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 and alternating decompositions

We prove that the genus of the Turaev surface of a link diagram is determined by a graph whose vertices correspond to the boundary components of the maximal alternating regions of the link diagram. Furthermore, we use these graphs to classify link diagrams whose Turaev surface has genus one or two, and we prove that similar classification theorems exist for all genera.Comment: 28 pages, 24 figures. Significant changes to the proofs of Theorem 1.5 and Theorem 3.

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

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

Development of television tubes for the large space telescope

Princeton Observatory has been working for several years under NASA sponsorship to develop television type sensors to use in place of photographic film for space astronomy. The performance of an SEC-vidicon with a 25 mm x 25 mm active area, MgF2 window, and bi-alkali photocathode is discussed. Results from ground based use on the Coude spectrograph of the 200-inch Hale telescope are included. The intended use of this tube in an echelle spectrograph sounding rocket payload and on Stratoscope 2 for direct high resolution imagery is also discussed. The paper also discusses the large space telescope image sensor requirements and the development of a larger television tube for this mission