Unknotting Nonorientable Surfaces of Genus 4 and 5

Mark Pencovitch

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Unknotting Nonorientable Surfaces of Genus 4 and 5

Mark Pencovitch

Geometric Topology · Submitted 28 Feb 2024

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

The authors' main claim This implies that closed, nonorientable, locally flatly embedded surfaces in the $4$-sphere with knot group $\mathbb{Z}_2$ of nonorientable genus $4$ and $5$ are topologically unknotted.

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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 4,439 input tokens.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.96 · human participants 0.01 · controlled comparison 0.04 · randomized assignment 0.01 · states quantitative result 0.71 · compares with prior results 0.79 · mentions code or data availability 0.07.

The abstract, sentence by sentence

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

Expanding on work by Conway, Orson, and Powell, we study the isotopy classes rel. boundary of nonorientable, compact, locally flatly embedded surfaces in $D^4$ with knot group $\mathbb{Z}_2$. In particular we show that if two such surfaces have fixed knot boundary $K$ in $S^4$ such that $\vert \det(K) \vert =1$, the same normal Euler number, and the same nonorientable genus $4$ or $5$, then they are ambiently isotopic rel. boundary. This implies that closed, nonorientable, locally flatly embedded surfaces in the $4$-sphere with knot group $\mathbb{Z}_2$ of nonorientable genus $4$ and $5$ are topologically unknotted. The proof relies on calculations, implemented in Sage, which imply that the modified surgery obstruction is elementary. Furthermore we show that this method fails for nonorientable genus $6$ and $7$.

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What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	Expanding on work by Conway, Orson, and Powell, we study the isotopy classes rel. boundary of nonorientable, compact, locally flatly embedded surfaces in $D^4$ with knot group $\mathbb{Z}_2$.	contribution 0.73	0.72	0.03	asserted 0.79
s1	In particular we show that if two such surfaces have fixed knot boundary $K$ in $S^4$ such that $\vert \det(K) \vert =1$, the same normal Euler number, and the same nonorientable genus $4$ or $5$, then they are ambiently isotopic rel. boundary.	result 0.99	0.95	0.53	asserted 1.00
s2	This implies that closed, nonorientable, locally flatly embedded surfaces in the $4$-sphere with knot group $\mathbb{Z}_2$ of nonorientable genus $4$ and $5$ are topologically unknotted.	interpretation 0.93	0.90	0.75	asserted 1.00
s3	The proof relies on calculations, implemented in Sage, which imply that the modified surgery obstruction is elementary.	method 1.00	0.92	0.03	asserted 1.00
s4	Furthermore we show that this method fails for nonorientable genus $6$ and $7$.	limitation 0.95	0.94	0.04	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 0.80.

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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

Expanding on work by Conway, Orson, and Powell, we study the isotopy classes rel. boundary of nonorientable, compact, locally flatly embedded surfaces in $D^4$ with knot group $\mathbb{Z}_2$. In particular we show that if two such surfaces have fixed knot boundary $K$ in $S^4$ such that $\vert \det(K) \vert =1$, the same normal Euler number, and the same nonorientable genus $4$ or $5$, then they are ambiently isotopic rel. boundary. This implies that closed, nonorientable, locally flatly embedded surfaces in the $4$-sphere with knot group $\mathbb{Z}_2$ of nonorientable genus $4$ and $5$ are topologically unknotted. The proof relies on calculations, implemented in Sage, which imply that the modified surgery obstruction is elementary. Furthermore we show that this method fails for nonorientable genus $6$ and $7$.

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What this paper does with the arXiv papers it cites

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

What it cites · 11 sentences without a confident label

Uses the method or tool of 1 paper

The results in [ref] use Kreck's modified surgery with respect to the normal $1$-type of the exteriors of the $_2$-surfaces.

Unknotting nonorientable surfaces · The modified surgery obstruction · 0.86

Cites as background 1 paper

In Section [ref] we give a brief background and outline of the proof given in [ref] .

Unknotting nonorientable surfaces · Introduction · 0.93

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