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Ozsváth and Szabó recently defined a knot concordance invariant that bounds the 4–ball genus of a knot. Here we discuss shortcuts to its computation. We include examples of Alexander polynomial one knots for which the invariant is nontrivial, including all iterated untwisted negative clasped doubles of knots with nonnegative ThurstonBennequin number, such… (More)

In his classification of the knot concordance groups, Levine [L1] defined the algebraic concordance groups, G±, of Witt classes of Seifert matrices and a homomorphism from the odd-dimensional knot concordance groups C4n±1 to G±. The homomorphism is induced by the function that assigns to a knot an associated Seifert matrix: it is an isomorphism on Ck, k ≥… (More)

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)

- JAMES F. Dvrs, James F. Davis, Charles Livingston
- 2001

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)

In this paper we will use the Casson-Walker invariant of three-manifolds to define a Vassiliev invariant of two-component links. This invariant, L, is shown to have a simple combinatorial definition, making it easy to compute. In addition to placing 1 in the general theory of Vassiliev invariants, we will consider applications. For example, it is seen that… (More)

The Gassner representation of the pure braid group to GLn(Z[Z ]) can be extended to give a representation of the concordance group of n-strand string links to GLn(F ), where F is the field of quotients of Z[Z ], F = Q(t1, · · · , tn); this was first observed by Le Dimet. Here we give a cohomological interpretation of this extension. Our first application is… (More)

A knot in S 3 whose complement contains an essential** torus is called a satellite knot. In this paper we discuss algebraic invariants of satellite knots, giving short proofs of some known results as well as new results. To each essential torus in the complement of an oriented satellite knot S , one may associate two oriented knots C and E (the companion… (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)