[Paper Review] Cyclotomic expansion and volume conjecture for superpolynomials of colored HOMFLY-PT homology and colored Kauffman homology

Qingtao Chen|arXiv (Cornell University)|Dec 24, 2015

Geometric and Algebraic Topology34 references3 citations

TL;DR

This paper proposes congruence relations and cyclotomic expansions for superpolynomials of colored HOMFLY-PT and Kauffman homologies, proving the N=1 case for torus knots and establishing a volume conjecture for SU(n)-specialized superpolynomials. It further links these invariants to smooth 4-ball genus via an invariant α, and confirms the conjecture for the figure-eight knot.

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 $α(T(m,n))=-(m-1)(n-1)/2$. 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 $α$ 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.

Motivation & Objective

• To establish congruence relations and cyclotomic expansions for superpolynomials of triply-graded reduced colored HOMFLY-PT homology.
• To investigate the relationship between the invariant α derived from cyclotomic expansion at N=1 and the smooth 4-ball genus.
• To extend the framework to superpolynomials of colored Kauffman homology and Heegaard-Floer knot homology.
• To propose and test a volume conjecture for SU(n)-specialized superpolynomials associated with reduced colored HOMFLY-PT homology.
• To verify the conjectured invariants and expansions across homologically thick knots and higher representations.

Proposed method

• Proposes a conjecture for congruence relations in superpolynomials of colored HOMFLY-PT homology, specifically for torus knots T(2,2p+1), with modular conditions involving (aq⁻¹ + t⁻¹a⁻¹q) and (t²aq^{N+k} + t⁻¹a⁻¹q^{-N-k}).
• Proves the N=1 case of the cyclotomic expansion conjecture for torus knots, deriving α(T(m,n)) = -(m−1)(n−1)/2.
• Applies the cyclotomic expansion framework to SU(n)-specialized superpolynomials, leading to a new volume conjecture for HOMFLY-PT homology.
• Tests the cyclotomic expansion conjecture on 41 homologically thick knots with 11 crossings, using known Poincaré polynomials.
• Adapts the same approach to Heegaard-Floer knot homology, deriving an expansion formula consistent with tested examples.
• Uses known closed-form expressions for superpolynomials of torus knots (from Fuji, Gukov, and Sulkowski) to verify conjectures through intensive computation.

Experimental results

Research questions

• RQ1 Does the superpolynomial of reduced colored HOMFLY-PT homology satisfy a cyclotomic expansion with congruence relations at N=1?
• RQ2 Can the invariant α derived from the N=1 cyclotomic expansion serve as a lower bound for the smooth 4-ball genus?
• RQ3 Is there a volume conjecture for SU(n)-specialized superpolynomials of reduced colored HOMFLY-PT homology?
• RQ4 Does the cyclotomic expansion framework extend to superpolynomials of colored Kauffman homology?
• RQ5 Can the same expansion method be applied to Heegaard-Floer knot homology, yielding consistent results?

Key findings

• The N=1 cyclotomic expansion is proven for torus knots T(2,2p+1), yielding α(T(m,n)) = -(m−1)(n−1)/2, which aligns with the Milnor conjecture.
• The invariant α derived from the N=1 cyclotomic expansion is conjectured to be a lower bound for the smooth 4-ball genus.
• The volume conjecture for SU(n)-specialized superpolynomials is verified for the figure-eight knot.
• The cyclotomic expansion conjecture is tested and holds for 41 homologically thick knots with 11 crossings.
• The framework successfully extends to Heegaard-Floer knot homology, producing consistent expansion formulas across all tested examples.
• The superpolynomial of colored Kauffman homology is shown to admit a cyclotomic expansion, supported by tested examples and higher representations.