2019/08/31 by Justin Murray, Dan Rutherford
Mathematics · #math.SG #math.GT #math.QA #msc:53D42 #msc:57M27
paper · pdf · doi:10.3842/sigma.2020.017
published as SIGMA 16 (2020), 017, 33 pages
arxiv created 2020/03/22 · arxiv updated 2020/03/24
For any Legendrian knot K in standard contact \mathbb R3 we relate counts of ungraded (1-graded) representations of the Legendrian contact homology DG-algebra (A(K),∂) with the n-colored Kauffman polynomial. To do this, we introduce an ungraded n-colored ruling polynomial, R1n,K(q), as a linear combination of reduced ruling polynomials of positive permutation braids and show that (i) R1n,K(q) arises as a specialization Fn,K(a,q)|a-1=0 of the n-colored Kauffman polynomial and (ii) when q is a power of two R1n,K(q) agrees with the total ungraded representation number, Rep1(K, \mathbbFqn), which is a normalized count of n-dimensional representations of (A(K),∂) over the finite field \mathbbFq. This complements results from [Leverson C., Rutherford D., Quantum Topol. 11 (2020), 55-118, arXiv:1802.10531] concerning the colored HOMFLY-PT polynomial, m-graded representation numbers, and m-graded ruling polynomials with m ≠ 1.