Mathematics Seminar: "On the AJ-conjecture, Part 2"
- Monday, September 17, 2018 at 4:00pm
- Wilson Hall, 1-144 - view map
Abstract: Over the past 35 years, knot theory has witnessed an explosion of powerful new invariants such as the colored Jones polynomial and the HOMFLY-PT polynomial. The origin of many of these invariants can be unified using quantum groups. One of the fundamental problems in modern knot theory is to relate these quantum knot invariants to the topology of a knot complement and the AJ-conjecture gives one possible connection. In this two part series of lectures, we will endeavor to understand the statement of the AJ-conjecture, its current states of affairs, and attempt to pinpoint the main difficulties in its resolution.
In part 2 of this series, we will define the A-polynomial of a knot using the SL(2, C) character variety machinery developed last time. We will briefly discuss the famous "boundary slopes are boundary slopes" theorem, then move on to the 'J' in the AJ-conjecture, which refers to the colored Jones polynomial of a knot. Next we'll turn to the recurrence ideal of a knot and frame our discussion in terms of skein modules. This perspective illuminates the plausibility of the AJ-conjecture. The lecture will finish with the statement of a theorem of Le and Zhang which highlights the primary difficulty in resolving the conjecture.
These lectures will be expository in nature and should be well suited to introduce graduate students to the topics described above.
- Department of Mathematical Sciences