In this note we observe how results of Gordon, Luecke, Feustel, and Whitten establish the precise extent to which knowledge of the knot group determines the knot, in both the oriented and the unoriented case. This enables us to say, in terms of the knot group, exactly when two spun knots are the same.

The problem of Josephus is the following. We are given two positive integers n, q. There are n places arranged around a circle, and numbered clockwise 1, 2, …, n. Each of n people takes one of the places, then (please excuse this, but we didn't invent the problem!) every qth one is executed, until just one remains. More precisely, the occupant of place q is ‘removed’ first, and in general, if some place j has just been vacated, then the qth one of the places clockwise around from j that are still occupied will be vacated next. One question is this: if you would like to be the last survivor, then into what place should you go initially? We denote the answer to this question by Jq(n). For example, if n = 5 and q = 2, the order of execution is 2, 4, 1, 5, 3, and J2(5) = 3. Other questions have been raised about the problem, and it has an extensive literature (see [1]–[10]). In this paper we deal with the Jq(n)'s.

A generalized tensor product of groups was introduced by R. Brown and J.-L. Loday [6], and has led to a substantial algebraic theory contained essentially in the following papers: [6, 7, 1, 5, 11, 12, 13, 14, 18, 19, 20, 23, 24] ([9, 27, 28] also contain results related to the theory, but are independent of Brown and Loday's work). It is clear that one should be able to develop an analogous theory of tensor products for other algebraic structures such as Lie algebras or commutative algebras. However to do so, many non-obvious algebraic identities need to be verified, and various topological proofs (which exist only in the group case) have to be replaced by purely algebraic ones. The work involved is sufficiently non-trivial to make it interesting.

Let S and T be inverse semigroups. Their free product S inv T is their coproduct in the category of inverse semigroups, defined by the usual commutative diagram. Previous descriptions of free products have been based, like that for the free product of groups, on quotients of the free semigroup product S sgp T. In that framework, a set of canonical forms for S inv T consists of a transversal of the classes of the congruence associated with the quotient. The general result [4] of Jones and previous partial results [3], [5], [6] take this approach.

1. Point (3) of the main theorem of our paper [3, Theorem 1.1] is incorrect: this note corrects the main and consequential errors, and shows that (after minor adjustments) almost all the other results of [3], including the remaining seven points of Theorem 1.1, remain correct.

2. The theme of [3] was a family of functors G,(–), defined on the category of rings with unity for each cardinal t. For t = 0, 1, the results of [3] are unchanged, but, for 2≤t<∞, major, and, for t infinite, less major, corrections are necessary; we therefore assume 2≤t. Terminology and notation are standard or as in [3], and I would like to thank A. W. Chatters and an anonymous referee for comments which prompted this correction.