LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-6-luna)verified 2026-10-22 (gpt-6.1-sol)

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Every smooth knot has a Morse representative for the fixed height

Statement

Identify the oriented standard $S^{3}$ with the unit sphere in $R^{4}$ and fix $h (x_{1}, x_{2}, x_{3}, x_{4}) = x_{4}$ . For every smooth oriented knot $K \subset S^{3}$ , there is an orientation-preserving ambient isotopy carrying $K$ to a smooth knot $K^{'}$ for which $h ∣_{K^{'}}$ has only nondegenerate critical points.

Statement references

The fixed coordinate height and Morse representative convention

Facts & Assumptions

Given: A smooth oriented knot $K \subset S^{3}$ and the fixed height $h (x) = x_{4}$ under the standard identification $S^{3} \subset R^{4}$ .

[F1]

For a compact embedded smooth manifold $M \subset R^{N}$ , a generic direction $u \in S^{N - 1}$ makes $x \mapsto u \cdot x$ a Morse function on $M$ . For a compact manifold embedded in Euclidean space, the restricted linear height is Morse for generic directions

Proof

Given: The smooth oriented knot $K$ and the fixed height in the statement.

1.1givenF1choose

Regard $K$ as a compact embedded smooth 1-manifold in $S^{3} \subset R^{4}$ . By [F1], the exceptional set $E$ of non-Morse directions is null in every smooth atlas of $S^{3}$ . It cannot contain all of $S^{3}$ , since a nonempty coordinate ball has positive Euclidean measure; choose $u \in S^{3} ∖ E$ . Then $x \mapsto u \cdot x$ is Morse on $K$ .

2.1givenstep 1.1construct∎

Choose $R \in S O (4)$ with $R^{T} e_{4} = u$ . Such a rotation is obtained in the plane spanned by $u$ and $e_{4}$ ; if $u = - e_{4}$ , use a half-turn in a plane containing $e_{4}$ . A rotation path $R_{t}$ from the identity to $R$ is an ambient isotopy of the oriented sphere. On $R (K)$ , the fixed height equals $e_{4} \cdot R x = u \cdot x$ , which is Morse by step 1.1. Thus $K^{'} = R (K)$ is the required representative.

Sources

Oriented links in the three-sphere and ambient isotopy

For a compact manifold embedded in Euclidean space, the restricted linear height is Morse for generic directions

Depends on

Used by

Bridge number and unknotting number of an oriented knot Definition
Bridge number one characterizes the unknot Theorem

Dependency tree · two levels

12 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Marco Gualtieri, Topology I: Smooth Manifolds, Part 10 (standard reference, not scraped)
Ozsváth, Stipsicz, and Szabó, Grid Homology for Knots and Links (standard reference, not scraped)