Seminar by Qingtao Chen (ETHZ, Zurich)

Title: Congruence relations, Cyclotomic Expansions and Volume Conjectures

2016.04.01 | Jane Jamshidi

Date Thu 07 Apr
Time 14:15 15:15
Location 1531-219 (Aud. D4)


In this talk, I will illustrate a new point of view that yields subtle relations for various quantum invariants of knots. I will first discuss congruence relations for colored Jones polynomials and their relationship with the cyclotomic expansion obtained by Habiro. A key observation is that the root of unity used in the Volume Conjecture of Kashaev-Murakami-Murakami naturally occurs in this context as the gap in the cyclotomic expansion of the colored Jones polynomials.

I will then apply the same ideas idea to colored SU(n) invariants of knots. In joint works with K. Liu, P. Peng and S. Zhu, we have discovered strong congruence relations for these invariants, which suggest the existence of certain cyclotomic expansions generalizing Habiro's as well as versions of the Volume Conjecture for these colored SU(n) invariants.

We then apply this viewpoint to the superpolynomial associated to reduced triply-graded HOMFLY-PT homology. This leads us to formulate a cyclotomic expansion for this superpolynomial, which involves a mysterious invariant of knots. This new invariant of knots has a close relationship to the smooth 4-ball genus and the Milnor Conjecture for torus knots. As before, the analysis of these gaps in the cyclotomic expansion enable us to formulate a Volume Conjecture in this context as well.