Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On knots, complements, and 6j-symbols   This paper investigates the relation between colored HOMFLY-PT and Kauffman
homology, $\text{SO}(N)$ quantum $6j$-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 $(\mathfrak{so}_6,[r]) \cong
(\mathfrak{sl}_4,[r^2])$. 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 $\text{SO}(N)(N\geq 4)$ quantum
$6j$-symbols for symmetric representations, and calculate the corresponding
$\text{SO}(N)$ fusion matrices for the cases when representations $R =
[1],[2]$. 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.

9
The categories of this paper are: nlin
27
The categories of this paper are: comp-gas
28
The categories of this paper are: alg-geom
20
The categories of this paper are: acc-phys
3
The categories of this paper are: physics
26
The categories of this paper are: bayes-an
12
The categories of this paper are: nucl-th
30
The categories of this paper are: q-alg
15
The categories of this paper are: nucl-ex
33
The categories of this paper are: plasm-ph
11
The categories of this paper are: hep-lat
21
The categories of this paper are: adap-org
35
The categories of this paper are: mtrl-th
5
The categories of this paper are: gr-qc
36
The categories of this paper are: cmp-lg
10
The categories of this paper are: q-bio