Symmetric union presentations for 2-bridge ribbon knots

Symmetric unions have been defined as generalizations of Kinoshita-Terasaka's construction in 1957. They are given by diagrams which look like the connected sum of a knot and its mirror image with additional twist tangles inserted near the symmetry axis. Because all symmetric unions are ribbon knots, we can ask how big a subfamily of ribbon knots they form. It is known that all 21 ribbon knots with crossing number less or equal 10 are symmetric unions. In this talk we extend our knowledge about symmetric unions: we prove that the family of symmetric unions contains all known 2-bridge ribbon knots. The question, however, whether the three families of 2-bridge ribbon knots, found by Casson and Gordon in 1974, are a complete list of all 2-bridge ribbon knots, is still open.Comment: 13 pages (notes for a talk at the Joint Meeting of AMS and DMV at Mainz, 2005-06-18

For which triangles is Pick's formula almost correct?

We present an intriguing question about lattice points in triangles where Pick's formula is "almost correct". The question has its origin in knot theory, but its statement is purely combinatorial. After more than 30 years the topological question was recently solved, but the lattice point problem is still open.Comment: 6 pages; v2 more background information; v3 update of recent developments; v4 final revision and reformattin

Equivalence of symmetric union diagrams

Motivated by the study of ribbon knots we explore symmetric unions, a beautiful construction introduced by Kinoshita and Terasaka 50 years ago. It is easy to see that every symmetric union represents a ribbon knot, but the converse is still an open problem. Besides existence it is natural to consider the question of uniqueness. In order to attack this question we extend the usual Reidemeister moves to a family of moves respecting the symmetry, and consider the symmetric equivalence thus generated. This notion being in place, we discuss several situations in which a knot can have essentially distinct symmetric union representations. We exhibit an infinite family of ribbon two-bridge knots each of which allows two different symmetric union representations.Comment: 19 pages, 20 figures; v2 corrected signs in section

Considerations for Mitigating VehicleMiles Traveled under SB 743

Pursuant to Senate Bill 743 (Steinberg, 2013), which reformed the process for California Environmental Quality Act (CEQA) review of transportation impacts to align with greenhouse gas emissions reduction goals, the Governor’s Office of Planning and Research identified vehicle miles traveled (VMT) as the key metric to measure transportation impacts of new developments under CEQA.As a result, project developers will now have to reduce VMT to mitigate significant transportation impacts. While methods for reducing VMT impacts are well understood, implementing VMT reduction measures thatare directly linked or near to individual developments may be difficult in some situations. As a result, broader and more flexible approaches to VMT mitigation may be necessary, such as VMT mitigation “banks” or “exchanges.” In a mitigation bank, developers would commit funds instead of undertaking specific on-site mitigation projects, and then a local or regional authority could aggregate funds and deploy them to top-priority projects throughout the jurisdiction. Similarly, in amitigation exchange, developers would be permitted to select from a list of pre-approved mitigation projects throughout the jurisdiction (or propose their own), without needing to mitigate their transportation impacts on-site.To understand how VMT banks or exchanges could be implemented in California, researchers from UC Berkeley assessed the structural and legal considerations of VMT banks and exchanges to determine which approach and scope would be most appropriate for each implementing jurisdiction (i.e., city, county, region, state). Key research findings are presented in this brief