Torus knot filtered embedded contact homology of the tight contact 3-sphere

Knot filtered embedded contact homology was first introduced by Hutchings in 2015; it has been computed for the standard transverse unknot in irrational ellipsoids by Hutchings and for the Hopf link in lens spaces L(n,n-1) via a quotient by Weiler. While straightforward toric constructions can be used to understand the ECH chain complexes of open books along the unknot and Hopf link, they do not readily adapt to general torus knots and links. In this paper, we generalize the definition and invariance of knot filtered embedded contact homology to allow for degenerate knots with rational rotation numbers. We then develop new methods for understanding the embedded contact homology chain complex of positive torus knotted fibrations of the standard tight contact 3-sphere in terms of their presentation as open books and as Seifert fibered spaces. We provide Morse-Bott methods, using a doubly filtered complex and the energy filtered perturbed Seiberg-Witten theory developed by Hutchings and Taubes, and use them to compute the T(2,q) knot filtered embedded contact homology, for q odd and positive. In the sequel we complete the computation for positive T(p,q) knots (where there is a nonvanishing differential) and use our results to deduce quantitative existence results for torus knotted Reeb dynamics on the tight 3-sphere and the mean action of area preserving diffeomorphisms of once punctured surfaces of arbitrary genus arising as Seifert surfaces of positive torus knots.Comment: 85 pages, arXiv insists the primary is GT rather than S

Embedded contact homology of prequantization bundles

The 2011 PhD thesis of Farris demonstrated that the ECH of a prequantization bundle over a Riemann surface is isomorphic as a Z/2Z-graded group to the exterior algebra of the homology of its base. We extend this result by computing the Z-grading on the chain complex, permitting a finer understanding of this isomorphism. We fill in some technical details, including the Morse-Bott direct limit argument and some writhe bounds. The former requires the isomorphism between filtered Seiberg-Witten Floer cohomology and filtered ECH as established by Hutchings--Taubes. The latter requires the work on higher asymptotics of pseudoholomorphic curves by Cristofaro-Gardiner--Hutchings--Zhang. In a subsequent paper, we plan to extend these methods for Seifert fiber spaces and relate the ECH U-map to orbifold Gromov-Witten invariants of the base.Comment: 85 page

A classification of infinite staircases for Hirzebruch surfaces

The ellipsoid embedding function of a symplectic manifold gives the smallest amount by which the symplectic form must be scaled in order for a standard ellipsoid of the given eccentricity to embed symplectically into the manifold. It was first computed for the standard four-ball (or equivalently, the complex projective plane) by McDuff and Schlenk, and found to contain the unexpected structure of an "infinite staircase," that is, an infinite sequence of nonsmooth points arranged in a piecewise linear stair-step pattern. Later work of Usher and Cristofaro-Gardiner--Holm--Mandini--Pires suggested that while four-dimensional symplectic toric manifolds with infinite staircases are plentiful, they are highly non-generic. This paper concludes the systematic study of one-point blowups of the complex projective plane, building on previous work of Bertozzi-Holm-Maw-McDuff-Mwakyoma-Pires-Weiler, Magill-McDuff, Magill-McDuff-Weiler, and Magill on these Hirzebruch surfaces. We prove a conjecture of Cristofaro-Gardiner--Holm--Mandini--Pires for this family: that if the blowup is of rational weight and the embedding function has an infinite staircase then that weight must be $1/3$. We show also that the function for this manifold does not have a descending staircase. Furthermore, we give a sufficient and necessary condition for the existence of an infinite staircase in this family which boils down to solving a quadratic equation and computing the function at one specific value. Many of our intermediate results also apply to the case of the polydisk (or equivalently, the symplectic product of two spheres).Comment: 56 pages, 7 figure

Four-periodic infinite staircases for four-dimensional polydisks

The ellipsoid embedding function of a symplectic four-manifold measures the amount by which its symplectic form must be scaled in order for it to admit an embedding of an ellipsoid of varying eccentricity. This function generalizes the Gromov width and ball packing numbers. In the one continuous family of symplectic four-manifolds that has been analyzed, one-point blowups of the complex projective plane, there is a full measure set of symplectic forms whose ellipsoid embedding functions are completely described by finitely many obstrutions, while there is simultaneously a Cantor set of symplectic forms for which an infinite number of obstructions are needed. The latter case is called an infinite staircase. In this paper we identify a new infinite staircase when the target is a four-dimensional polydisk, extending a countable family identified by Usher in 2019. We use almost toric fibrations to compute key upper bounds and outline an analogy with the case of the one-point blowup of the complex projective plane. Our work indicates a method of proof for a conjecture of Usher. We further describe the continued fractions of ellipsoid eccentricities which provide key lower bounds on the ellipsoid embedding function, and we explain Python code for efficiently exploring the space of symplectic embeddings.Comment: 55 pages, 15 figure

