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Reverse mirror of an oriented knot
Definition
Fix the orientation-reversing reflection chosen for the mirror convention in the oriented connected-sum definition. If is a smooth oriented knot, its mirror carries the orientation induced from by . The reverse mirror of , denoted , is the image with its knot orientation reversed. Thus changes both the ambient orientation convention and the orientation of the knot.
Definition references
Depends on
Used by
- The square knot is slice despite its nonunit Alexander polynomial Counterexample
- A knot and its reverse mirror sum to a slice knot Example
- The square knot has a nontrivial character vanishing on a linking metabolizer Example
- The square knot is a nontrivial ribbon knot Example
- The square-knot Seifert form is metabolic Example
- A knot summed with its reverse mirror bounds a ribbon disk Lemma
- A reverse-mirror sum admits an explicit ribbon immersion Lemma
- Epsilon is stable-equivalence invariant but not a homomorphism Lemma
- Knot Floer duality and exchange of the two filtrations Lemma
- Rasmussen's invariant changes sign under mirror image Lemma
- The reverse-mirror Seifert matrix is the negative original Lemma
- Epsilon and Upsilon concordance data Theorem
- Smooth and topological knot concordance classes form abelian groups Theorem
- Smooth concordance implies stable filtered equivalence of knot Floer complexes Theorem
- Upsilon additivity, mirror symmetry, and four-ball genus bound Theorem
Dependency tree · two levels
7 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
- Arunima Ray, Slice Knots and Knot Concordance, Winter Braids Lecture Notes 8 (standard reference, not scraped)
- Dale Rolfsen, Knots and Links (standard reference, not scraped)