  We first study superpolynomial associated to triply-graded reduced colored
HOMFLY-PT homology. We propose conjectures of congruent relations and
cyclotomic expansion for it. We prove conjecture of $N=1$ for torus knot case,
through which we obtain the corresponding invariant
$\alpha(T(m,n))=-(m-1)(n-1)/2$. This is closely related to the Milnor
conjecture. Many examples including homologically thick knots and higher
representations are also tested. Based on these examples, we further propose a
conjecture that invariant $\alpha$ determined in cyclotomic expansion at $N=1$
is a lower bound for smooth 4-ball genus. According to the structure of
cyclotomic expansion, we propose a volume conjecture for $SU(n)$ specialized
superpolynomial associated to reduced colored HOMFLY homology. We also prove
the figure eight case for this new volume conjecture.
  Then we study superpolynomial associated to triply-graded reduced colored
Kauffman homology. We propose a conjecture of cyclotomic expansion for it.
Homologically thick examples and higher representations are tested. Finally we
apply the same idea to the Heegaard-Floer knot homology and also obtain an
expansion formula for all the examples we tested.
