Charles Livingston

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The concordance group of algebraically slice knots is the subgroup of the classical knot concordance group formed by algebraically slice knots. Results of Casson and Gordon and of Jiang showed that this group contains in infinitely generated free (abelian) subgroup. Here it is shown that the concordance group of algebraically slice knots also contain(More)
Given a three-manifold M and a cohomology class τ ∈ H(M,Z/nZ), there is a naturally defined invariant of singular knots in M with exactly one double point, Vτ . It has been known that for some manifolds Vτ is integrable and that in these cases it defines an easily computed and highly effective knot invariant. This paper provides necessary and sufficient(More)
In 1926 Artin [3] described the construction of knotted 2–spheres in R. The intersection of each of these knots with the standard R ⊂ R is a nontrivial knot in R. Thus a natural problem is to identify which knots can occur as such slices of knotted 2–spheres. Initially it seemed possible that every knot is such a slice knot and it wasn’t until the early(More)
A KNOT K in S’ is said to have period n > I if there is a transformation T of S’ of order n such that K is invariant under T and the fixed point set of T is a circle B, disjoint from K. (The positive solution of the Smith conjecture implies that B is unknotted and the transformation is equivalent to the one-point compactification of rotation about the(More)
The classical knot concordance group, C1, was defined in 1961 by Fox [F]. He proved that it is nontrivial by finding elements of order two; details were presented in [FM]. Since then one of the most vexing questions concerning the concordance group has been whether it contains elements of finite order other than 2–torsion. Interest in this question was(More)