Knots with Signature 4 and Determinant 4k+1: An Answer to a Question of Shinohara

By a congruence of Murasugi, the determinant and the signature of a knot K satisfy det K ≡ (−1)^(σ(K)/2) (mod 4); moreover, a knot of determinant 1 has signature divisible by 8, because its symmetrized Seifert form is even and unimodular. In 1971 Shinohara realized all pairs (det K, |σ(K)|) allowed by these conditions, except those with det K ≡ 1 (mod 8) and |σ(K)| ≡ 4 (mod 8); he stated that the case det K ≢ 1 (mod 24) can be treated as well and left the case det K ≡ 1 (mod 24) open, expecting an affirmative answer. The question reappeared as Problem 12.21 of Ohtsuki's list of problems on invariants of knots and 3-manifolds: is there, for every n = 4k + 1 with k > 0, a knot with determinant n and signature 4? Stoimenow showed that every prime factor of a counterexample is ≡ 1 (mod 24) and at least 33049, and asked whether {det K : σ(K) = 4} = 5 + 4ℕ. We answer these questions affirmatively. For every n ≡ 1 (mod 4) with n ≥ 5 there is a knot of determinant n and signature 4; it can be taken of genus 2 unless n is a square all of whose prime factors are ≡ 1 (mod 4), and of genus at most 3 in that case. Consequently, an odd positive integer d and an even integer s are the determinant and the signature of some knot if and only if d ≡ (−1)^(s/2) (mod 4), where d = 1 requires s ≡ 0 (mod 8). The proof combines Seifert's realization theorem with an explicit family of genus-two Seifert matrices and a lemma based on Dirichlet's theorem on primes in arithmetic progressions and quadratic reciprocity. We also give explicit knots, among them a two-bridge knot of determinant 33049, and report exact computer checks. This is an unrefereed note. Unrefereed preprint released for independent mathematical scrutiny. Publication on Zenodo does not constitute peer review. AI-assisted tools supported research, computation, proof development, and manuscript preparation. The author remains responsible for all claims and the final text. Corpus identifier: AMR-103-0226 (T. Ohtsuki (ed.), Problems on invariants of knots and 3-manifolds, Geom. Topol. Monogr. 4 (2002), Problem 12.21, attributed to Y. Shinohara; origin: Y. Shinohara, Trans. Amer. Math. Soc. 156 (1971), Remark after Theorem 6).