Satellite operators as group actions on knot concordance

Any knot in a solid torus, called a pattern or satellite operator, acts on knots in the 3-sphere via the satellite construction. We introduce a generalization of satellite operators which form a group (unlike traditional satellite operators), modulo a generalization of concordance. This group has an action on the set of knots in homology spheres, using which we recover the recent result of Cochran and the authors that satellite operators with strong winding number $\pm 1$ give injective functions on topological concordance classes of knots, as well as smooth concordance classes of knots modulo the smooth 4--dimensional Poincare Conjecture. The notion of generalized satellite operators yields a characterization of surjective satellite operators, as well as a sufficient condition for a satellite operator to have an inverse. As a consequence, we are able to construct infinitely many non-trivial satellite operators P such that there is a satellite operator $\overline{P}$ for which $\overline{P}(P(K))$ is concordant to K (topologically as well as smoothly in a potentially exotic $S^3\times [0,1]$) for all knots K; we show that these satellite operators are distinct from all connected-sum operators, even up to concordance, and that they induce bijective functions on topological concordance classes of knots, as well as smooth concordance classes of knots modulo the smooth 4--dimensional Poincare Conjecture.Comment: 20 pages, 9 figures; in the second version, we have added several new results about surjectivity of satellite operators, and inverses of satellite operators, and the exposition and structure of the paper have been improve

Concordance of knots in S1×S2S^1\times S^2

We establish a number of results about smooth and topological concordance of knots in $S^1\times S^2$. The winding number of a knot in $S^1\times S^2$ is defined to be its class in $H_1(S^1\times S^2;\mathbb{Z})\cong \mathbb{Z}$. We show that there is a unique smooth concordance class of knots with winding number one. This improves the corresponding result of Friedl-Nagel-Orson-Powell in the topological category. We say a knot in $S^1\times S^2$ is slice (resp. topologically slice) if it bounds a smooth (resp. locally flat) disk in $D^2\times S^2$. We show that there are infinitely many topological concordance classes of non-slice knots, and moreover, for any winding number other than $\pm 1$, there are infinitely many topological concordance classes even within the collection of slice knots. Additionally we demonstrate the distinction between the smooth and topological categories by constructing infinite families of slice knots that are topologically but not smoothly concordant, as well as non-slice knots that are topologically slice and topologically concordant, but not smoothly concordant.Comment: 25 pages, 19 figures, final version, to appear in Journal of London Mathematical Societ

The Effectiveness of a Simple Helmholtz coil-like Magnetic Shield at Reducing X-ray-like Background in Space-based X-ray Detectors

Both active and passive magnetic shielding have been used extensively during past and current X-ray astronomy missions to shield detectors from soft protons and electrons entering through telescope optics. However, simulations performed throughout the past decade have discovered that a significant proportion of X-ray-like background originates from secondary electrons produced in spacecraft shielding surrounding X-ray detectors, which hit detectors isotropically from all directions. Here, the results from Geant4 simulations of a simple Helmholtz coil-like magnetic field surrounding a detector are presented, and it is found that a Helmholtz coil-like magnetic field is extremely effective at preventing secondary electrons from reaching the detector. This magnetic shielding method could remove almost all background associated with both backscattering electrons and fully absorbed soft electrons, which together are expected to account for approximately two thirds of the expected off-axis background in silicon-based X-ray detectors of several hundred microns in thickness. The magnetic field structure necessary for doing this could easily be produced using a set of solenoids or neodymium magnets providing that power requirements can be sufficiently optimised or neodymium fluorescence lines can be sufficiently attenuated, respectively.Comment: 20 pages, 6 figure

Plant Community Composition and Structure Monitoring for Scotts Bluff National Monument, 2011-2015 Summary Report

Executive Summary The Northern Great Plains Inventory & Monitoring Program and Fire Effects Program have been monitoring vegetation in Scotts Bluff National Monument for over 18 years. While methods have changed slightly, this report summarizes data from over 80 locations from 1998-2015. Below, we list the questions we asked using these data and provide a summarized answer. For more details see the full report. A summary of the current condition (2011-2015) and trends (based on 1988-2015) in plant communities at Scotts Bluff is found in Table ES-1. 1. What is the current status of plant community composition and structure of SCBL grasslands (species richness, cover, and diversity) and how has this changed from 1998 to 2015? SCBL plays a vital role in protecting and managing some of the last remnants of native mixed-grass prairie in the area. Native plant diversity is at a moderate level compared to other grasslands in the region (Table 10), but diversity is spatially variable. We found no significant trends in native diversity or evenness from 1998 to 2015, but both are threatened by the increasing cover of annual bromes (Figure 9). There has been an increase in annual brome abundance since the 1990s and continued control efforts will be necessary to maintain native prairie within SCBL. 2. How do trends in grassland condition correlate with climate and fire history? The large variability in SCBL’s climate makes it difficult to discern strong patterns linking temperature, precipitation, and plant community structure (e.g. exotic cover, diversity). Native diversity increased in plots with longer times since burning. There is an adaptive management program planned for 2017 which should provide better guidance to the park on the role of prescribed fire in managing annual bromes. 3. What, if any, rare plants were found in SCBL long-term monitoring plots? We identified 35 rare plant species in SCBL between 1998 and 2015; eight of these are considered critically imperiled within Nebraska. These plants are found in such low abundance and in such few plots, it is unlikely that plant community monitoring will be able to detect any trends in rare plant abundance. We recommend more targeted surveys of rare plant species of concern be completed when funds are available. 4. Was the SCBL golf course restoration effective at creating a grassland community dominated by native species? The golf course restoration project had mixed results. While some native grasses were established in one of the monitoring plots, establishment was poor in the other. To improve the rates of success and the establishment of native species, future projects should include funds to cover invasive plant control for many years (~10) after planting. 5. What is the composition and structure of riparian forests at SCBL? The riparian forest in SCBL is a fairly diverse assemblage of cottonwood, willow species, green ash, and box elder. Exotic grasses and forbs are common in the understory of the riparian forest, viii and continuing control efforts will be necessary to prevent their spread. The large abundance of green ash and box elder seedlings suggests that a transition to ash-dominated forests is underway