## 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.

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.

Original language | English |
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Pages (from-to) | 1-31 |

Number of pages | 31 |

Journal | arXiv e-prints |

Volume | math.QA/1512.07906 |

Publication status | Published - 27 Jan 2016 |