1,989 research outputs found
Surgery description of colored knots
The pair (K,r) consisting of a knot K and a surjective map r from the knot
group onto a dihedral group is said to be a p-colored knot. D. Moskovich
conjectured that for any odd prime p there are exactly p equivalence classes of
p-colored knots up to surgery along unknots in the kernel of the coloring. We
show that there are at most 2p equivalence classes. This is a vast improvement
upon the previous results by Moskovich for p=3, and 5, with no upper bound
given in general. T. Cochran, A. Gerges, and K. Orr, in "Dehn surgery
equivalence relations of 3-manifolds", define invariants of the surgery
equivalence class of a closed 3-manifold M in the context of bordisms. By
taking M to be 0-framed surgery of the 3-sphere along K we may define
Moskovich's colored untying invariant in the same way as the Cochran-Gerges-Orr
invariants. This bordism definition of the colored untying invariant will be
then used to establish the upper bound.Comment: 41 pages, 23 figures (Version 3) Minor revisions and typos fixed.
Proofs of Propositions 4.1 and 4.8 revise
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There has been a burst of research on immigrant health in the United States and an increasing attention to the broad range of state and local policies that are social determinants of immigrant health. Many of these policies criminalize immigrants by regulating the "legality" of their day-to-day lives while others function to integrate immigrants through expanded rights and eligibility for health care, social services, and other resources.Research on the health impact of policies has primarily focused on the extremes of either criminalization or integration. Most immigrants in the United States, however, live in states that possess a combination of both criminalizing and integrating policies, resulting in distinct contexts that may influence their well-being.We present data describing the variations in criminalization and integration policies across states and provide a framework that identifies distinct but concurrent mechanisms of deportability and inclusion that can influence health. Future public health research and practice should address the ongoing dynamics created by both criminalization and integration policies as these likely exacerbate health inequities by citizenship status, race/ethnicity, and other social hierarchies
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Surgery description of colored knots
By a knot, or link, we mean a circle, or a collection of circles, embedded in the three-sphere S3. The study of knots is a very rich subject and plays a key role in the area of low-dimensional topology. In fact, a theorem of W.B.R. Lickorish and A.D. Wallace states that any three-dimensional manifold may be described by Dehn surgery along a link which is the process of removing the link from S3 and then gluing it back in a way that possibly changes the resulting manifold. In this dissertation, we will be interested in the pair (K, ρ) consisting of a knot K and a surjective map ρ from the knot group onto a dihedral group of order 2p called a coloring. Such an object is said to be a p-colored knot. In Surgery untying of colored knots , D. Moskovich conjectures that for any odd prime p there are exactly p equivalence classes of p-colored knots up to surgery which preserves colorability. This is an analog to the classical result that every knot has a surgery description or equivalently that every knot is surgery equivalent to the unknot if we place fewer restrictions on the allowed surgery curves. We show that there are at most 2p equivalence classes for p any odd number. This is an improvement upon the previous results by Moskovich for p = 3, and 5, with no upper bound given in general. We do this by defining a new invariant, or an algebraic object associated to a p-colored knot, which is complete in the sense that two p-colored knots are surgery equivalent if and only if they both have the same value of this invariant. The complete invariant consists of Moskovich’s colored untying invariant redefined in the same way as the three-manifold invariants developed by T. Cochran, A. Gerges, and K. Orr, and another object we call the η invariant. We also extend these methods to give similar results for A4-colored knots which have representations onto the alternating group on four letters
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