On Knots, Complements, and 6j-Symbols

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