Branched twist spins and knot determinants (Q1687140)

Given a knot \(K \subset S^3\) and a pair of coprime integers \((m, n) \in \mathbb Z \times \mathbb N\), with \(m \neq 0\), the author gives the notion of branched twist spin of \(K\). This is a smooth 2-sphere in \(S^4\), and denoted by \(K^{m,n}\). This generalizes famous constructions of 2-knots in \(S^4\) starting from a classical knot in \(S^3\), such as the spun and twist spun knots of Artin and Zeeman. The main theorem of this paper provides a criterion for ambient non-isotopy: two branched twist spins \(K_1^{m_1,n_1}\) and \(K_2^{m_2,n_2}\) are smoothly non-isotopic in \(S^4\) if either \(|\Delta_{K_1}(-1)| \neq |\Delta_{K_2}(-1)|\) and \(m_1\), \(m_2\) are even, or \(|\Delta_{K_1}(-1)| \neq 1\) and \(m_1\) is even and \(m_2\) is odd, where \(\Delta_{K}\) denotes the Alexander polynomial of a knot \(K \subset S^3\). The proof is based on a lemma that expresses a presentation of \(\pi_1(S^4 - K^{m,n})\) in terms of a presentation of the knot group of \(K\). Then, the proof of the main theorem proceeds by an explicit computation of the Alexander ideals of a branched twist spin of a knot \(K\) by means of the Fox calculus, and showing how the first ideal is related to the Alexander polynomial of \(K\).