Given a knot K we describe a modification of K which leaves the double branched cover of S 3 branched along K unchanged. We then modify certain pretzel knots in this way to produce arbitrarily large families of distinct knots having the property that all of the associated double branched covers are homeomorphic.1. Introduction. This paper concerns the relationship between a knot and its associated double branched cover. A brief review of the history of this problem will indicate some of the known results.Given This result can be interpreted in a different manner as follows. We note that the manifold constructed is a Seifert fibered manifold which then admits n distinct involutions where distinct here means non-conjugate in the automorphism group. For more on this see Plotnick [11].
We continue the program of Bedient et al.1 by investigating some of the ways of embedding IFS with 1-step memory into IFS with 2-step memory, and 1- and 2-step memory into IFS with 3-step memory. This reveals a hierarchy of attractors of m-step memory IFS as subsets of attractors of n-step memory IFS.
By applying a result from the theory of subshifts of finite type,1 we generalize the result of Frame and Lanski2 to IFS with multistep memory. Specifically, we show that for an IFS [Formula: see text] with m-step memory, there is an IFS with 1-step memory (though in general with many more transformations than [Formula: see text]) having the same attractor as [Formula: see text].
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