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We first study superpolynomial associated to triply-graded reduced coloredHOMFLY-PT homology. We propose conjectures of congruent relations andcyclotomic 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 Milnorconjecture. Many examples including homologically thick knots and higherrepresentations are also tested. Based on these examples, we further propose aconjecture that invariant $\alpha$ determined in cyclotomic expansion at $N=1$is a lower bound for smooth 4-ball genus. According to the structure ofcyclotomic expansion, we propose a volume conjecture for $SU(n)$ specializedsuperpolynomial associated to reduced colored HOMFLY homology. We also provethe figure eight case for this new volume conjecture.  Then we study superpolynomial associated to triply-graded reduced coloredKauffman homology. We propose a conjecture of cyclotomic expansion for it.Homologically thick examples and higher representations are tested. Finally weapply the same idea to the Heegaard-Floer knot homology and also obtain anexpansion formula for all the examples we tested.

Cyclotomic expansion and volume conjecture for superpolynomials of colored HOMFLY-PT homology and colored Kauffman homology