arXiv subjects
Publications and source records attributed to Colin Adams.
We define a class of links in handlebodies called ``charm bracelets," which are a subset of staked links. We provide tools to construct infinitely many such hyperbolic links and bound the corresponding volumes from below in terms of volumes corresponding to the individual charms.
Let $M_0$ be a compact and orientable 3-manifold. After capping off spherical boundaries with balls and removing any torus boundaries, we prove that the resulting manifold $M$ contains handlebodies of arbitrary genus such that the closure of their complement is hyperbolic. We then extend the octahedral decomposition to obtain bounds on volume for some of these handlebody complements.
We consider a simple but infinite class of staked links known as bongles. We provide necessary and sufficient conditions for these bongles to be hyperbolic. Then, we prove that all balanced hyperbolic $n$-bongles have the same volume and the corresponding volume is an upper bound on the volume of any hyperbolic $n$-bongle for $n$ even. Moreover, all hyperbolic $n$-bongles have volume strictly less than $5n(1.01494\dots)$. We also include explicit volume calculations for all hyperbolic 3-bongles through 6-bongles.
VERITAS is an imaging atmospheric Cherenkov telescope (IACT) array most sensitive to gamma rays in the very-high-energy (VHE) energy band (85 GeV - 30 TeV). As a part of its active galactic nuclei (AGN) program, VERITAS focuses on the identification and follow-up of AGN flares reported by other multiwavelength observatories. Between October 15th and October 19th, 2020, VERITAS followed up on the Fermi-LAT and MAGIC detections of a flare of the intermediate-frequency-peaked BL Lacertae (IBL) object, B2 1811+31, located at a redshift of z=0.117. In this work, we present preliminary scientific results from the analysis of B2 1811+31's 2020 flare, including the corresponding Fermi-LAT light curve and VERITAS detection analysis.
Let $F$ be a compact orientable surface with nonempty boundary other than a disk. Let $L$ be a link in $F \times I$ with a connected weakly prime cellular alternating projection to $F$. We provide simple conditions that determine exactly when $(F \times I) \setminus N(L)$ is hyperbolic. We also consider suitable embeddings of $F \times I$ in an ambient manifold $Y$ with boundary and provide conditions on links $L \subset F \times I$ which guarantee tg-hyperbolicity of $Y \setminus N(L)$. These results provide many examples of hyperbolic links in handlebodies and fiber bundles. It also provides many examples of staked links that are hyperbolic.
The Breakthrough Listen Initiative is conducting a program using multiple telescopes around the world to search for "technosignatures": artificial transmitters of extraterrestrial origin from beyond our solar system. The VERITAS Collaboration joined this program in 2018, and provides the capability to search for one particular technosignature: optical pulses of a few nanoseconds duration detectable over interstellar distances. We report here on the analysis and results of dedicated VERITAS observations of Breakthrough Listen targets conducted in 2019 and 2020 and of archival VERITAS data collected since 2012. Thirty hours of dedicated observations of 136 targets and 249 archival observations of 140 targets were analyzed and did not reveal any signals consistent with a technosignature. The results are used to place limits on the fraction of stars hosting transmitting civilizations. We also discuss the minimum-pulse sensitivity of our observations and present VERITAS observations of CALIOP: a space-based pulsed laser onboard the CALIPSO satellite. The detection of these pulses with VERITAS, using the analysis techniques developed for our technosignature search, allows a test of our analysis efficiency and serves as an important proof-of-principle.
We report the detection of very high energy gamma-ray emission from the blazar S3 1227+25 (VER J1230+253) with the Very Energetic Radiation Imaging Telescope Array System (VERITAS). VERITAS observations of the source were triggered by the detection of a hard-spectrum GeV flare on May 15, 2015 with the Fermi-Large Area Telescope (LAT). A combined five-hour VERITAS exposure on May 16th and May 18th resulted in a strong 13$\sigma$ detection with a differential photon spectral index, $\Gamma$ = 3.8 $\pm$ 0.4, and a flux level at 9% of the Crab Nebula above 120 GeV. This also triggered target of opportunity observations with Swift, optical photometry, polarimetry and radio measurements, also presented in this work, in addition to the VERITAS and Fermi-LAT data. A temporal analysis of the gamma-ray flux during this period finds evidence of a shortest variability timescale of $\tau_{obs}$ = 6.2 $\pm$ 0.9 hours, indicating emission from compact regions within the jet, and the combined gamma-ray spectrum shows no strong evidence of a spectral cut-off. An investigation into correlations between the multiwavelength observations found evidence of optical and gamma-ray correlations, suggesting a single-zone model of emission. Finally, the multiwavelength spectral energy distribution is well described by a simple one-zone leptonic synchrotron self-Compton radiation model.
