Symmetric Criticality for Tight Knots

We prove a version of symmetric criticality for ropelength-critical knots. Our theorem implies that a knot or link with a symmetric representative has a ropelength-critical configuration with the same symmetry. We use this to construct new examples of ropelength critical configurations for knots and links which are different from the ropelength minima for these knot and link types.Comment: This version adds references, and most importantly an acknowledgements section which should have been in the original postin

Intrinsic symmetry groups of links with 8 and fewer crossings

We present an elementary derivation of the "intrinsic" symmetry groups for knots and links of 8 or fewer crossings. The standard symmetry group for a link is the mapping class group \MCG(S^3,L) or \Sym(L) of the pair $(S^3,L)$. Elements in this symmetry group can (and often do) fix the link and act nontrivially only on its complement. We ignore such elements and focus on the "intrinsic" symmetry group of a link, defined to be the image $\Sigma(L)$ of the natural homomorphism \MCG(S^3,L) \rightarrow \MCG(S^3) \cross \MCG(L). This different symmetry group, first defined by Whitten in 1969, records directly whether $L$ is isotopic to a link $L'$ obtained from $L$ by permuting components or reversing orientations. For hyperbolic links both \Sym(L) and $\Sigma(L)$ can be obtained using the output of \texttt{SnapPea}, but this proof does not give any hints about how to actually construct isotopies realizing $\Sigma(L)$. We show that standard invariants are enough to rule out all the isotopies outside $\Sigma(L)$ for all links except $7^2_6$, $8^2_{13}$ and $8^3_5$ where an additional construction is needed to use the Jones polynomial to rule out "component exchange" symmetries. On the other hand, we present explicit isotopies starting with the positions in Cerf's table of oriented links which generate $\Sigma(L)$ for each link in our table. Our approach gives a constructive proof of the $\Sigma(L)$ groups.Comment: 72 pages, 66 figures. This version expands the original introduction into three sections; other minor changes made for improved readabilit

Biodiversity Assessment of Champion Creek Utilizing DNA Barcoding

Biodiversity, the variety and number of species in a particular habitat, can be an important indicator of the overall health and functioning of an ecosystem. In this regard, to determine the diversity of an environment requires the ability to accurately identify what species are present and determine their relative abundance in the habitat. The accurate identification of species based on morphology can be problematic as it often overlooks cryptic species, relies on taxonomic expertise and often leads to incorrect identifications. An alternative that overcomes these limitations is DNA barcoding. DNA barcoding is a system of species identification that utilizes an ~650 bp fragment of the mitochondrial cytochrome oxidase one gene (COI) to identify and delineate species. In this project, students in Molecular Ecology (BIOL 4951) set out to assess the freshwater diversity of Champion Creek at Georgia College’s East Campus. To do this, ~60 fish were collected from Champion Creek through seining and were brought back to the laboratory for biological identification via DNA barcoding. In the laboratory, high molecular weight DNA was extracted from 40 specimens and the COI gene was amplified via the polymerase chain reaction (PCR). Successfully amplified products were then sent off to Eurofins Genomics for sequencing. Analysis of DNA sequences identified 16 unique species of fish. The high number of species recovered, along with the high diversity and richness indices calculated, suggest that Lake Laurel is an overall healthy ecosystem. Information regarding the biodiversity of Lake Laurel from this project could be valuable to subsequent field studies regarding the ecology of this area

The 27 Possible Intrinsic Symmetry Groups of Two-Component Links

We consider the “intrinsic” symmetry group of a two-component link L, defined to be the image ∑(L) of the natural homomorphism from the standard symmetry group MCG(S3, L) to the product MCG(S3) × MCG(L). This group, first defined by Whitten in 1969, records directly whether L is isotopic to a link L′ obtained from L by permuting components or reversing orientations; it is a subgroup of Γ2, the group of all such operations. For two-component links, we catalog the 27 possible intrinsic symmetry groups, which represent the subgroups of Γ2 up to conjugacy. We are able to provide prime, nonsplit examples for 21 of these groups; some are classically known, some are new. We catalog the frequency at which each group appears among all 77,036 of the hyperbolic two-component links of 14 or fewer crossings in Thistlethwaite’s table. We also provide some new information about symmetry groups of the 293 non-hyperbolic two-component links of 14 or fewer crossings in the table

Dynamics of the December 2020 Ash‐Poor Plume Formed by Lava‐Water Interaction at the Summit of Kīlauea Volcano, Hawaiʻi

Abstract On 20 December 2020, after more than 2 years of quiescence at Kīlauea Volcano, Hawaiʻi, renewed volcanic activity in the summit crater caused boiling of the water lake over a period of ∼90 min. The resulting water‐rich, electrified plume rose to 11–13 km above sea level, which is among the highest plumes on record for Kīlauea. Although conventional models would infer a high mass flux from explosive magma‐water interaction, the plume was not associated with an infrasound signal indicative of “explosive” activity, nor did it produce a measurable ash‐fall deposit. We use multisensor data to characterize lava‐water interaction and plume generation during this opening phase of the 2020–21 eruption. Satellite, weather radar, and eyewitness observations revealed that the plume was rich in water vapor and hydrometeors but transported less ash than expected from its maximum height. Volcanic lightning flashes detected by ground‐based cameras were confined to freezing altitudes of the upper cloud, suggesting that the ice formation drove the electrification of this plume. The low acoustic energy from lava‐water interaction points to a weakly explosive style of hydrovolcanism. Heat transfer calculations show that the lava to water heat flux was sufficient to boil the lake within 90 min. Limited mixing of lava and water inhibited major steam explosions and fine fragmentation. Results from one‐dimensional plume modeling suggest that the models may underpredict plume height due to overestimation of crosswind air‐entrainment. Our findings shed light on an unusual style of volcanism in which weakly explosive lava‐water interaction generated an outsized plume