's intended value convention for $A$; that convention must be recovered before this becomes a fully self-contained statement. The supplied curation identifies this target as Conjecture 4. The quoted original passage asks whether $A$ agrees with the Arf invariant but does not itself state the torsion restriction; that restriction is retained from the supplied clean statement and its verification metadata.

Short English statement

The conjecture asks whether the source's involutive invariant $A$ equals the Arf invariant for every knot whose smooth concordance class has finite order. The supplied record does not define $A$, so its construction and value convention require source review.

Sources

• Supplied local UnsolvedMath problem registry, permanent ID $30005388$, source problem code catalogue entry, title: Equality of Involutive and Arf Knot Invariants. The record supplies the status open and statement status corrected_verified; its verification metadata reports comparison with the official report. These are supplied curation claims, not independent verification here. No individual proposer is specified; the proposed year is 2023.
• Morphisms in Low Dimensions (2023), Oberwolfach Reports, report 4/2023, pp. 215--259, DOI 10.4171/owr/2023/4. Supplied report provenance: workshop documentation and open-problem discussions. https://doi.org/10.4171/owr/2023/4.
• Kang and Park, Torsion in the knot concordance group and cabling. Supplied literature reference: https://arxiv.org/abs/2207.11870.
• Literature assessment checked at 2026-08-22: the supplied curation describes the equality for torsion knots as Conjecture 4 and reports that no proof or counterexample was located. This is supplied dated curation derived from the OWR review, not an independently verified literature assessment. Time-sensitive status and cited claims require independent verification.