Senior Recital: Jack Cazet, guitar

https://digitalwindow.vassar.edu/musi_senior/1017/thumbnail.jp

Quandles with one non-trivial column

The axioms of a quandle imply that the columns of its Cayley table are permutations. This paper studies quandles with exactly one non-trivially permuted column. Their automorphism groups, quandle polynomials, (symmetric) cohomology groups, and $Hom$ quandles are studied. The quiver and cocycle invariant of links using these quandles are shown to relate to linking number.Comment: 14 pages, 4 figure

The stick number of rail arcs

Consider two parallel lines $\ell_1$ and $\ell_2$ in $\mathbb{R}^3$. A rail arc is an embedding of an arc in $\mathbb{R}^3$ such that one endpoint is on $\ell_1$, the other is on $\ell_2$, and its interior is disjoint from $\ell_1\cup\ell_2$. Rail arcs are considered up to rail isotopies, ambient isotopies of $\mathbb{R}^3$ with each self-homeomorphism mapping $\ell_1$ and $\ell_2$ onto themselves. When the manifolds and maps are taken in the piecewise linear category, these rail arcs are called stick rail arcs. The stick number of a rail arc class is the minimum number of sticks, line segments in a p.l. arc, needed to create a representative. This paper will calculate the stick numbers of rail arcs classes with a crossing number at most 2 and use a winding number invariant to calculate the stick numbers of infinitely many rail arc classes. Each rail arc class has two canonically associated knot classes, its under and over companions. This paper also introduces the rail stick number of knot classes, the minimum number of sticks needed to create a rail arcs whose under or over companion is the knot class. The rail stick number is calculated for all knot classes with crossing number at most 9. The stick number of multi-component rail arcs classes is considered as well as the lattice stick number of rail arcs.Comment: 19 pages, 22 figure

Quandle Theoretic Knot Invariants

This dissertation is based on the following three publications of the author [15,16,17], focusing on quandles and their applications in knot theory.Chapter 2 defines a family of quandles and studies their algebraic invariants. The axioms of a quandle imply that the columns of its Cayley table are permutations. The chapter studies quandles with exactly one non-trivially permuted column. Their automorphism groups, quandle polynomials, (symmetric) cohomology groups, and Hom quandles are studied. The quiver and cocycle invariant of links using these quandles are shown to relate to linking number. Chapter 3 describes the triple point number of non-orientable surface-links using symmetric quandles. Analogous to a classical knot diagram, a surface-link can be generically projected to 3-space and given crossing information to create a broken sheet diagram. The triple point number of a surface-link is the minimal number of triple points among all broken sheet diagrams that lift to that surface-link. The chapter generalizes a family of Oshiro to show that there are non- split surface-links of arbitrarily many trivial components whose triple point number can be made arbitrarily large. Chapter 4 focuses on the triple point number of surface-links found in Yoshikawa’s table. Yoshikawa made an enumeration of knotted surfaces in R4 with ch-index 10 or less. This remark- able table is the first to tabulate knotted surfaces analogous to the classical prime knot table. This chapter compiles the known triple point numbers of the surface-links represented in Yoshikawa’s table and calculates or provides bounds on the triple point number of the remaining surface-links. Chapter 5 is included to study quandle invariants of knotoids. The chapter focuses on the chirality of knotoids using shadow quandle colorings and the shadow quandle cocycle invariant. The shadow coloring number and the shadow quandle cocycle invariant is shown to distinguish infinitely many knotoids from their mirrors. Specifically, the knot-type knotoid 31 is shown to be chiral. The weight of a quandle 3-cocycle is used to calculate the crossing numbers of infinitely many multi-linkoids

New key-tools for pollen identification in research and education

Pollen ID offers a free and easy access to various palynological information and compiles in the same web-space a pollen database and different services through a friendly user interface. Pollen ID proposes, or will propose, pollen and plant descriptions, terminology learning with an illustrated glossary and interactive images, identification keys, pollen analysis, pollen diagram construction, links with vegetation and climate data. The Pollen ID project is presently restricted to the European and Mediterranean geographical area, but it will be extended to other regions as well. This project is still in progress; its content and user interface – presently in French - will be soon available in English. In its final shape, the Pollen ID project will include palynological applications such as pollen determination tests, several original pollen analysis exercises with representations in diagrams and an easy interpretation of vegetation and climate. Pollen ID is accessible on http://lisupmc. snv.jussieu.fr/pollen/

