Unknotting Numbers are not Realized in Minimal Projections for a Class of Rational Knots

In previous results, Bleiler and Nakanishi produced an example of a knot where the unknotting number was not realized in a minimal projection of the knot. Bernhard generalied this example to an infi{}nite class of examples with Conway notation $\left(2j+1,1,2j\right)$ with j $\geq$ 2. In this paper we examine the entire class of knots given in Conway notation by (2j + 1, 2k + 1, 2j) where j $\geq$ 1 and k $\geq$ 0 and we determine that a large class of knots of this form have the unknotting number not realized in a minimal projection. We also produce an infi{}nite class of two component links with unknotting number gap arbitrarily large

Chaotic Examples in Low-Dimensional Topology

Abstract. Our earlier paper provide

Homogeneity groups of ends of open 3-manifolds

For every finitely generated abelian group G, we construct an irreducible open 3-manifold M[subscript G] whose end set is homeomorphic to a Cantor set and whose homogeneity group is isomorphic to G. The end homogeneity group is the group of self-homeomorphisms of the end set that extend to homeomorphisms of the 3-manifold. The techniques involve computing the embedding homogeneity groups of carefully constructed Antoine-type Cantor sets made up of rigid pieces. In addition, a generalization of an Antoine Cantor set using infinite chains is needed to construct an example with integer homogeneity group. Results about the local genus of points in Cantor sets and about the geometric index are also used.This is the publisher’s final pdf. The published article is copyrighted by the Mathematical Sciences Publishers and can be found at: http://msp.org/pjm/2014/269-1/index.xhtml.Keywords: Manifold end, Homogeneity group, Defining sequence, Geometric index, Cantor set, Abelian group, Rigidity, Open 3-manifol

Inequivalent Cantor sets in ³ whose complements have the same fundamental group

For each Cantor set C in R³, all points of which have bounded local genus, we show that there are infinitely many inequivalent Cantor sets in R³ with the complement having the same fundamental group as the complement of C. This answers a question from Open Problems in Topology and has as an application a simple construction of nonhomeomorphic open 3-manifolds with the same fundamental group. The main techniques used are analysis of local genus of points of Cantor sets, a construction for producing rigid Cantor sets with simply connected complement, and manifold decomposition theory. The results presented give an argument that for certain groups G, there are uncountably many nonhomeomorphic open 3-manifolds with fundamental group G.First published in Proceedings of the American Mathematical Society in Vol. 141 no. 8, published by the American Mathematical Society. This is the publisher’s final pdf. The published article is copyrighted by the American Mathematical Society and can be found at: http://www.ams.org/publications/journals/journalsframework/procKeywords: Cantor set, Defining sequence, End, Local genus, Open 3-manifold, Fundamental group, Rigidit

Topology and Chaos

Abstract. We discuss some basic topological techniques used in the study of chaotic dynamical systems. This paper is partially motivated by a talk given by the second author at the 7th international summer school and conference Chaos 2008: Let's Face Chaos Through Nonlinear Dynamic