arXiv subjects

Bridget D. Franklin

Publications and source records attributed to Bridget D. Franklin.

3 recordsLinked to original sources

arXiv2011

The effect of infecting curves on knot concordance

Various obstructions to knot concordance have been found using Casson-Gordon invariants, higher-order Alexander polynomials, as well as von-Neumann rho-invariants. Examples have been produced using (iterated) doubling operations K=R(c,J), and considering these as parametrized by invariants of the base knot J and doubling operator R. In this paper, we introduce a new mew method to obstruct concordance. We show that infinitely many distinct concordance classes may be constructed by varying the infecting curve c in S^3-R. Distinct concordance classes are found even while fixing the base knot, the doubling operator, and the order of c in the Alexander module.

math.GT↗

arXiv2011

Knot concordance and homology cobordism

We consider the question: "If the zero-framed surgeries on two oriented knots in the 3-sphere are integral homology cobordant, preserving the homology class of the positive meridians, are the knots themselves concordant?" We show that this question has a negative answer in the smooth category, even for topologically slice knots. To show this we first prove that the zero-framed surgery on K is Z-homology cobordant to the zero-framed surgery on many of its winding number one satellites P(K). Then we prove that in many cases the tau and s-invariants of K and P(K) differ. Consequently neither tau nor s is an invariant of the smooth homology cobordism class of the zero-framed surgery. We also show, that a natural rational version of this question has a negative answer in both the topological and smooth categories, by proving similar results for K and its (p,1)-cables.

math.GT↗

arXiv2010

Rational knot concordance and homology cobordism

The following is a long-standing open question: "If the zero-framed surgeries on two knots in the 3-sphere are integral homology cobordant, are the knots themselves concordant?" We show that an obvious rational version of this question has a negative answer. Namely, we give examples of knots whose zero-framed surgeries are rational homology cobordant 3-manifolds, wherein the knots are not rationally concordant (that is not concordant in any rational homology S^3 x [0,1]). Specifically, we prove that, for any positive integer p and any knot K, the zero framed surgery on K is Z[1/p]-homology cobordant to the zero framed surgery on its (p,1) cable. Then we observe that most knots are not rationally concordant to their (p,1) cables.

math.GT↗