Legendrian DGA Representations and the Colored Kauffman Polynomial

Abstract

For any Legendrian knot 𝛫 in standard contact ℝ³, we relate counts of ungraded (1-graded) representations of the Legendrian contact homology DG-algebra (A(𝛫), ∂) with the n-colored Kauffman polynomial. To do this, we introduce an ungraded n-colored ruling polynomial, R¹ₙ, 𝛫(q), as a linear combination of reduced ruling polynomials of positive permutation braids and show that (i) R¹ₙ, 𝛫(q) arises as a specialization 𝘍ₙ, 𝛫(a, q)∣ₐ⁻¹₌₀ of the n-colored Kauffman polynomial and (ii) when q is a power of two R¹ₙ, 𝛫(q) agrees with the total ungraded representation number, Rep₁(𝛫, 𝔽ⁿq), which is a normalized count of n-dimensional representations of (A(𝛫),∂) over the finite field 𝔽q. This complements results from [Leverson C., Rutherford D., Quantum Topol. 11 (2020), 55-118] concerning the colored HOMFLY-PT polynomial, m-graded representation numbers, and m-graded ruling polynomials with m≠1.