Characterising slopes for hyperbolic knots and Whitehead doubles

This paper establishes explicit conditions under which Dehn surgery slopes are characterising for hyperbolic and certain satellite knots by combining JSJ decomposition analysis with geodesic length techniques, while also refining these results for Whitehead doubles using hyperbolic volume inequalities and constructing counterexamples for specific multiclasped cases.

The Gist

Imagine you have a magical knot in a 3D space. You can perform a specific operation on this knot called Dehn surgery. Think of this like taking a piece of the knot out and gluing a solid tube back in, but you can twist the tube in different ways before gluing it. Each way of twisting is called a slope (represented by a fraction like $p/q$).

The big question this paper asks is: If I give you the resulting shape after the surgery, can you tell me exactly which knot I started with?

If the answer is "Yes, that shape only comes from this specific knot," then that slope is called characterising. If the answer is "No, that shape could have come from two different knots," then the slope is non-characterising.

Here is a breakdown of Laura Wakelin's research using simple analogies:

1. The Detective Work: Identifying the Knot

Think of a knot complement (the space around the knot) as a unique fingerprint. When you do surgery, you are essentially modifying that fingerprint.

• The Goal: The author wants to find a "safe zone" of slopes. If you pick a slope with a large enough denominator (a specific type of twist), you are guaranteed that the resulting shape uniquely identifies the original knot.
• The Problem: For some knots, certain twists create shapes that look identical to shapes made by other knots. It's like two different people wearing the same coat; if you only see the coat, you can't tell who is underneath.

2. The Tools: Measuring the "Shortest Path"

To solve this, the author uses two main detective tools:

A. The "Shortest Path" Ruler (Geodesics)
Imagine the space around a knot is a complex, bumpy landscape. A geodesic is the shortest possible path you can walk through that landscape.

• The Analogy: Think of the knot complement as a room filled with furniture. The "shortest path" is the tightest squeeze you can get through.
• The Discovery: The author proves that if you twist the tube enough (a large denominator), the "shortest path" in the new shape becomes the core of the tube you just glued in. Because this core path is unique to the specific knot's geometry, it acts like a beacon. If two shapes have the same shortest path, they must have come from the same knot.
• The Result: For hyperbolic knots (knots with a specific, complex geometry), if the twist is strong enough, the knot is uniquely identified.

B. The "Volume" Scale (For Whitehead Doubles)
Some knots are special "satellite" knots, made by wrapping a pattern around a companion knot. A famous example is the Whitehead double.

• The Analogy: Imagine the space around a knot has a specific "volume" (like the amount of air in a balloon).
• The Discovery: The author uses a computer database (SnapPy) that lists all known 2-holed balloon shapes (2-cusped manifolds) sorted by size. She knows that the Whitehead link creates the smallest possible volume for this type of shape.
• The Logic: If you perform a surgery and the resulting shape has a volume that is too small to be anything else except the Whitehead link, then you know exactly what pattern you started with. By calculating how much volume surgery can "shrink" a shape, she sets a limit: if the twist is strong enough, the volume will be so small that no other pattern could possibly fit.

3. The "Gotcha": When the Trick Works

The paper also constructs a specific scenario where the "fingerprint" fails.

• The Analogy: Imagine two different pairs of shoes (Knot A and Knot B). Usually, they leave different footprints. But the author found a specific way to twist them (slope $1/q$) where they leave the exact same footprint.
• The Construction: She takes two different "double twist" knots and wraps them in Whitehead patterns. By twisting them in a specific way, the resulting 3D shapes become identical, even though the original knots were different.
• Why it matters: This proves that not every slope is characterising. It shows the limits of the method and explains why the author had to be careful to exclude certain slopes (like $1/q$) in her main rules.

Summary of the Main Findings

1. For most complex knots: If you twist the surgery tube enough (specifically, if the denominator of the slope is large enough), you can always tell the knot apart from any other. The author provides a formula to calculate exactly how "large" the twist needs to be based on the knot's geometry.
2. For Whitehead Doubles: She found an even better rule using volume. If the computer census of 3D shapes is complete up to a certain size, she can guarantee that specific twists will uniquely identify these knots.
3. The Exception: She built a "counter-example" showing that for a specific type of twist ($1/q$), two different knots can produce the exact same result. This is like finding two different keys that open the same lock.

