A Turaev surface approach to Khovanov homology

We introduce Khovanov homology for ribbon graphs and show that the Khovanov homology of a certain ribbon graph embedded on the Turaev surface of a link is isomorphic to the Khovanov homology of the link (after a grading shift). We also present a spanning quasi-tree model for the Khovanov homology of a ribbon graph.Comment: 30 pages, 18 figures, added sections on virtual links and Reidemeister move

Turaev genus, knot signature, and the knot homology concordance invariants

We give bounds on knot signature, the Ozsvath-Szabo tau invariant, and the Rasmussen s invariant in terms of the Turaev genus of the knot.Comment: 15 pages, 5 figure

Extremal Khovanov homology of Turaev genus one links

The Turaev genus of a link can be thought of as a way of measuring how non-alternating a link is. A link is Turaev genus zero if and only if it is alternating, and in this viewpoint, links with large Turaev genus are very non-alternating. In this paper, we study Turaev genus one links, a class of links which includes almost alternating links. We prove that the Khovanov homology of a Turaev genus one link is isomorphic to $\mathbb{Z}$ in at least one of its extremal quantum gradings. As an application, we compute or nearly compute the maximal Thurston Bennequin number of a Turaev genus one link.Comment: 30 pages, 18 figure

The 2-Factor Polynomial Detects Even Perfect Matchings

In this paper, we prove that the 2-factor polynomial, an invariant of a planar trivalent graph with a perfect matching, counts the number of 2- factors that contain the the perfect matching as a subgraph. Consequently, we show that the polynomial detects even perfect matchings.Comment: 16 pages, 17 figure

The Disadvantaged Status in the American Medical College Application Service: Medical Students Reflect on Their Experiences

The option to identify as disadvantaged in the American Medical College Application Service (AMCAS) primary application may lead to unclear expectations for applicants. Despite its inception more than a decade ago, there are no published materials containing either a comprehensive definition, or explanation of the role of the Disadvantaged Status in student selection. Research on the Disadvantaged Status is similarly lacking. The purpose of this interpretivist case study was to explore how current students made meaning of the option to identify as disadvantaged when they were applying to medical schools. Through this research I uncovered meanings applicants ascribed to the disadvantaged term, how they determined whether they were disadvantaged, and how they decided whether or not to apply as such. Through open-ended interviews with 15 students at a medical school in the Northeast, document analysis of their AMCAS files, and with a theoretical framework that included symbolic interactionism, social comparison theory, stigma, and impression management, it became clear that deciding whether or not to apply as a disadvantaged applicant in the AMCAS primary application is both complex and fragile. Simply having experienced hardships during childhood was insufficient for many participants in this study to determine whether or not they were disadvantaged or should apply as such. The process of determining whether or not to apply as a disadvantaged applicant was confounded by a myriad of factors represented by the following nine themes: experience with disadvantage, resources, ambiguity, audience, pride, stigma, ethics, right to identify, and impression management. I concluded this dissertation with what I considered to be the most significant implications, in particular, that not all applicants are using the Disadvantaged Status consistently. I made recommendations for staff at the Association of American Medical Colleges, faculty and staff at medical schools and undergraduate institutions, and future applicants. I closed this dissertation with my final thoughts on this research experience

The Jones polynomial of an almost alternating link

A link is almost alternating if it is non-alternating and has a diagram that can be transformed into an alternating diagram via one crossing change. We give formulas for the first two and last two potential coefficients of the Jones polynomial of an almost alternating link. Using these formulas, we show that the Jones polynomial of an almost alternating link is nontrivial. We also show that either the first two or last two coefficients of the Jones polynomial of an almost alternating link alternate in sign. Finally, we describe conditions that ensure an almost alternating diagram has the fewest number of crossings among all almost alternating diagrams of the link.Comment: 24 pages, 16 figures; minor changes throughou

Invariants for turaev genus one links

The Turaev genus defines a natural filtration on knots where Turaev genus zero knots are precisely the alternating knots. We show that the signature of a Turaev genus one knot is determined by the number of components in its all-A Kaufiman state, the number of positive crossings, and its determinant. We also show that either the leading or trailing coeficient of the Jones polynomial of a Turaev genus one link (or an almost alternating link) has absolute value one