Seifert forms and concordance (Q1865109)

Two results about classical knots in \(S^{3}\) are presented in this short article. One relates an algebraic property to a geometric property: the Alexander polynomial \(\Delta _{K}(t)\) of a knot \(K\) has a nontrivial irreducible factor which is not a cyclotomic polynomial \(\phi _{n}(t)\) with \(n\) divisible by three distinct primes if and only if some prime power cyclic branched cover of \(K\) is not a homology sphere. The second result is a construction which shows that if \(K\) is a knot that satisfies the equivalent conditions of the first result and \(V_{K}\) is a Seifert matrix for \(K\) then there is an infinite family of non-concordant knots each of which has \(V_{K}\) as a Seifert matrix.