[Paper Review] Denominators and Differences of Boundary Slopes for (1,1)-Knots

Jason Callahan|arXiv (Cornell University)|Jan 26, 2013

Geometric and Algebraic Topology13 references4 citations

TL;DR

This paper demonstrates that every nonzero integer appears in the denominator of a boundary slope for infinitely many (1,1)-knots, using Montesinos knots of the form $K(-1/2, m/(2m\pm1), 1/n)$ with odd $n > 1$. It further proves that for such knots, boundary slopes can have arbitrarily small differences by increasing $n$, leveraging essential surfaces with controlled Euler characteristic and boundary components via the algorithm of Hatcher and Oertel.

ABSTRACT

We show that every nonzero integer occurs in the denominator of a boundary slope for infinitely many (1,1)-knots and that infinitely many (1,1)-knots have boundary slopes of arbitrarily small difference. Specifically, we prove that for any integers m, n > 1 with n odd the exterior of the Montesinos knot K(-1/2, m/(2m \pm 1), 1/n) in S^3 contains an essential surface with boundary slope r = 2(n-1)^2/n if m is even and 2(n+1)^2/n if m is odd. If n > 4m, we prove that K(-1/2, m/(2m+1), 1/n) also has a boundary slope whose difference with r is (8m-2)/(n^2-4mn+n), which decreases to 0 as n increases. All of these knots are (1,1)-knots.

Motivation & Objective

• To show that every nonzero integer occurs as a denominator in the boundary slope of some essential surface for infinitely many (1,1)-knots.
• To demonstrate that infinitely many (1,1)-knots admit boundary slopes with arbitrarily small differences.
• To extend and correct prior results on boundary slopes for Montesinos knots, particularly addressing an error in [8] regarding surface orientability and boundary component count.
• To verify that the constructed knots are (1,1)-knots using the characterization from [10].
• To analyze the sharpness of denominator bounds for boundary slopes in Montesinos knot exteriors as established in [9].

Proposed method

• Applies the Hatcher-Oertel algorithm for computing boundary slopes via admissible edgepath systems in the $uv$-plane graph $\mathcal{D}$, which encodes curve systems on the 4-punctured sphere.
• Constructs essential surfaces with specified number of sheets, Euler characteristic, and boundary components by analyzing edgepath systems corresponding to rational tangles.
• Uses the boundary slope formula derived from the edgepath system: $r = 2(n-1)^2/n$ for even $m$, and $2(n+1)^2/n$ for odd $m$, with a second slope given by rational functions in $n$ and $m$.
• Proves that the difference between two boundary slopes is $ (8m-2)/(n^2 - 4mn + n) $, which tends to zero as $n$ increases.
• Verifies that the knots $K(-1/2, m/(2m\pm1), 1/n)$ are (1,1)-knots using Theorem 2.1 from [10], which characterizes (1,1)-knots among Montesinos knots.
• Corrects an error in [8] by showing that the essential surface has two boundary components (hence can be orientable), not one, and thus the surface is not necessarily non-orientable.

Experimental results

Research questions

• RQ1 Can every nonzero integer appear as a denominator in the boundary slope of an essential surface for infinitely many (1,1)-knots?
• RQ2 Do there exist (1,1)-knots with boundary slopes whose difference can be made arbitrarily small?
• RQ3 Is the boundary slope formula for Montesinos knots $K(-1/2, m/(2m+1), 1/n)$ valid and correctly interpreted when $n \geq 4m+1$?
• RQ4 Does the essential surface constructed in the proof have one or two boundary components, and what does this imply about orientability?
• RQ5 How do the boundary slope denominators relate to the bounds in Theorem 4.1 from [9], and which knots achieve the optimal bounds?

Key findings

• For any integers $m,n > 1$ with $n$ odd, the Montesinos knot $K(-1/2, m/(2m\pm1), 1/n)$ admits an essential surface with boundary slope $2(n-1)^2/n$ if $m$ is even and $2(n+1)^2/n$ if $m$ is odd.
• When $n \geq 4m+1$, the knot $K(-1/2, m/(2m+1), 1/n)$ has a second boundary slope, and the difference between the two slopes is $ (8m-2)/(n^2 - 4mn + n) $, which decreases to 0 as $n$ increases.
• For any $m > 1$ and $\epsilon > 0$, there exists $N$ such that for all $n \geq N$, the boundary slope difference is less than $\epsilon$, proving arbitrarily small differences exist.
• The denominator of the boundary slope in lowest terms is $ (n-4m+1)/2 $, which is an integer, confirming the surface has two boundary components and may be orientable.
• The construction generalizes and corrects the result in [8], which incorrectly claimed a single boundary component and non-orientability; the surface is actually orientable with two components.
• The paper confirms that for $m=1$, the knot $K(-1/2,1/3,1/n)$ achieves the second bound in Theorem 4.1 from [9] (for $(-2,3,n)$-pretzel knots), while for $m>1$, the knots achieve the first bound, and for $m>2$, they achieve neither bound but still realize the denominator $ (n-4m+1)/2 $.