[Paper Review] Knot Cabling and the Degree of the Colored Jones Polynomial

Efstratia Kalfagianni, Anh T. Tran|arXiv (Cornell University)|Jan 7, 2015

Geometric and Algebraic Topology24 references3 citations

TL;DR

This paper establishes conditions under which the Slope Conjecture holds for $(p,q)$-cables of knots, showing that if a knot $K$ satisfies the conjecture and $p/q$ is not a Jones slope of $K$, then its cable $K_{p,q}$ also satisfies the conjecture, provided the degree of the colored Jones polynomial has bounded periodicity. The authors further prove the conjecture for iterated cables of adequate knots and iterated torus knots, and introduce a topological interpretation of the linear term in the degree of the colored Jones polynomial.

ABSTRACT

We study the behavior of the degree of the colored Jones polynomial and the boundary slopes of knots under the operation of cabling. We show that, under certain hypothesis on this degree, if a knot $K$ satisfies the Slope Conjecture then a $(p, q)$-cable of $K$ satisfies the conjecture, provided that $p/q$ is not a Jones slope of $K$. As an application we prove the Slope Conjecture for iterated cables of adequate knots and for iterated torus knots. Furthermore we show that, for these knots, the degree of the colored Jones polynomial also determines the topology of a surface that satisfies the Slope Conjecture. We also state a conjecture suggesting a topological interpretation of the linear terms of the degree of the colored Jones polynomial (Conjecture ef{conj}), and we prove it for the following classes of knots:iterated torus knots and iterated cables of adequate knots, iterated cables of several non-alternating knots with up to nine crossings, pretzel knots of type $(-2, 3, p)$ and their cables, and two-fusion knots.

Motivation & Objective

• To investigate the behavior of the degree of the colored Jones polynomial and boundary slopes under the cabling operation.
• To verify the Slope Conjecture for iterated cables of adequate knots and iterated torus knots.
• To formulate and verify a conjecture linking the linear term in the degree of the colored Jones polynomial to topological invariants of the knot complement.
• To show that for certain classes of knots, the degree of the colored Jones polynomial determines the topology of a surface realizing the Slope Conjecture.

Proposed method

• Analyzes the asymptotic behavior of the colored Jones polynomial $J_K(n)$, focusing on the highest and lowest degrees $d_+(J_K(n))$ and $d_-(J_K(n))$.
• Uses the fact that $d_+(J_K(n))$ and $d_-(J_K(n))$ are quadratic quasi-polynomials with period $\leq 2$, and studies their coefficients under cabling.
• Applies the cabling formula for boundary slopes: $q^2 \cdot bs_K \cup \{pq\} \subset bs_{K_{p,q}}$.
• Derives the transformation of Jones slopes under cabling: $js_{K_{p,q}} \subset q^2 \cdot js_K \cup \{pq/4\}$, under the condition $p/q \notin js_K$.
• Introduces Conjecture 5.1, proposing that the linear coefficient $b(n)$ in $d_+(J_K(n))$ detects essential annuli in the knot complement.
• Employs explicit computations for 2-fusion knots and pretzel knots to verify Conjecture 5.1 and the Slope Conjecture in specific families.

Experimental results

Research questions

• RQ1 Under what conditions does the Slope Conjecture hold for a $(p,q)$-cable of a knot $K$?
• RQ2 How do the degrees of the colored Jones polynomial transform under cabling, particularly the leading and linear terms?
• RQ3 Can the linear coefficient in the degree of the colored Jones polynomial be given a topological interpretation in terms of essential surfaces?
• RQ4 Do iterated cables of adequate knots and iterated torus knots satisfy the Slope Conjecture?
• RQ5 For which classes of knots is the linear term $b(n)$ in $d_+(J_K(n))$ non-positive and $b^*(n)$ non-negative, and what does this imply?

Key findings

• If $K$ satisfies the Slope Conjecture and $p/q$ is not a Jones slope of $K$, then the $(p,q)$-cable $K_{p,q}$ satisfies the Slope Conjecture provided the degree of $J_K(n)$ has period $\leq 2$.
• The Jones slopes of $K_{p,q}$ are contained in $q^2 \cdot js_K \cup \{pq/4\}$, and the boundary slopes of $K_{p,q}$ contain $q^2 \cdot bs_K \cup \{pq\}$.
• For 2-fusion knots, the linear coefficient $b_K(n)$ in $d_+(J_K(n))$ is non-positive, and $b_K(n) = 0$ if and only if the knot is a torus knot.
• The Slope Conjecture holds for iterated cables of adequate knots and for iterated torus knots.
• Conjecture 5.1 is verified for iterated torus knots, iterated cables of adequate knots, several non-alternating knots with up to nine crossings, pretzel knots $(-2,3,p)$ and their cables, and 2-fusion knots.
• For the $(-2,3,2m+3)$-pretzel knot $K(m,1)$, the conditions for the Slope Conjecture to hold on its cables are explicitly given in terms of $p$ and $q$.