A new arcade-based planarization technique plus the Kuperberg bracket yields a closed system of classical relations toward su3 A-polynomials, demonstrated on the trefoil.
Skein theory for SU(n)-quantum invariants
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abstract
For any n>1 we define an isotopy invariant, <Gamma>_n, for a certain set of n-valent ribbon graphs Gamma in R^3, including all framed oriented links. We show that our bracket coincides with the Kauffman bracket for n=2 and with the Kuperberg's bracket for n=3. Furthermore, we prove that for any n, our bracket of a link L is equal, up to normalization, to the SU_n-quantum invariant of L. We show a number of properties of our bracket extending those of the Kauffman's and Kuperberg's brackets, and we relate it to the bracket of Murakami-Ohtsuki-Yamada. Finally, on the basis of the skein relations satisfied by <.>_n, we define the SU_n-skein module of any 3-manifold M and we prove that it determines the SL_n-character variety of pi_1(M).
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On geometric bases for A-polynomials II: $\mathfrak{su}_3$ and Kuberberg bracket
A new arcade-based planarization technique plus the Kuperberg bracket yields a closed system of classical relations toward su3 A-polynomials, demonstrated on the trefoil.