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

A new cohomological formula for helicity in R2k+1\R^{2k+1} reveals the effect of a diffeomorphism on helicity

The helicity of a vector field is a measure of the average linking of pairs of integral curves of the field. Computed by a six-dimensional integral, it is widely useful in the physics of fluids. For a divergence-free field tangent to the boundary of a domain in 3-space, helicity is known to be invariant under volume-preserving diffeomorphisms of the domain that are homotopic to the identity. We give a new construction of helicity for closed $(k+1)$-forms on a domain in $(2k+1)$-space that vanish when pulled back to the boundary of the domain. Our construction expresses helicity in terms of a cohomology class represented by the form when pulled back to the compactified configuration space of pairs of points in the domain. We show that our definition is equivalent to the standard one. We use our construction to give a new formula for computing helicity by a four-dimensional integral. We provide a Biot-Savart operator that computes a primitive for such forms; utilizing it, we obtain another formula for helicity. As a main result, we find a general formula for how much the value of helicity changes when the form is pushed forward by a diffeomorphism of the domain; it relies upon understanding the effect of the diffeomorphism on the homology of the domain and the de Rham cohomology class represented by the form. Our formula allows us to classify the helicity-preserving diffeomorphisms on a given domain, finding new helicity-preserving diffeomorphisms on the two-holed solid torus, and proving that there are no new helicity-preserving diffeomorphisms on the standard solid torus. We conclude by defining helicities for forms on submanifolds of Euclidean space. In addition, we provide a detailed exposition of some standard `folk' theorems about the cohomology of the boundary of domains in $\R^{2k+1}$.Comment: 51 pages, 5 figures. For v2, references updated, typos corrected, and a new appendix explaining how the Hodge Decomposition Theorem for forms on manifolds with boundary affects our theorems added. For v3, corrected an error in the caption to Figure 3 and updated reference

The Biot-Savart operator and electrodynamics on subdomains of the three-sphere

We study steady-state magnetic fields in the geometric setting of positive curvature on subdomains of the three-dimensional sphere. By generalizing the Biot-Savart law to an integral operator BS acting on all vector fields, we show that electrodynamics in such a setting behaves rather similarly to Euclidean electrodynamics. For instance, for current J and magnetic field BS(J), we show that Maxwell's equations naturally hold. In all instances, the formulas we give are geometrically meaningful: they are preserved by orientation-preserving isometries of the three-sphere. This article describes several properties of BS: we show it is self-adjoint, bounded, and extends to a compact operator on a Hilbert space. For vector fields that act like currents, we prove the curl operator is a left inverse to BS; thus the Biot-Savart operator is important in the study of curl eigenvalues, with applications to energy-minimization problems in geometry and physics. We conclude with two examples, which indicate our bounds are typically within an order of magnitude of being sharp.Comment: 24 pages (was 28 pages) Revised to include a new introduction, a detailed example, and results about helicity; other changes for readabilit

An infinite family of convex Brunnian links in RnR^n

This paper proves that convex Brunnian links exist for every dimension $n \geq 3$ by constructing explicit examples. These examples are three-component links which are higher-dimensional generalizations of the Borromean rings.Comment: 10 pages, 4 figure

The Shapes of Tight Composite Knots

We present new computations of tight shapes obtained using the constrained gradient descent code RIDGERUNNER for 544 composite knots with 12 and fewer crossings, expanding our dataset to 943 knots and links. We use the new data set to analyze two outstanding conjectures about tight knots, namely that the ropelengths of composite knots are at least 4\pi-4 less than the sums of the prime factors and that the writhes of composite knots are the sums of the writhes of the prime factors.Comment: Summary text file of tight knot lengths and writhing numbers stored in anc/ropelength_data.txt. All other data freely available at http:://www.jasoncantarella.com/ and through Data Conservanc

A pathway-specific microarray analysis highlights the complex and co-ordinated transcriptional networks of the developing grain of field-grown barley

