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

Abstract

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.