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

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Knot genus is invariant under mirroring and reversal

Statement

Assume the Axiom of Countable Choice. Let $K$ be a smooth oriented knot in $S^{3}$ , and let $r : S^{3} \to S^{3}$ be an orientation-reversing reflection. Give $r (K)$ the image orientation, and let $- K$ denote reversal of the knot orientation. Then $g (K) = g (r (K)) = g (- K) = g (- r (K))$ .

Statement references

Definition of knot genus by minimum spanning-surface genus

Facts & Assumptions

Given: AC $_{ω}$ , a smooth oriented knot $K$ , and a smooth reflection $r$ of $S^{3}$ .

[F1]

Knot genus is the minimum genus among connected compact smooth oriented surfaces whose oriented boundary is the knot. (Seifert surfaces and smooth knot genus)

[F2]

For a connected compact orientable surface with one boundary component, reversing its orientation or applying a diffeomorphism preserves its genus. (Finite surface normal forms, Jordan disks, and torsion control)

[A1]

AC $_{ω}$ is the sequence-choice principle. (The Axiom of Countable Choice ( ${A C}_{ω}$ ))

Proof

Given: The knot $K$ , reflection $r$ , and AC $_{ω}$ .

1.1givenF1F2A1construct

If $F$ is any spanning surface for $K$ , its image $r (F)$ is a connected compact smooth surface with boundary $r (K)$ . Push the orientation of $F$ forward to orient $r (F)$ ; reversing the boundary orientation instead is achieved by reversing the orientation on $r (F)$ . By [F2], all these surfaces have genus $g (F)$ . Thus every genus admissible for $K$ is admissible with the same value for $r (K)$ and $- K$ .

2.1givenstep 1.1algebra∎

Apply the same argument using $r^{- 1} = r$ and orientation reversal again. This gives the reverse inclusion of the sets of admissible genera, so their minima agree for all four knots.

Sources

Seifert surfaces and smooth knot genus

Finite surface normal forms, Jordan disks, and torsion control

The Axiom of Countable Choice ( ${A C}_{ω}$ )

Depends on

Used by

Connected-sum genus and ribbon examples Example
Torus-knot Alexander polynomial and genus formulas Theorem

Dependency tree · two levels

32 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

Burde and Zieschang, Knots, 2nd revised and extended edition (standard reference, not scraped)