The aim of the study was to describe the molecular and biochemical interactions associated with amino acid biosynthesis and storage protein accumulation in the developing grains of field-grown barley. Our strategy was to analyse the transcription of genes associated with the biosynthesis of storage products during the development of field-grown barley grains using a grain-specific microarray assembled in our laboratory. To identify co-regulated genes, a distance matrix was constructed which enabled the identification of three clusters corresponding to early, middle, and late grain development. The gene expression pattern associated with the clusters was investigated using pathway-specific analysis with specific reference to the temporal expression levels of a range of genes involved mainly in the photosynthesis process, amino acid and storage protein metabolism. It is concluded that the grain-specific microarray is a reliable and cost-effective tool for monitoring temporal changes in the transcriptome of the major metabolic pathways in the barley grain. Moreover, it was sensitive enough to monitor differences in the gene expression profiles of different homologues from the storage protein families. The study described here should provide a strong complement to existing knowledge assisting further understanding of grain development and thereby provide a foundation for plant breeding towards storage proteins with improved nutritional quality

A NEW COHOMOLOGICAL FORMULA FOR HELICITY IN R 2k+1 REVEALS THE EFFECT OF A DIFFEOMORPHISM ON HELICITY

Abstract. The helicity of a vector field is a measure of the average linking of pairs of integral curves of the field. Computed by a six-dimensional integral, it is widely useful in the physics of fluids. For a divergence-free field tangent to the boundary of a domain in 3-space, helicity is known to be invariant under volume-preserving diffeomorphisms of the domain that are homotopic to the identity. We give a new construction of helicity for closed (k + 1)-forms on a domain in (2k + 1)-space that vanish when pulled back to the boundary of the domain. Our construction expresses helicity in terms of a cohomology class represented by the form when pulled back to the compactified configuration space of pairs of points in the domain. We show that our definition is equivalent to the standard one. We use our construction to give a new formula for computing helicity by a four-dimensional integral. We provide a Biot-Savart operator that computes a primitive for such forms; utilizing it, we obtain another formula for helicity. As a main result, we find a general formula for how much the value of helicity changes when the form is pushed forward by a diffeomorphism of the domain; it relies upon understanding the effect of the diffeomorphism on the homology of the domain and the de Rham cohomology class represented by the form. Our formula allows us to classify the helicity-preserving diffeomorphisms on a given domain, finding new helicity-preserving diffeomorphisms on the two-holed solid torus and proving that there are no new helicity-preserving diffeomorphisms on the standard solid torus. We conclude by defining helicities for forms on submanifolds of Euclidean space. In addition, we provide a detailed exposition of some standard ‘folk ’ theorems about the cohomology of the boundary of domains in R 2k+1. 1

Graduate Group ChairpersonAcknowledgements

Most graduate students search for one good advisor; I am fortunate to have two wonderful advisors, Dennis DeTurck and Herman Gluck. Maybe I needed twice the guidance of an average student. They both have been extremely generous with their time, insights, and patience. Herman taught me that math simply cannot be done without tea and cookies. He also taught me that, no matter how hectic academic life may be, there is always time to play tennis or at least talk wistfully about it. Dennis is probably the most efficient person I know; he is excellent at multi-tasking. Dennis taught me a lot about math when I was teaching with him and taught me a lot about teaching when I was working on math with him. My years at Penn would not nearly have been as pleasant or productive without the great office staff here. Janet Burns has gone out of her way to assist me whenever I needed it; she encouraged me and kept me out of trouble for six years, even as I procrastinated my way past deadline after deadline. Janet, I appreciate everything you have done for me! Paula Airey, Monica Pallanti, and Robin Toney have been terrifically helpful; Paula and Monica can cheer up even the dreariest day. I want to acknowledge the continued support of my parents, who have always believed in me and helped me succeed. There are many others at Penn whose wisdom and camaraderie are much appreciated. Among the faculty, I especially want to mention Chris Croke, John Etnyre, and Steve Shatz. So many graduate students have been encouraging, including a great list of officemates: Darren Glass, Nadia Masri, Nakia Rimmer, Sukhendu Mehrotra, and Fred Butler. My girlfriend Sarah has inspired me, comforted me, kept me sane, and forced me to enjoy life during a hectic time. Thanks for all your understanding Sarah! The Biot-Savart operator and electrodynamics on bounded subdomains of the three-spher