Concordance and Mutation

We provide a framework for studying the interplay between concordance and positive mutation and identify some of the basic structures relating the two. The fundamental result in understanding knot concordance is the structure theorem proved by Levine: for n>1 there is an isomorphism phi from the concordance group C_n of knotted (2n-1)-spheres in S^{2n+1} to an algebraically defined group G_{+-}; furthermore, G__{+-} is isomorphic to the infinite direct sum Z^infty direct sum Z_2^infty direct sum Z_4^infty. It was a startling consequence of the work of Casson and Gordon that in the classical case the kernel of phi on C_1 is infinitely generated. Beyond this, little has been discovered about the pair (C_1,phi). In this paper we present a new approach to studying C_1 by introducing a group, M, defined as the quotient of the set of knots by the equivalence relation generated by concordance and positive mutation, with group operation induced by connected sum. We prove there is a factorization of phi, C_1-->M-->G_-. Our main result is that both maps have infinitely generated kernels. Among geometric constructions on classical knots, the most subtle is positive mutation. Positive mutants are indistinguishable using classical abelian knot invariants as well as by such modern invariants as the Jones, Homfly or Kauffman polynomials. Distinguishing positive mutants up to concordance is a far more difficult problem; only one example has been known until now. The results in this paper provide, among other results, the first infinite families of knots that are distinct from their positive mutants, even up to concordance.Comment: Published in Geometry and Topology at http://www.maths.warwick.ac.uk/gt/GTVol5/paper26.abs.htm

Multivariate Measures of Concordance for Copulas and their Marginals

Building upon earlier work in which axioms were formulated for multivariate measures of concordance, we examine properties of such measures. In particular, we examine the relations between the measure of concordance of an $n$-copula and the measures of concordance of the copula's marginals

Inertia Groups and Smooth Structures on Quaternionic Projective Spaces

For a quarternionic projective space, the homotopy inertia group and the concordance inertia group are isomorphic, but the inertia group might be different. We show that the concordance inertia group is trivial in dimension 20, but there are many examples in high dimensions where the concordance inertia group is non-trivial. We extend these to computations of concordance classes of smooth structures. These have applications to $3$-sphere actions on homotopy spheres and tangential homotopy structures.Comment: 13 page

Knot concordance and Heegaard Floer homology invariants in branched covers

By studying the Heegaard Floer homology of the preimage of a knot K in S^3 inside its double branched cover, we develop simple obstructions to K having finite order in the classical smooth concordance group. As an application, we prove that all 2-bridge knots of crossing number at most 12 for which the smooth concordance order was previously unknown have infinite smooth concordance order.Comment: Expanded references; 25 pages, 5 figure

Coefficient of Structural Concordance and an Example of its Application: Labour Productivity and Wages in Slovenia

The article presents the underlying principles, derivation and properties of a simple descriptive measure of concordance between two analogous rank structures that we call the coefficient of structural concordance. It is based upon the idea of Kendall’s coefficient of concordance, which we extend to two rank structures. As the coefficient of structural concordance is a pure intergroup measure of concordance, it is designed to complement the Kendall’s intragroup coefficient of concordance. We apply this descriptive measure by exploring the relationship between wages and labour productivity in Slovenia for the period 1998–2007. We are able to confirm the hypothesis of high concordance between wages and labour productivity, which indicates a stimulative role of wages in production of market traded goods and services.coefficient of structural concordance, Kendall’s coefficient of concordance, labour productivity, Slovenia, value added per employee, wages

Sex differences in emotional concordance.

Emotions involve response synchronization across experiential, physiological, and behavioral systems, referred to as concordance or coherence. Women are thought to be more emotionally aware and expressive than men and may therefore display stronger response concordance; however, research on this topic is scant. Using a random-order film-average design, we assessed concordance among experiential (arousal, valence), autonomic (electrodermal activity, heart rate, preejection period, respiratory sinus arrhythmia), respiratory (respiratory rate), and behavioral (corrugator and zygomatic electromyography) responses to 15 two-minute films varying in valence and arousal. We then calculated for each participant and pair of measures a within-subject correlation index using averages from the 15 films. Pronounced individual concordance of up to 0.9 was observed. Arousal-physiology and valence-behavior concordances were particularly pronounced. Women displayed higher concordance than men for almost all pairs of measures. Findings indicate stronger psychophysiological response coupling in women than men and provide novel insights into affective differences between the sexes