In a nutshell: This paper is a guide for mathematicians on how to "lock" a knot so that no other knot can mimic its shape after surgery. It uses geometry (shortest paths) and volume (size of the shape) to set the rules, while also showing exactly where those rules break down.

Technical Summary

1. Problem Statement

The paper addresses the Dehn surgery characterisation problem in knot theory. Specifically, it investigates whether a specific Dehn surgery slope $p/q$ on a knot $K$ uniquely determines the knot.

• Definition: A slope $p/q \in \mathbb{Q}$ is characterising for a knot $K$ if, for any other knot $K'$, an orientation-preserving homeomorphism $S^3_K(p/q) \cong S^3_{K'}(p/q)$ implies $K = K'$.
• Context: While it is known that every knot has infinitely many characterising slopes (Lackenby, 2019), the existing proofs for the existence of a bound $C(K)$ (such that all slopes with $|q| \ge C(K)$ are characterising) are non-constructive for general knots.
• Goal: The author aims to provide explicit, computable values for $C(K)$ for two specific classes of knots:
  1. Hyperbolic knots.
  2. Satellite knots constructed using a hyperbolic pattern (specifically focusing on Whitehead doubles).
• Secondary Goal: To construct explicit counterexamples (pairs of distinct knots sharing a surgery) to demonstrate the necessity of conditions on the numerator $p$ of the slope.

2. Methodology

The paper employs a multi-faceted approach combining geometric topology, hyperbolic geometry, and computational census data.

A. JSJ Decomposition Analysis

The author uses the Jaco-Shalen-Johannson (JSJ) decomposition to analyze the structure of the manifolds resulting from Dehn surgery.

• Reduction: The problem of comparing $S^3_K(p/q)$ and $S^3_{K'}(p/q)$ is reduced to comparing the "outermost" JSJ pieces containing the surgery curve.
• Creation/Destruction of Tori: The paper analyzes how Dehn filling affects the JSJ tori. It establishes that for sufficiently large $|q|$, the JSJ decomposition of the surgered manifold is determined by the JSJ decomposition of the original knot complement, with at most one torus disappearing (specifically in the case of cable spaces).
• Pattern Matching: For satellite knots $K = P(J)$, the author proves that if $S^3_K(p/q) \cong S^3_{K'}(p/q)$, then $K'$ must be a satellite of the same companion $J$ by a pattern $P'$, and there must be a slope-preserving homeomorphism between the filled pattern pieces $S^3_P(p/q) \cong S^3_{P'}(p/q)$.

B. Geometric Length Arguments (Geodesics)

To prove that the patterns $P$ and $P'$ are identical, the paper utilizes the geometry of hyperbolic 3-manifolds.

• Shortest Geodesics: The core strategy relies on identifying the shortest geodesic in the surgered manifold.
• The 6-Theorem and Quantitative Bounds: Using results from Agol, Lackenby, and Futer-Purcell-Schleimer (FPS), the author derives explicit bounds. If the denominator $|q|$ is large enough, the core curve of the Dehn filling becomes the unique shortest geodesic in the resulting manifold.
• Rigidity: By Mostow Rigidity, a homeomorphism between hyperbolic manifolds is homotopic to an isometry, which must preserve the length and identity of the shortest geodesic. If the shortest geodesic in both manifolds corresponds to the core of the filling, the isometry must map the core of one to the core of the other, implying the underlying link complements (and thus the knots) are identical.
• Explicit Formula: The author defines a quantity $q(Y)$ based on the systole (shortest geodesic length) of the knot complement $Y$:
  $$q(Y) := \left\lceil \sqrt{\frac{6}{\sqrt{3}}} \left( \frac{1.9793}{2\pi \cdot \text{sys}(Y)} + 28.78 \right) \right\rceil$$
  This allows for the calculation of a concrete $C(K)$.

C. Hyperbolic Volume Inequalities (Whitehead Doubles)

For the special case of Whitehead doubles (where the pattern is the Whitehead link), the author employs an alternative strategy based on hyperbolic volume.

