Microsatellite mutations identified in pedigrees confirm that most changes involve the gain or loss of single repeats. However, an unexpected pattern is revealed when the resulting data are plotted on standardized scales that range from the shortest to longest allele at a locus. Both mutation rate and mutation bias reveal a strong dependency on allele length relative to other alleles at the same locus. We show that models in which alleles mutate independently cannot explain these patterns. Instead, both mutation probability and direction appear to involve interactions between homologues in heterozygous individuals. Simple models in which the longer homologue in heterozygotes is more likely to mutate and/or biased towards contraction readily capture the observed trends. The exact model remains unclear in all its details but inter-allelic interactions are a vital component, implying a link between demographic history and the mode and tempo of microsatellite evolution.
Given a knot s : S k−1 → ∂M with a framed dual sphere G : S d−k → ∂M , we describe the homotopy type of the space of neat embeddings D k → M d , with boundary s, as the loop space of neatly embedded (k − 1)-disks in the manifold obtained from M by attaching a handle to G. As a consequence, we conclude that the Dax invariant gives a complete isotopy classification of such disks in a 4-manifold. Moreover, we compute the resulting group structure on the set of these isotopy classes and show that it is nonabelian in most cases. Finally, we recover all previous results for spheres and prove that for them the Dax invariant reduces to the Freedman-Quinn invariant.
We compute in many classes of examples the first potentially interesting homotopy group of the space of embeddings of either an arc or a circle into a manifold M M of dimension d ≥ 4 d\geq 4 . In particular, if M M is a simply connected 4-manifold the fundamental group of both of these embedding spaces is isomorphic to the second homology group of M M , answering a question posed by Arone and Szymik. The case d = 3 d=3 gives isotopy invariants of knots in a 3-manifold that are universal of Vassiliev type ≤ 1 \leq 1 , and reduce to Schneiderman’s concordance invariant.
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