To each knot $K\subset S^3$ one can associated its knot Floer homology $\hat{HFK}(K)$, a finitely generated bigraded abelian group. In general, the nonzero ranks of these homology groups lie on a finite number of slope one lines with respect to the bigrading. The width of the homology is, in essence, the largest horizontal distance between two such lines. Also, for each diagram $D$ of $K$ there is an associated Turaev surface, and the Turaev genus is the minimum genus of all Turaev surfaces for $K$. We show that the width of knot Floer homology is bounded by Turaev genus plus one. Skein relations for genus of the Turaev surface and width of a complex that generates knot Floer homology are given.Comment: 15 pages, 15 figure

Purpose To add to the limited research on the Disadvantaged Status, a component in the American Medical College Application Service (AMCAS) primary application, the authors explored how applicants to a medical school between 2014 and 2016 determined whether they were disadvantaged and whether to apply as such. Method The authors used case study methodology to explore the experiences of students at a medical school in the Northeast. The authors derived data from transcripts of semistructured interviews with students and the students’ AMCAS applications. Transcripts and applications were analyzed using a constant comparative approach and considered in the context of social comparison and impression management theories. Results Overall, the 15 student participants (8 used the Disadvantaged Status) had difficulty determining whether they were disadvantaged and how applying as such would affect their prospects. Contributing factors included ambiguity around both the term disadvantaged and its use in the admissions process. Simply experiencing hardship during childhood was insufficient for most participants to deem themselves disadvantaged. Participants’ decision processes were confounded by the need to rely on social comparisons to determine whether they were disadvantaged and impression management to decide whether to apply as such. Conclusions The ambiguous nature of the Disadvantaged Status, comparisons with even more disadvantaged peers, and uncertainty about how shared information might affect admission decisions distorted participants’ understandings of identity within the context of the application. The authors believe that many applicants who have experienced significant hardships/barriers are not using the Disadvantaged Status.

In the first few homological gradings, there is an isomorphism between the Khovanov homology of a link and the categorification of the chromatic polynomial of a graph related to the link. In this article, we show that all torsion in the categorification of the chromatic polynomial is of order two, and hence all torsion in Khovanov homology in the gradings where the isomorphism is defined is of order two. We also prove that odd Khovanov homology is torsion-free in its first few homological gradings.

Abstract. An alternating distance is a link invariant that measures how far away a link is from alternating. We study several alternating distances and demonstrate that there exist families of links for which the difference between certain alternating distances is arbitrarily large. We also show that two alternating distances, the alternation number and the alternating genus, are not comparable.

Abstract. Khovanov homology is a bigraded Z-module that categorifies the Jones polynomial. The support of Khovanov homology lies on a finite number of slope two lines with respect to the bigrading. The Khovanov width is essentially the largest horizontal distance between two such lines. We show that it is possible to generate infinite families of links with the same Khovanov width from link diagrams satisfying certain conditions. Consequently, we compute the Khovanov width for all closed 3-braids.Mathematics Subject Classification (2010). 57M25, 57M27.

Abstract. In this paper we introduce a representation of knots and links called a cube diagram. We show that a property of a cube diagram is a link invariant if and only if the property is invariant under five cube diagram moves. A knot homology is constructed from cube diagrams and shown to be equivalent to knot Floer homology.

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 2factors that contain the the perfect matching as a subgraph. Consequently, we show that the polynomial detects even perfect matchings.

Abstract. 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.