# Documentation

Mathlib.Algebra.Quandle

# Racks and Quandles #

This file defines racks and quandles, algebraic structures for sets that bijectively act on themselves with a self-distributivity property. If R is a rack and act : R → (R ≃ R)→ (R ≃ R)≃ R) is the self-action, then the self-distributivity is, equivalently, that

act (act x y) = act x * act y * (act x)⁻¹
⁻¹


where multiplication is composition in R ≃ R≃ R as a group. Quandles are racks such that act x x = x for all x.

One example of a quandle (not yet in mathlib) is the action of a Lie algebra on itself, defined by act x y = Ad (exp x) y.

Quandles and racks were independently developed by multiple mathematicians. David Joyce introduced quandles in his thesis [Joyce1982] to define an algebraic invariant of knot and link complements that is analogous to the fundamental group of the exterior, and he showed that the quandle associated to an oriented knot is invariant up to orientation-reversed mirror image. Racks were used by Fenn and Rourke for framed codimension-2 knots and links in [FennRourke1992]. Unital shelves are discussed in [crans2017].

The name "rack" came from wordplay by Conway and Wraith for the "wrack and ruin" of forgetting everything but the conjugation operation for a group.

## Main definitions #

• Shelf is a type with a self-distributive action
• UnitalShelf is a shelf with a left and right unit
• Rack is a shelf whose action for each element is invertible
• Quandle is a rack whose action for an element fixes that element
• Quandle.conj defines a quandle of a group acting on itself by conjugation.
• ShelfHom is homomorphisms of shelves, racks, and quandles.
• Rack.EnvelGroup gives the universal group the rack maps to as a conjugation quandle.
• Rack.oppositeRack gives the rack with the action replaced by its inverse.

## Main statements #

• Rack.EnvelGroup is left adjoint to Quandle.Conj (toEnvelGroup.map). The universality statements are toEnvelGroup.univ and toEnvelGroup.univ_uniq.

## Implementation notes #

"Unital racks" are uninteresting (see Rack.assoc_iff_id, UnitalShelf.assoc), so we do not define them.

## Notation #

The following notation is localized in quandles:

• x ◃ y◃ y is Shelf.act x y
• x ◃⁻¹ y◃⁻¹ y⁻¹ y is Rack.inv_act x y
• S →◃ S'→◃ S'◃ S' is ShelfHom S S'

Use open quandles to use these.

## Todo #

• If g is the Lie algebra of a Lie group G, then (x ◃ y) = Ad (exp x) x◃ y) = Ad (exp x) x forms a quandle.
• If X is a symmetric space, then each point has a corresponding involution that acts on X, forming a quandle.
• Alexander quandle with a ◃ b = t * b + (1 - t) * b◃ b = t * b + (1 - t) * b, with a and b elements of a module over Z[t,t⁻¹]⁻¹].
• If G is a group, H a subgroup, and z in H, then there is a quandle (G/H;z) defined by yH ◃ xH = yzy⁻¹xH◃ xH = yzy⁻¹xH⁻¹xH. Every homogeneous quandle (i.e., a quandle Q whose automorphism group acts transitively on Q as a set) is isomorphic to such a quandle. There is a generalization to this arbitrary quandles in [Joyce's paper (Theorem 7.2)][Joyce1982].

## Tags #

rack, quandle

class Shelf (α : Type u) :
• The action of the Shelf over α

act : ααα
• A verification that act is self-distributive

self_distrib : ∀ {x y z : α}, act x (act y z) = act (act x y) (act x z)

A Shelf is a structure with a self-distributive binary operation. The binary operation is regarded as a left action of the type on itself.

Instances
class UnitalShelf (α : Type u) extends , :
• one_act : ∀ (a : α), = a
• act_one : ∀ (a : α), = a

A unital shelf is a shelf equipped with an element 1 such that, for all elements x, we have both x ◃ 1◃ 1 and 1 ◃ x◃ x equal x.

