2018/02/28 by Caitlin Leverson, Leverson, Caitlin, Dan Rutherford +1 · 1 citation
Mathematics · #53D42 #57M25 (Secondary) #57R17 (Primary) #FOS: Mathematics #Geometric Topology (math.GT) #Symplectic Geometry (math.SG) #math.GT #math.SG #msc:53D42 #msc:57M25 #msc:57R17
paper · pdf · doi:10.48550/arxiv.1802.10531
38 pages, 8 figures. Minor revisions. To appear in Quantum Topology
arxiv created 2019/10/08 · arxiv updated 2019/10/10
We establish relationships between two classes of invariants of Legendrian knots in ℝ3: Representation numbers of the Chekanov-Eliashberg DGA and satellite ruling polynomials. For positive permutation braids, β⊂ J1S1, we give a precise formula in terms of representation numbers for the m-graded ruling polynomial RmS(K,β)(z) of the satellite of K with β specialized at z=q1/2-q-1/2 with q a prime power, and we use this formula to prove that arbitrary m-graded satellite ruling polynomials, RmS(K,L), are determined by the Chekanov-Eliashberg DGA of K. Conversely, for m≠ 1, we introduce an n-colored m-graded ruling polynomial, Rmn,K(q), in strict analogy with the n-colored HOMFLY-PT polynomial, and show that the total n-dimensional m-graded representation number of K to \mathbbFqn, Repm(K,\mathbbFqn), is exactly equal to Rmn,K(q). In the case of 2-graded representations, we show that R2n,K=Rep2(K, \mathbbFqn) arises as a specialization of the n-colored HOMFLY-PT polynomial.