Topology Atlas | Conferences

The Gram determinant of type Mb
by
Dionne Kunkel
The George Washington University
Coauthors: Rhea Palak Bakshi, Sujoy Mukherjee and Józef H. Przytycki

In 1995, a general closed formula for the Gram determinant of Type A was discovered in order to prove the existence and uniqueness of Lickorish's construct of the Witten - Reshetikhin - Turaev invariants of 3 - manifolds. This determinant is of a matrix given by a bilinear form on crossingless connections in the disc with 2n boundary points. Thirtheen years later, a general closed formula for the Gram determinant of Type B was solved. In this case, the determinant is of a matrix given by a bilinear form on crossingless connections in the annulus with 2n boundary points. The idea to work in the Möbius band, was formulated in October 2008. By April 2009, Qi Chen conjectured a general closed formula for the Gram determinant of the Möbius band, that is,

D(Mb)n(d, x, y, z, w) = Πi=0n Dn, iΠj=1nOn, j( 2n || (n-j )).

Where we let D_{n, 0} = ∏_k=1ⁿ (T_k(d)²-z²)^{(2n || (n-k ))}, and for i > 0, we let  D_{n, i} = ∏_k=1+iⁿ (T_2k(d)-2)^{( 2n || (n-k ))}, O_{n, 2i} = T_2i(w) - [(2(2-z))/(T_2i(d) - z)], and O_{n, 2i+1} = T_2i+1(w)- [2xy/(T_2i+1(d) +z)].

Where (a || b) means a choose b and the i represents the number of curves passing through the cross cap.

In this talk, we will discuss the bilinear form on crossingless connections in the Möbius band with 2n boundary points then give insight to our progress in proving Qi Chen's conjecture.

Date received: May 7, 2019

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