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Sign changes of torus-knot parameters give mirror and reversal

Statement

Assume Countable Choice. For coprime nonzero integers $p, q$ , the fixed orientation-reversing reflection used for the mirror convention sends the oriented curve $T (p, q)$ to the image-oriented curve $T (p, - q)$ , and reversing the orientation of $T (p, q)$ gives $T (- p, - q)$ .

Statement references

The torus-knot parameterization and mirror convention

Cattabriga and Francaviglia, Proposition 5.1.4 and Lemma 5.1.5

Facts & Assumptions

Given: Countable Choice, coprime nonzero integers $p, q$ , and the parameterization in the torus-knot definition.

[A1]

Countable Choice is the choice-function principle for sequences of nonempty sets (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

[F1]

Coprime torus knot parameterizes $T (p, q)$ by $(r e^{2 π i p s}, \sqrt{1 - r^{2}} e^{2 π i q s})$ and fixes reflection in the second coordinate as the mirror convention.

Proof

1.1A1F1givenalgebra

Applying $(z_{1}, z_{2}) \mapsto (z_{1}, \overline{z_{2}})$ to the parameterization changes its second angular coordinate from $q s$ to $- q s$ and leaves the first coordinate and parameter direction unchanged. Its image is therefore exactly the oriented curve $T (p, - q)$ .

2.1A1F1givenalgebra∎

Replacing the parameter $s$ by $- s$ traverses the same geometric curve in the opposite direction and changes the angular winding pair $(p, q)$ to $(- p, - q)$ . Hence the reversed orientation is the orientation specified for $T (- p, - q)$ .

Depends on

Used by

Torus-knot Alexander polynomial and genus formulas Theorem

Dependency tree · two levels

9 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

Alessia Cattabriga and Stefano Francaviglia, Knot Theory and Applications: Lecture Notes (standard reference, not scraped)