Pattern Avoidance in Poset Permutations

We extend the concept of pattern avoidance in permutations on a totally ordered set to pattern avoidance in permutations on partially ordered sets. The number of permutations on $P$ that avoid the pattern $\pi$ is denoted $Av_P(\pi)$. We extend a proof of Simion and Schmidt to show that $Av_P(132) \leq Av_P(123)$ for any poset $P$, and we exactly classify the posets for which equality holds.Comment: 13 pages, 1 figure; v2: corrected typos; v3: corrected typos and improved formatting; v4: to appear in Order; v5: corrected typos; v6: updated author email addresse

A Gene for Autosomal Recessive Limb-Girdle Muscular Dystrophy in Manitoba Hutterites Maps to Chromosome Region 9q31-q33: Evidence for Another Limb-Girdle Muscular Dystrophy Locus

SummaryCharacterized by proximal muscle weakness and wasting, limb-girdle muscular dystrophies (LGMDs) are a heterogeneous group of clinical disorders. Previous reports have documented either autosomal dominant or autosomal recessive modes of inheritance, with genetic linkage studies providing evidence for the existence of at least 12 distinct loci. Gene products have been identified for five genes responsible for autosomal recessive forms of the disorder. We performed a genome scan using pooled DNA from a large Hutterite kindred in which the affected members display a mild form of autosomal recessive LGMD. A total of 200 markers were used to screen pools of DNA from patients and their siblings. Linkage between the LGMD locus and D9S302 (maximum LOD score 5.99 at recombination fraction .03) was established. Since this marker resides within the chromosomal region known to harbor the gene causing Fukuyama congenital muscular dystrophy (FCMD), we expanded our investigations, to include additional markers in chromosome region 9q31-q34.1. Haplotype analysis revealed five recombinations that place the LGMD locus distal to the FCMD locus. The LGMD locus maps close to D9S934 (maximum multipoint LOD score 7.61) in a region that is estimated to be ∼4.4 Mb (Genetic Location Database composite map). On the basis of an inferred ancestral recombination, the gene may lie in a 300-kb region between D9S302 and D9S934. Our results provide compelling evidence that yet another gene is involved in LGMD; we suggest that it be named “LGMD2H.

VLBI observations of SN 2008iz: I. Expansion velocity and limits on anisotropic expansion

We present observations of the recently discovered supernova 2008iz in M82 with the VLBI High Sensitivity Array at 22 GHz, the Very Large Array at frequencies of 1.4, 4.8, 8.4, 22 and 43 GHz, and the Chandra X-ray observatory. The supernova was clearly detected on two VLBI images, separated by 11 months. The source shows a ring-like morphology and expands with a velocity of ~23000 km/s. The most likely explosion date is in mid February 2008. The measured expansion speed is a factor of ~2 higher than expected under the assumption that synchrotron self-absorption dominates the light curve at the peak, indicating that this absorption mechanism may not be important for the radio emission. We find no evidence for an asymmetric explosion. The VLA spectrum shows a broken power law, indicating that the source was still optically thick at 1.4 GHz in April 2009. Finally, we report upper limits on the X-ray emission from SN 2008iz and a second radio transient recently discovered by MERLIN observations.Comment: accepted Astronomy & Astrophysics, 9 pages, 8 figures, also available at http://www.mpifr-bonn.mpg.de/staff/abrunthaler/pub.shtm

Multi-epoch very long baseline interferometric observations of the nuclear starburst region of NGC 253: Improved modelling of the supernova and star-formation rates