• Volume Bounds: The Whitehead link complement has the smallest volume among all 2-cusped orientable hyperbolic 3-manifolds (jointly with its sister).
• Filling Inequalities: Using theorems on how Dehn filling decreases volume (Theorem 5.4), the author establishes an upper bound $V_{max}$ on the volume of any potential pattern complement $S^3_{P'}$ that could yield a homeomorphic surgery.
• Census Verification: By assuming the completeness of the SnapPy census of 2-cusped hyperbolic manifolds up to a certain volume, the author systematically eliminates all other manifolds in the census that do not match the topological constraints (specifically, having winding number 0). This yields a tighter bound for Whitehead doubles than the general geodesic method.

D. Counterexample Construction

To demonstrate that conditions on the numerator $p$ are necessary, the author constructs pairs of distinct knots $K$ and $K'$ that share a surgery of slope $1/q$.

• Mechanism: The construction involves "switching" the roles of the companion and the pattern in a satellite knot construction. Specifically, $K = W^n(T^m_q)$ and $K' = W^m(T^n_q)$ (Whitehead doubles of double twist knots).
• Gluing Maps: The proof relies on analyzing the JSJ decomposition and showing that the gluing maps between the JSJ pieces (double twist knot complements) are compatible under the surgery $1/q$, effectively swapping the components.

3. Key Contributions and Results

Main Theorems

1. Theorem 1.2 (Hyperbolic Knots): For any hyperbolic knot $K$, every slope $p/q$ with $|q| \ge \max\{35, q(S^3_K)\}$ is characterising.
   • Significance: Provides the first explicit, computable bound for hyperbolic knots, replacing non-constructive existence proofs.
2. Theorem 1.4 (Satellite Knots): For a satellite knot $K = P(J)$ with a hyperbolic pattern $P$ and winding number $w$, if $\gcd(p, w) \neq 1$ and $|q| \ge \max\{35, q(S^3_P)\}$, the slope is characterising.
   • Significance: Extends the result to satellite knots, provided the numerator condition is met (which prevents the pattern from collapsing into a knot complement).
3. Theorem 1.6 (Whitehead Doubles): For $\pm 1$-clasped Whitehead doubles $K = W^{\pm}_t(J)$, slopes with $|p| \neq 1$ and $|q| \ge q_{min}$ are characterising.
   • Refinement: If the SnapPy census is complete up to stage $k$, $q_{min}$ can be taken as $q_k$ (values listed in Table 1, e.g., $q_1 = 43$, $q_8 = 24$). This provides significantly tighter bounds than the general geodesic method.
4. Theorem 1.7 (Non-Characterising Slopes): For $K = W^n(T^m_q)$ and $K' = W^m(T^n_q)$, there exists a homeomorphism $S^3_K(1/q) \cong S^3_{K'}(1/q)$.
   • Significance: Proves that slopes of the form $1/q$are **not** characterising for these families, justifying the$|p| \neq 1$ condition in Theorem 1.6.

Specific Corollaries

• Twist Knots: For hyperbolic twist knots $T^{\pm}_t$, if $|t| \le 3$, $|q| \ge 35$ suffices. If $|t| \ge 4$, the bound depends on the systole of the specific knot.
• Whitehead Doubles: For $n$-clasped Whitehead doubles, if $|n| \le 4$, $|q| \ge 35$ suffices. If $|n| \ge 5$, the bound depends on the systole of the pattern.

4. Significance

• Constructive Bounds: The paper moves the field from knowing that characterising slopes exist to being able to calculate them for specific, important classes of knots. This is a major step toward a complete classification of characterising slopes.
• Interplay of Geometry and Topology: The work elegantly combines JSJ decomposition (topological structure) with hyperbolic geometry (geodesic lengths and volumes) to solve a purely topological uniqueness problem.
• Computational Reliance: By leveraging the SnapPy census and volume inequalities, the paper demonstrates how computational data can be rigorously integrated into theoretical proofs to improve bounds.
• Clarification of Conditions: The construction of non-characterising slopes for $1/q$ clarifies the limitations of the results and highlights the critical role of the numerator in the characterisation problem, particularly for satellite knots.

In summary, Wakelin provides a robust framework for determining when Dehn surgery uniquely identifies a knot, offering explicit numerical thresholds for hyperbolic and satellite knots, while simultaneously delineating the precise boundaries where this uniqueness fails.