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Legendrian Dga Representations And The Colored Kauffman Polynomial, Justin Murray, Dan Rutherford

Faculty Publications

For any Legendrian knot K in standard contact R-3 we relate counts of ungraded (1-graded) representations of the Legendrian contact homology DG-algebra (A(K), partial derivative) with the n-colored Kauffman polynomial. To do this, we introduce an ungraded n-colored ru-ling polynomial, R-n,K(1)(q), as a linear combination of reduced ruling polynomials of positive permutation braids and show that (i) R-n,K(1)(q) arises as a specialization F-n,F-K(a, q)vertical bar(a-1) = 0 of the n-colored Kauffman polynomial and (ii) when q is a power of two R-n,K(1)(q) agrees with the total ungraded representation number, Rep(1) (K, F-q(n)), which is a normalized count of n-dimensional representations …