Instances
theorem ShelfHom.ext_iff {S₁ : Type u_1} {S₂ : Type u_2} :
∀ {inst : Shelf S₁} {inst_1 : Shelf S₂} (x y : ShelfHom S₁ S₂), x = y x.toFun = y.toFun
theorem ShelfHom.ext {S₁ : Type u_1} {S₂ : Type u_2} :
∀ {inst : Shelf S₁} {inst_1 : Shelf S₂} (x y : ShelfHom S₁ S₂), x.toFun = y.toFunx = y
structure ShelfHom (S₁ : Type u_1) (S₂ : Type u_2) [inst : Shelf S₁] [inst : Shelf S₂] :
Type (maxu_1u_2)
• The function under the Shelf Homomorphism

toFun : S₁S₂
• The homomorphism property of a Shelf Homomorphism

map_act' : ∀ {x y : S₁}, toFun () = Shelf.act (toFun x) (toFun y)

The type of homomorphisms between shelves. This is also the notion of rack and quandle homomorphisms.

Instances For
class Rack (α : Type u) extends :
• The inverse actions of the elements

invAct : ααα
• Proof of left inverse

left_inv : ∀ (x : α), Function.LeftInverse (invAct x) ()
• Proof of right inverse

right_inv : ∀ (x : α), Function.RightInverse (invAct x) ()

A rack is an automorphic set (a set with an action on itself by bijections) that is self-distributive. It is a shelf such that each element's action is invertible.

The notations x ◃ y◃ y and x ◃⁻¹ y◃⁻¹ y⁻¹ y denote the action and the inverse action, respectively, and they are right associative.

Instances

Action of a Shelf

Equations

Inverse Action of a Rack

Equations

Shelf Homomorphism

Equations
theorem UnitalShelf.act_act_self_eq {S : Type u_1} [inst : ] (x : S) (y : S) :
Shelf.act () x =

A monoid is graphic if, for all x and y, the graphic identity (x * y) * x = x * y holds. For a unital shelf, this graphic identity holds.

theorem UnitalShelf.act_idem {S : Type u_1} [inst : ] (x : S) :
= x
theorem UnitalShelf.act_self_act_eq {S : Type u_1} [inst : ] (x : S) (y : S) :
Shelf.act x () =
theorem UnitalShelf.assoc {S : Type u_1} [inst : ] (x : S) (y : S) (z : S) :

def Rack.act' {R : Type u_1} [inst : Rack R] (x : R) :
R R

A rack acts on itself by equivalences.

Equations
• One or more equations did not get rendered due to their size.
@[simp]
theorem Rack.act'_apply {R : Type u_1} [inst : Rack R] (x : R) (y : R) :
↑() y =
@[simp]
theorem Rack.act'_symm_apply {R : Type u_1} [inst : Rack R] (x : R) (y : R) :
↑() y =
@[simp]
theorem Rack.invAct_apply {R : Type u_1} [inst : Rack R] (x : R) (y : R) :
()⁻¹ y =
@[simp]
theorem Rack.invAct_act_eq {R : Type u_1} [inst : Rack R] (x : R) (y : R) :
Rack.invAct x () = y
@[simp]
theorem Rack.act_invAct_eq {R : Type u_1} [inst : Rack R] (x : R) (y : R) :
Shelf.act x () = y
theorem Rack.left_cancel {R : Type u_1} [inst : Rack R] (x : R) {y : R} {y' : R} :
= Shelf.act x y' y = y'
theorem Rack.left_cancel_inv {R : Type u_1} [inst : Rack R] (x : R) {y : R} {y' : R} :
= Rack.invAct x y' y = y'
theorem Rack.self_distrib_inv {R : Type u_1} [inst : Rack R] {x : R} {y : R} {z : R} :
theorem Rack.ad_conj {R : Type u_1} [inst : Rack R] (x : R) (y : R) :

The adjoint action of a rack on itself is op', and the adjoint action of x ◃ y◃ y is the conjugate of the action of y by the action of x. It is another way to understand the self-distributivity axiom.

This is used in the natural rack homomorphism toConj from R to Conj (R ≃ R)≃ R) defined by op'.

instance Rack.oppositeRack {R : Type u_1} [inst : Rack R] :

The opposite rack, swapping the roles of ◃◃ and ◃⁻¹◃⁻¹⁻¹.

