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feat(algebra/quandle): racks and quandles (#4247)
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This adds the algebraic structures of racks and quandles, defines a few examples, and provides the universal enveloping group of a rack.

A rack is a set that acts on itself bijectively, and sort of the point is that the action `act : α → (α ≃ α)` satisfies
```
act (x ◃ y) = act x * act y * (act x)⁻¹
```
where `x ◃ y` is the usual rack/quandle notation for `act x y`.  (Note: racks do not use `has_scalar` because it's convenient having `x ◃⁻¹ y` for the inverse action of `x` on `y`.  Plus, associative racks have a trivial action.)

In knot theory, the universal enveloping group of the fundamental quandle is isomorphic to the fundamental group of the knot complement.  For oriented knots up to orientation-reversed mirror image, the fundamental quandle is a complete invariant, unlike the fundamental group, which fails to distinguish non-prime knots with chiral summands.
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kmill committed Sep 29, 2020
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27 changes: 27 additions & 0 deletions docs/references.bib
Expand Up @@ -397,3 +397,30 @@ @book{cassels1967algebraic
year={1967},
publisher={Academic Pr}
}

@Article{Joyce1982,
author = {David Joyce},
title = {A classifying invariant of knots, the knot quandle},
journal = {Journal of Pure and Applied Algebra},
year = {1982},
volume = {23},
number = {1},
month = {1},
pages = {37--65},
doi = {10.1016/0022-4049(82)90077-9},
publisher = {Elsevier {BV}}
}

@Article{FennRourke1992,
author = {Fenn, Roger and Rourke, Colin},
journal = {Journal of Knot Theory and its Ramifications},
title = {Racks and links in codimension two},
year = {1992},
issn = {0218-2165},
number = {4},
pages = {343--406},
volume = {1},
doi = {10.1142/S0218216592000203},
keywords = {57M25 (57N10)},
mrnumber = {1194995}
}

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