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Triple linking numbers and surface systems

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Christopher William Davis, Matthias Nagel, Patrick Orson, Mark Powell

Geometric Topology · Submitted 25 Sep 2017

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim Given two links with the same pairwise linking numbers we show that they have the same refined triple linking number collection if and only if the links admit homeomorphic surface systems.

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arXiv ID1709.08478 (version 1)
Pith IDUHKDXUTW, Pith's permanent record of this paper and its review
CitationChristopher William Davis, Mark Powell, Matthias Nagel, Patrick Orson. "Triple linking numbers and surface systems." arXiv:1709.08478 (2017). https://pith.science/paper/UHKDXUTW
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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 3,696 input tokens.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.08 · human participants 0.01 · controlled comparison 0.03 · randomized assignment 0.01 · states quantitative result 0.08 · compares with prior results 0.08 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; .

We give a refined value group for the collection of triple linking numbers of links in the 3-sphere. Given two links with the same pairwise linking numbers we show that they have the same refined triple linking number collection if and only if the links admit homeomorphic surface systems. Moreover these two conditions hold if and only if the link exteriors are bordant over $B \mathbb{Z}^n$, and if and only if the third lower central series quotients $\pi/\pi_3$ of the link groups are isomorphic preserving meridians and longitudes. We also show that these conditions imply that the link groups have isomorphic fourth lower central series quotients $\pi/\pi_4$, preserving meridians.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 We give a refined value group for the collection of triple linking numbers of links in the 3-sphere. contribution 0.94 0.95 0.15 asserted 1.00
s1 Given two links with the same pairwise linking numbers we show that they have the same refined triple linking number collection if and only if the links admit homeomorphic surface systems. result 0.98 0.93 0.55 asserted 1.00
s2 Moreover these two conditions hold if and only if the link exteriors are bordant over $B \mathbb{Z}^n$, and if and only if the third lower central series quotients $\pi/\pi_3$ of the link groups are isomorphic preserving meridians and longitudes. result 0.99 0.87 0.47 asserted 1.00
s3 We also show that these conditions imply that the link groups have isomorphic fourth lower central series quotients $\pi/\pi_4$, preserving meridians. result 0.93 0.93 0.11 asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

The paper
Original abstract

We give a refined value group for the collection of triple linking numbers of links in the 3-sphere. Given two links with the same pairwise linking numbers we show that they have the same refined triple linking number collection if and only if the links admit homeomorphic surface systems. Moreover these two conditions hold if and only if the link exteriors are bordant over $B \mathbb{Z}^n$, and if and only if the third lower central series quotients $\pi/\pi_3$ of the link groups are isomorphic preserving meridians and longitudes. We also show that these conditions imply that the link groups have isomorphic fourth lower central series quotients $\pi/\pi_4$, preserving meridians.

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