Equations
• One or more equations did not get rendered due to their size.
@[simp]
theorem Rack.op_act_op_eq {R : Type u_1} [inst : Rack R] {x : R} {y : R} :
Shelf.act { unop := x } { unop := y } = { unop := }
@[simp]
theorem Rack.op_invAct_op_eq {R : Type u_1} [inst : Rack R] {x : R} {y : R} :
Rack.invAct { unop := x } { unop := y } = { unop := }
@[simp]
theorem Rack.self_act_act_eq {R : Type u_1} [inst : Rack R] {x : R} {y : R} :
Shelf.act () y =
@[simp]
theorem Rack.self_invAct_invAct_eq {R : Type u_1} [inst : Rack R] {x : R} {y : R} :
@[simp]
theorem Rack.self_act_invAct_eq {R : Type u_1} [inst : Rack R] {x : R} {y : R} :
@[simp]
theorem Rack.self_invAct_act_eq {R : Type u_1} [inst : Rack R] {x : R} {y : R} :
Shelf.act () y =
theorem Rack.self_act_eq_iff_eq {R : Type u_1} [inst : Rack R] {x : R} {y : R} :
= x = y
theorem Rack.self_invAct_eq_iff_eq {R : Type u_1} [inst : Rack R] {x : R} {y : R} :
= x = y
def Rack.selfApplyEquiv (R : Type u_1) [inst : Rack R] :
R R

The map x ↦ x ◃ x↦ x ◃ x◃ x is a bijection. (This has applications for the regular isotopy version of the Reidemeister I move for knot diagrams.)

Equations
• One or more equations did not get rendered due to their size.
def Rack.IsInvolutory (R : Type u_1) [inst : Rack R] :

An involutory rack is one for which Rack.oppositeRack R x is an involution for every x.

Equations
• = ∀ (x : R),
theorem Rack.involutory_invAct_eq_act {R : Type u_1} [inst : Rack R] (h : ) (x : R) (y : R) :
=
def Rack.IsAbelian (R : Type u_1) [inst : Rack R] :

An abelian rack is one for which the mediality axiom holds.

Equations
theorem Rack.assoc_iff_id {R : Type u_1} [inst : Rack R] {x : R} {y : R} {z : R} :
Shelf.act x () = Shelf.act () z = z

Associative racks are uninteresting.

instance ShelfHom.instCoeFunShelfHomForAll {S₁ : Type u_1} {S₂ : Type u_2} [inst : Shelf S₁] [inst : Shelf S₂] :
CoeFun (ShelfHom S₁ S₂) fun x => S₁S₂
Equations
• ShelfHom.instCoeFunShelfHomForAll = { coe := ShelfHom.toFun }
@[simp]
theorem ShelfHom.map_act {S₁ : Type u_1} {S₂ : Type u_2} [inst : Shelf S₁] [inst : Shelf S₂] (f : ShelfHom S₁ S₂) {x : S₁} {y : S₁} :
def ShelfHom.id (S : Type u_1) [inst : ] :

The identity homomorphism

Equations
instance ShelfHom.inhabited (S : Type u_1) [inst : ] :
Equations
• = { default := }
def ShelfHom.comp {S₁ : Type u_1} {S₂ : Type u_2} {S₃ : Type u_3} [inst : Shelf S₁] [inst : Shelf S₂] [inst : Shelf S₃] (g : ShelfHom S₂ S₃) (f : ShelfHom S₁ S₂) :
ShelfHom S₁ S₃

The composition of shelf homomorphisms

Equations
• One or more equations did not get rendered due to their size.
@[simp]
theorem ShelfHom.comp_apply {S₁ : Type u_3} {S₂ : Type u_1} {S₃ : Type u_2} [inst : Shelf S₁] [inst : Shelf S₂] [inst : Shelf S₃] (g : ShelfHom S₂ S₃) (f : ShelfHom S₁ S₂) (x : S₁) :
class Quandle (α : Type u_1) extends :
Type u_1
• The fixing property of a Quandle

fix : ∀ {x : α}, = x

A quandle is a rack such that each automorphism fixes its corresponding element.