Bounds in simple hexagonal lattice and classification of 11-stick knots

The stick number and the edge length of a knot type in the simple hexagonal lattice (sh-lattice) are the minimal numbers of sticks and edges required, respectively, to construct a knot of the given type in sh-lattice. By introducing a linear transformation between lattices, we prove that for any given knot both values in the sh-lattice are strictly less than the values in the cubic lattice. Moreover, we find lower bounds for any given knot's stick number and edge length in sh-lattice using these properties in the cubic lattice. Finally, we show that the only non-trivial 11-stick knots in the sh-lattice are the trefoil knot ($3_1$) and the figure-eight knot ($4_1$).Comment: 25 pages, 17 figure

Three-dimensional organotypic matrices from alternative collagen sources as pre-clinical models for cell biology.

Organotypic co-cultures bridge the gap between standard two-dimensional culture and mouse models. Such assays increase the fidelity of pre-clinical studies, to better inform lead compound development and address the increasing attrition rates of lead compounds within the pharmaceutical industry, which are often a result of screening in less faithful two-dimensional models. Using large-scale acid-extraction techniques, we demonstrate a step-by-step process to isolate collagen I from commercially available animal byproducts. Using the well-established rat tail tendon collagen as a benchmark, we apply our novel kangaroo tail tendon collagen as an alternative collagen source for our screening-ready three-dimensional organotypic co-culture platform. Both collagen sources showed equal applicability for invasive, proliferative or survival assessment of well-established cancer models and clinically relevant patient-derived cancer cell lines. Additional readouts were also demonstrated when comparing these alternative collagen sources for stromal contributions to stiffness, organization and ultrastructure via atomic force microscopy, second harmonic generation imaging and scanning electron microscopy, among other vital biological readouts, where only minor differences were found between the preparations. Organotypic co-cultures represent an easy, affordable and scalable model to investigate drug responses within a physiologically relevant 3D platform

Cryopreservation of human cancers conserves tumour heterogeneity for single-cell multi-omics analysis

Background: High throughput single-cell RNA sequencing (scRNA-Seq) has emerged as a powerful tool for exploring cellular heterogeneity among complex human cancers. scRNA-Seq studies using fresh human surgical tissue are logistically difficult, preclude histopathological triage of samples, and limit the ability to perform batch processing. This hindrance can often introduce technical biases when integrating patient datasets and increase experimental costs. Although tissue preservation methods have been previously explored to address such issues, it is yet to be examined on complex human tissues, such as solid cancers and on high throughput scRNA-Seq platforms. Methods: Using the Chromium 10X platform, we sequenced a total of ~ 120,000 cells from fresh and cryopreserved replicates across three primary breast cancers, two primary prostate cancers and a cutaneous melanoma. We performed detailed analyses between cells from each condition to assess the effects of cryopreservation on cellular heterogeneity, cell quality, clustering and the identification of gene ontologies. In addition, we performed single-cell immunophenotyping using CITE-Seq on a single breast cancer sample cryopreserved as solid tissue fragments. Results: Tumour heterogeneity identified from fresh tissues was largely conserved in cryopreserved replicates. We show that sequencing of single cells prepared from cryopreserved tissue fragments or from cryopreserved cell suspensions is comparable to sequenced cells prepared from fresh tissue, with cryopreserved cell suspensions displaying higher correlations with fresh tissue in gene expression. We showed that cryopreservation had minimal impacts on the results of downstream analyses such as biological pathway enrichment. For some tumours, cryopreservation modestly increased cell stress signatures compared to freshly analysed tissue. Further, we demonstrate the advantage of cryopreserving whole-cells for detecting cell-surface proteins using CITE-Seq, which is impossible using other preservation methods such as single nuclei-sequencing. Conclusions: We show that the viable cryopreservation of human cancers provides high-quality single-cells for multiomics analysis. Our study guides new experimental designs for tissue biobanking for future clinical single-cell RNA sequencing studies

Tumour-associated carbohydrate antigens in breast cancer

Glycosylation changes that occur in cancer often lead to the expression of tumour-associated carbohydrate antigens. In breast cancer, these antigens are usually associated with a poor prognosis and a reduced overall survival. Cellular models have shown the implication of these antigens in cell adhesion, migration, proliferation and tumour growth. The present review summarizes our current knowledge of glycosylation changes (structures, biosynthesis and occurrence) in breast cancer cell lines and primary tumours, and the consequences on disease progression and aggressiveness. The therapeutic strategies attempted to target tumour-associated carbohydrate antigens in breast cancer are also discussed