Browsing by Author "Leidy, Constance"

Now showing 1 - 2 of 2
  • Thumbnail Image
    Item
    Higher-order linking forms
    (2004) Leidy, Constance; Cochran, Tim D.
    Trotter [T] found examples of knots that have isomorphic classical Alexander modules, but non-isomorphic classical Blanchfield linking forms. T. Cochran [C] defined higher-order Alexander modules, An , (K), of a knot, K, and higher-order linking forms, Bℓn (K), which are linking forms defined on An , (K). When n = 0, these invariants are just the classical Alexander module and Blanchfield linking form. The question was posed in [C] whether Trotter's result generalized to the higher-order invariants. We show that it does. That is, we construct examples of knots that have isomorphic nth-order Alexander modules, but non-isomorphic nth-order linking forms. Furthermore, we define new higher-order linking forms on the Alexander modules for 3-manifolds considered by S. Harvey [H]. We construct examples of 3-manifolds with isomorphic nth-order Alexander modules, but non-isomorphic nth-order linking forms.
  • Thumbnail Image
    Item
    Infection By A String Link
    (2015-04-23) Vela, Diego; Hassett, Brendan; Harvey, Shelly; Cox, Steve; Leidy, Constance
    Satellite constructions on a knot can be thought of as taking some strands of a knot and then tying in another knot. Using satellite constructions one can construct many distinct isotopy classes of knots. Pushing this further one can construct distinct concordance classes of knots which preserve some algebraic invariants. Infection is a generalization of satellite operations which has been previously studied. An infection by a string link can be thought of as grabbing a knot at multiple locations and then tying in a link. Cochran, Friedl and Teichner showed that any algebraically slice knot is the result of infecting a slice knot by a string link(1). In this paper we use the infection construction to show that there exist knots which arise from infections by n-component string links that cannot be obtained by (n − 1)-component string links.