The results of multi-epoch observations of the southern starburst galaxy, NGC 253, with the Australian Long Baseline Array (LBA) at 2.3 GHz are presented. As with previous radio interferometric observations of this galaxy, no new sources were discovered. By combining the results of this survey with Very Large Array observations at higher frequencies from the literature, spectra were derived and a free-free absorption model was fitted of 20 known sources in NGC 253. The results were found to be consistent with previous studies. The supernova remnant, 5.48−43.3, was imaged with the highest sensitivity and resolution to date, revealing a two-lobed morphology. Comparisons with previous observations of similar resolution give an upper limit of 104 km s−1 for the expansion speed of this remnant. We derive a supernova rate of <0.2 yr−1 for the inner 300 pc using a model that improves on previous methods by incorporating an improved radio supernova peak luminosity distribution and by making use of multi-wavelength radio data spanning 21 years. A star formation rate of SFR(M ≥ 5 M) < 4.9 M yr−1 was also estimated using the standard relation between supernova and star-formation rates. Our improved estimates of supernova and star-formation rates are consistent with studies at other wavelengths. The results of our study point to the possible existence of a small population of undetected supernova remnants, suggesting a low rate of radio supernovae production in NGC 253

SN 2011ht: Confirming a Class of Interacting Supernovae with Plateau Light Curves (Type IIn-P)

We present photometry and spectroscopy of the Type IIn supernova (SN) 2011ht, identified previously as a SN impostor. The light curve exhibits an abrupt transition from a well-defined ~120 day plateau to a steep bolometric decline. Leading up to peak brightness, a hot emission-line spectrum exhibits signs of interaction with circumstellar material (CSM), in the form of relatively narrow P-Cygni features of H I and He I superimposed on broad Lorentzian wings. For the remainder of the plateau phase the spectrum exhibits strengthening P-Cygni profiles of Fe II, Ca II, and H-alpha. By day 147, after the plateau has ended, the SN entered the nebular phase, heralded by the appearance of forbidden transitions of [O I], [O II], and [Ca II] over a weak continuum. At this stage, the light curve exhibits a low luminosity that is comparable to that sub-luminous Type II-P supernovae, and a relatively fast visual-wavelength decline that is significantly steeper than the Co-56 decay rate. However, the total bolometric decline, including the IR luminosity, is consistent with Co-56 decay, and implies a low Ni-56 mass of ~0.01 M(Sun). We therefore characterize SN 2011ht as a bona-fide core-collapse SN very similar to the peculiar SNe IIn 1994W and 2009kn. These three SNe define a subclass, which are Type IIn based on their spectrum, but that also exhibit well-defined plateaus and produce low Ni-56 yields. We therefore suggest Type IIn-P as a name for this subclass. Possible progenitors of SNe IIn-P, consistent with the available data, include 8-10 M(Sun) stars, which undergo core collapse as a result of electron capture after a brief phase of enhanced mass loss, or more massive M>25 M(Sun) progenitors, which experience substantial fallback of the metal-rich radioactive ejecta. In either case, the energy radiated by these three SNe during their plateau must be dominated by CSM interaction (abridged).Comment: accepted, post-proof version (includes new data

Republicanism and Markets

The republican tradition has long been ambivalent about markets and commercial society more generally: from the contrasting positions of Rousseau and Smith in the eighteenth century to recent neorepublican debates about capitalism, republicans have staked out diverse positions on fundamental issues of political economy. Rather than offering a systematic historical survey of these discussions, this chapter will instead focus on the leading neo-republican theory—that of Philip Pettit—and consider its implications for market society. As I will argue, Pettit’s theory is even friendlier to markets than most have believed: far from condemning commercial society, his theory recognizes that competitive markets and their institutional preconditions are an alternative means to limit arbitrary power across the domestic, economic, and even political spheres. While most republican theorists have focused on political means to limit such power—including both constitutional means (e.g., separation of powers, judicial review, the rule of law, federalism) and participatory ones (democratic elections and oversight)—I will examine here an economic model of republicanism that can complement, substitute for, and at times displace the standard political model. Whether we look at spousal markets, labor markets, or residential markets within federal systems, state policies that heighten competition among their participants and resource exit from abusive relationships within them can advance freedom as non-domination as effectively or even more effectively than social-democratic approaches that have recently gained enthusiasts among republicans. These conclusions suggest that democracy, be it social or political, is just one means among others for restraining arbitrary power and is consequently less central to (certain versions of) republicanism than we may have expected. So long as they counteract domination, economic inroads into notionally democratic territory are no more worrisome than constitutional ones