Dehn fillings of knot manifolds containing essential once-punctured tori
Bibliographic record
Abstract
In this paper we study exceptional Dehn fillings on hyperbolic knot manifolds which contain an essential once-punctured torus. Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M"> <mml:semantics> <mml:mi>M</mml:mi> <mml:annotation encoding="application/x-tex">M</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be such a knot manifold and let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="beta"> <mml:semantics> <mml:mi> β </mml:mi> <mml:annotation encoding="application/x-tex">\beta</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be the boundary slope of such an essential once-punctured torus. We prove that if Dehn filling <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M"> <mml:semantics> <mml:mi>M</mml:mi> <mml:annotation encoding="application/x-tex">M</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with slope <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="alpha"> <mml:semantics> <mml:mi> α </mml:mi> <mml:annotation encoding="application/x-tex">\alpha</mml:annotation> </mml:semantics> </mml:math> </inline-formula> produces a Seifert fibred manifold, then <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Delta left-parenthesis alpha comma beta right-parenthesis less-than-or-equal-to 5"> <mml:semantics> <mml:mrow> <mml:mi mathvariant="normal"> Δ </mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi> α </mml:mi> <mml:mo>,</mml:mo> <mml:mi> β </mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo> ≤ </mml:mo> <mml:mn>5</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">\Delta (\alpha ,\beta )\leq 5</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . Furthermore we classify the triples <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis upper M semicolon alpha comma beta right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>M</mml:mi> <mml:mo>;</mml:mo> <mml:mi> α </mml:mi> <mml:mo>,</mml:mo> <mml:mi> β </mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(M; \alpha ,\beta )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> when <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Delta left-parenthesis alpha comma beta right-parenthesis greater-than-or-equal-to 4"> <mml:semantics> <mml:mrow> <mml:mi mathvariant="normal"> Δ </mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi> α </mml:mi> <mml:mo>,</mml:mo> <mml:mi> β </mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo> ≥ </mml:mo> <mml:mn>4</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">\Delta (\alpha ,\beta )\geq 4</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . More precisely, when <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Delta left-parenthesis alpha comma beta right-parenthesis equals 5"> <mml:semantics> <mml:mrow> <mml:mi mathvariant="normal"> Δ </mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi> α </mml:mi> <mml:mo>,</mml:mo> <mml:mi> β </mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:mn>5</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">\Delta (\alpha ,\beta )=5</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , then <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M"> <mml:semantics> <mml:mi>M</mml:mi> <mml:annotation encoding="application/x-tex">M</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is the (unique) manifold <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper W h left-parenthesis negative 3 slash 2 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>W</mml:mi> <mml:mi>h</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mo> − </mml:mo> <mml:mn>3</mml:mn> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">Wh(-3/2)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> obtained by Dehn filling one boundary component of the Whitehead link exterior with slope <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="negative 3 slash 2"> <mml:semantics> <mml:mrow> <mml:mo> − </mml:mo> <mml:mn>3</mml:mn> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/
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How this classification was reachedexpand
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.002 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".