Fixing the functoriality of Khovanov homology

We describe a modification of Khovanov homology (math.QA/9908171), in the spirit of Bar-Natan (math.GT/0410495), which makes the theory properly functorial with respect to link cobordisms. This requires introducing `disorientations' in the category of smoothings and abstract cobordisms between them used in Bar-Natan's definition. Disorientations have `seams' separating oppositely oriented regions, coming with a preferred normal direction. The seams satisfy certain relations (just as the underlying cobordisms satisfy relations such as the neck cutting relation). We construct explicit chain maps for the various Reidemeister moves, then prove that the compositions of chain maps associated to each side of each of Carter and Saito's movie moves (MR1238875, MR1445361) always agree. These calculations are greatly simplified by following arguments due to Bar-Natan and Khovanov, which ensure that the two compositions must agree, up to a sign. We set up this argument in our context by proving a result about duality in Khovanov homology, generalising previous results about mirror images of knots to a `local' result about tangles. Along the way, we reproduce Jacobsson's sign table (math.GT/0206303) for the original `unoriented theory', with a few disagreements.Comment: 91 pages. Added David Clark as co-author. Further detail on variations of third Reidemeister moves, to allow treatment of previously missing cases of movie move six. See changelog section for more detai

Spinning knots about submanifolds; spinning knots about projections of knots

AbstractA method is defined and discussed for constructing higher dimensional codimension two knots. These methods generalize the various known methods of spinning of knots. Although the focus is on knotted sphere pairs, the methods more generally provide a method of producing a variety of knottings from any given codimension two pair of manifolds

When Plans Go Unpaid: A Look at the Defalcation Exception as Applied to Pension-Plan Sponsors and Underfunded Plans

Article published in the Michigan State University School of Law Student Scholarship Collection

Ukraine\u27s Developing Mortgage Market

Gary Roseman, Ph.D., is assistant professor of economics, Department of Economics, Campbell School of Business, Berry College, Mount Berry, GA 30149-5024

Order Behavior In High Frequency Markets

In Part 1, I study the characteristics of short orders in stock markets. Fleeting orders are quick limit orders that remain on the limit order book for only a few seconds before being canceled, and are significantly different than more patient, static, limit orders that are added to the limit order book and await execution. I investigate the impact that fleeting orders have on spread and depth measures of market quality, and how fleeting orders differ from static orders. Attention is also given to the extent that total depth can be decomposed into the two components of fleeting and static depth. The results suggest that static orders have a positive impact on both spread and depth. However, fleeting orders have little impact on total liquidity. The results suggest that fleeting orders contribute noise to markets, and do not positively impact the spread and depth components of liquidity. This result is robust to the simultaneous issue that order submission strategies depend on current market quality conditions. In Part 2, I investigate the link between orders and trades in equity markets. A substantial body of research on limit order markets investigates the characteristics of orders and the characteristics of trades. However, there has been little research on how the characteristics of orders impact the characteristics of trades. I investigate the impact that marketable orders and limit orders have on the resulting trade characteristics. In addition, we test theoretical predictions on how market characteristics, like time of day and depth, impact order and trade characteristics. Lastly, in Part 3, I investigate the causes, and effects of intraday flash crashes. Breakdowns in financial markets occur when the market is not able to facilitate its principal responsibilities of liquidity provision and price discovery. In this paper we look at flash crashes, a special type of market breakdown. These crashes are generally non-fundamental in nature, and the market making responsibilities of liquidity and price discovery are only temporarily suspended for a short period before rebounding to pre-crash levels. This paper analyzes intraday flash crashes, primarily focusing on three aspects of flash crashes: crash frequency, crash triggers, and the impact on market quality once the crash has seceded