Geodesic
Legendrian DGA Representations and the Colored Kauffman Polynomial
Symmetry, integrability and geometry: methods and applications, Tome 16 (2020) Cet article a éte moissonné depuis la source Math-Net.Ru

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For any Legendrian knot $K$ in standard contact $\mathbb{R}^3$ we relate counts of ungraded ($1$-graded) representations of the Legendrian contact homology DG-algebra $(\mathcal{A}(K),\partial)$ with the $n$-colored Kauffman polynomial. To do this, we introduce an ungraded $n$-colored ruling polynomial, $R^1_{n,K}(q)$, as a linear combination of reduced ruling polynomials of positive permutation braids and show that (i) $R^1_{n,K}(q)$ arises as a specialization $F_{n,K}(a,q)\big|_{a^{-1}=0}$ of the $n$-colored Kauffman polynomial and (ii) when $q$ is a power of two $R^1_{n,K}(q)$ agrees with the total ungraded representation number, $\operatorname{Rep}_1\big(K, \mathbb{F}_q^n\big)$, which is a normalized count of $n$-dimensional representations of $(\mathcal{A}(K),\partial)$ over the finite field $\mathbb{F}_q$. 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 $m \neq 1$.
Keywords: Legendrian knots, Kauffman polynomial, ruling polynomial
Mots-clés : augmentations. 
@article{SIGMA_2020_16_a16,
     author = {Justin Murray and Dan Rutherford},
     title = {Legendrian {DGA} {Representations} and the {Colored} {Kauffman} {Polynomial}},
     journal = {Symmetry, integrability and geometry: methods and applications},
     year = {2020},
     volume = {16},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2020_16_a16/}
}
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Justin Murray; Dan Rutherford. Legendrian DGA Representations and the Colored Kauffman Polynomial. Symmetry, integrability and geometry: methods and applications, Tome 16 (2020). http://geodesic.mathdoc.fr/item/SIGMA_2020_16_a16/

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