Abstract. For any Legendrian knot Λ in pR 3 , ker pdz´ydxqq, we show that the existence of an augmentation to any field of the Chekanov-Eliashberg differential graded algebra over Z rt, t´1s is equivalent to the existence of a ruling of the front diagram, generalizing results of Fuchs, Ishkhanov, and Sabloff. We also show that any even graded augmentation must send t to´1.
We establish relationships between two classes of invariants of Legendrian knots in R 3 : Representation numbers of the Chekanov-Eliashberg DGA and satellite ruling polynomials. For positive permutation braids, β ⊂ J 1 S 1 , we give a precise formula in terms of representation numbers for the m-graded ruling polynomial R m S(K,β) (z) of the satellite of K with β specialized at z = q 1/2 − q −1/2 with q a prime power, and we use this formula to prove that arbitrary m-graded satellite ruling polynomials, R m S(K,L) , are determined by the Chekanov-Eliashberg DGA of K. Conversely, for m = 1, we introduce an n-colored m-graded ruling polynomial, R m n,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 F n q , Rep m (K, F n q ), is exactly equal to R m n,K (q). In the case of 2-graded representations, we show that R 2 n,K = Rep 2 (K, F n q ) arises as a specialization of the n-colored HOMFLY-PT polynomial.
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