On Knots, Complements, and 6j-Symbols

This paper investigates the relation between colored HOMFLY-PT and Kauffman homology, $${{\,\mathrm{SO}\,}}(N)$$ quantum 6j-symbols, (a, t)-deformed $$F_K$$ . First, we present a simple rule of grading change which allows us to obtain [r]-colored quadruply graded homology from $$[r^2]$$ -colored for thin knots. stems isomorphism representations $$(\mathfrak {so}_6,[r]) \cong (\mathfrak {sl}_4,[r^2])$$ Also, find relationship among A-polynomials $${{\,\mathrm{SO}\,}}$$ $${{\,\mathrm{SU}\,}}$$ type coming differential on homology. Second, put forward closed-form expression $${{\,\mathrm{SO}\,}}(N)(N\ge 4)$$ 6j-symbols symmetric calculate corresponding fusion matrices cases when Third, conjecture expressions complements double twist knots with positive braids. Using conjectural expressions, derive t-deformed ADO polynomials.