We consider knots and links in handlebodies that have hyperbolic complements and operations akin to composition. Cutting the complements of two such open along separating twice-punctured disks such that each of the four resulting handlebodies has positive genus, and gluing a pair of pieces together along the twice-punctured disks in their boundaries, we show the result is also hyperbolic. This should be contrasted with composition of any pair of knots in the 3-sphere, which is never hyperbolic. Similar results are obtained when both twice-punctured disks are in the same handlebody and we glue a resultant piece to itself along copies of the twice-punctured disks on its boundary. We include applications to staked links.
In 2010, Turaev introduced knotoids as a variation on knots that replaces the embedding of a circle with the embedding of a closed interval with two endpoints. A variety of knot invariants have been extended to knotoids. Here we provide definitions of hyperbolicity for both spherical and planar knotoids. We prove that the product of hyperbolic spherical knotoids is hyperbolic and the volumes add. We also determine the least volume of a rational spherical knotoid and provide various classes of hyperbolic knotoids. We also include tables of hyperbolic volumes for both spherical and planar knotoids.
In 2010, Turaev introduced knotoids as a variation on knots that replaces the embedding of a circle with the embedding of a closed interval with two endpoints which here we call poles. We define generalized knotoids to allow arbitrarily many poles, intervals, and circles, each pole corresponding to any number of interval endpoints, including zero. This theory subsumes a variety of other related topological objects and introduces some particularly interesting new cases. We explore various analogs of knotoid invariants, including height, index polynomials, bracket polynomials and hyperbolicity. We further generalize to knotoidal graphs, which are a natural extension of spatial graphs that allow both poles and vertices.
In a variety of settings we provide a method for decomposing a 3-manifold $M$ into pieces. When the pieces have the appropriate type of hyperbolicity, then the manifold $M$ is hyperbolic and its volume is bounded below by the sum of the appropriately defined hyperbolic volumes of the pieces. A variety of examples of appropriately hyperbolic pieces and volumes are provided, with many examples from link complements in the 3-sphere.
The composition of any two nontrivial classical knots is a satellite knot, and thus, by work of Thurston, is not hyperbolic. In this paper, we explore the composition of virtual knots, which are an extension of classical knots that generalize the idea of knots in $S^3$ to knots in $S \times I$ where $S$ is a closed orientable surface. We prove that for any two hyperbolic virtual knots, there is a composition that is hyperbolic. We then obtain strong lower bounds on the volume of the composition using information from the original knots.
The recent discovery of electromagnetic signals in coincidence with gravitational waves from neutron-star mergers has solidified the importance of multimessenger campaigns for studying the most energetic astrophysical events. Pioneering multimessenger observatories, such as the LIGO/Virgo gravitational wave detectors and the IceCube neutrino observatory, record many candidate signals that fall short of the detection significance threshold. These sub-threshold event candidates are promising targets for multimessenger studies, as the information provided by these candidates may, when combined with time-coincident gamma-ray observations, lead to significant detections. In this contribution, I describe our use of sub-threshold binary neutron star merger candidates identified in Advanced LIGO's first observing run (O1) to search for transient events in very-high-energy gamma rays using archival observations from the VERITAS imaging atmospheric Cherenkov telescope array. I describe the promise of this technique for future joint sub-threshold searches.
Menasco proved that nontrivial links in the 3-sphere with connected prime alternating non-2-braid projections are hyperbolic. This was further extended to augmented alternating links wherein non-isotopic trivial components bounding disks punctured twice by the alternating link were added. Lackenby proved that the first and second collections of links together form a closed subset of the set of all finite volume hyperbolic 3-manifolds in the geometric topology. Adams showed hyperbolicity for generalized augmented alternating links, which include additional trivial components that bound n-punctured disks for $n \geq 2$. Here we prove that generalized augmented cellular alternating links in I-bundles over closed surfaces are also hyperbolic and that in $S \times I$, the cellular alternating links and the augmented cellular alternating together form a closed subset of finite volume hyperbolic 3-manifolds in the geometric topology. Explicit examples of additional links in $S \times I$ to which these results apply are included.
Multicrossings, which have previously been defined for classical knots and links, are extended to virtual knots and links. In particular, petal diagrams are shown to exist for all virtual knots.
We determine all non-edge-to-edge tilings of the sphere by regular spherical polygons of three or more sides.
We improve the upper bound on the superbridge index $sb[K]$ of a knot type $[K]$ in terms of the bridge index $b[K]$ from $sb[K] \leq 5b -3$ to $sb[K]\leq 3b[k] - 1$.
We construct for every connected surface $S$ of finite negative Euler characteristic and every $H \in [0,1)$, a hyperbolic 3-manifold $N(S,H)$ of finite volume and a proper, two-sided, totally umbilic embedding $f\colon S\to N(S,H)$ with mean curvature $H$. Conversely, we prove that a complete, totally umbilic surface with mean curvature $H \in [0,1)$ embedded in a hyperbolic 3-manifold of finite volume must be proper and have finite negative Euler characteristic.