Instances
@[simp]
theorem Quandle.fix_inv {Q : Type u_1} [inst : ] {x : Q} :
= x
instance Quandle.oppositeQuandle {Q : Type u_1} [inst : ] :
Equations
def Quandle.Conj (G : Type u_1) :
Type u_1

The conjugation quandle of a group. Each element of the group acts by the corresponding inner automorphism.

Equations
instance Quandle.Conj.quandle (G : Type u_1) [inst : ] :
Equations
@[simp]
theorem Quandle.conj_act_eq_conj {G : Type u_1} [inst : ] (x : ) (y : ) :
= x * y * x⁻¹
theorem Quandle.conj_swap {G : Type u_1} [inst : ] (x : ) (y : ) :
= y = x
def Quandle.Conj.map {G : Type u_1} {H : Type u_2} [inst : ] [inst : ] (f : G →* H) :
ShelfHom () ()

Conj is functorial

Equations
• = { toFun := f, map_act' := (_ : ∀ (a a_1 : ), f (a * a_1 * a⁻¹) = f a * f a_1 * (f a)⁻¹) }

The dihedral quandle. This is the conjugation quandle of the dihedral group restrict to flips.

Used for Fox n-colorings of knots.

Equations
def Quandle.dihedralAct (n : ) (a : ZMod n) :
ZMod nZMod n

The operation for the dihedral quandle. It does not need to be an equivalence because it is an involution (see dihedralAct.inv).

Equations
theorem Quandle.dihedralAct.inv (n : ) (a : ZMod n) :
Equations
def Rack.toConj (R : Type u_1) [inst : Rack R] :

This is the natural rack homomorphism to the conjugation quandle of the group R ≃ R≃ R that acts on the rack.

Equations

### Universal enveloping group of a rack #

The universal enveloping group EnvelGroup R of a rack R is the universal group such that every rack homomorphism R →◃ conj G→◃ conj G◃ conj G is induced by a unique group homomorphism EnvelGroup R →* G→* G. For quandles, Joyce called this group AdConj R.

The EnvelGroup functor is left adjoint to the Conj forgetful functor, and the way we construct the enveloping group is via a technique that should work for left adjoints of forgetful functors in general. It involves thinking a little about 2-categories, but the payoff is that the map EnvelGroup R →* G→* G has a nice description.

Let's think of a group as being a one-object category. The first step is to define PreEnvelGroup, which gives formal expressions for all the 1-morphisms and includes the unit element, elements of R, multiplication, and inverses. To introduce relations, the second step is to define PreEnvelGroupRel', which gives formal expressions for all 2-morphisms between the 1-morphisms. The 2-morphisms include associativity, multiplication by the unit, multiplication by inverses, compatibility with multiplication and inverses (congr_mul and congr_inv), the axioms for an equivalence relation, and, importantly, the relationship between conjugation and the rack action (see Rack.ad_conj).

None of this forms a 2-category yet, for example due to lack of associativity of trans. The PreEnvelGroupRel relation is a Prop-valued version of PreEnvelGroupRel', and making it Prop-valued essentially introduces enough 3-isomorphisms so that every pair of compatible 2-morphisms is isomorphic. Now, while composition in PreEnvelGroup does not strictly satisfy the category axioms, PreEnvelGroup and PreEnvelGroupRel' do form a weak 2-category.

Since we just want a 1-category, the last step is to quotient PreEnvelGroup by PreEnvelGroupRel', and the result is the group EnvelGroup.

For a homomorphism f : R →◃ Conj G→◃ Conj G◃ Conj G, how does EnvelGroup.map f : EnvelGroup R →* G→* G work? Let's think of G as being a 2-category with one object, a 1-morphism per element of G, and a single 2-morphism called eq.refl for each 1-morphism. We define the map using a "higher Quotient.lift" -- not only do we evaluate elements of PreEnvelGroup as expressions in G (this is toEnvelGroup.mapAux), but we evaluate elements of PreEnvelGroup' as expressions of 2-morphisms of G (this is toEnvelGroup.mapAux.well_def). That is to say, toEnvelGroup.mapAux.well_def recursively evaluates formal expressions of 2-morphisms as equality proofs in G. Now that all morphisms are accounted for, the map descends to a homomorphism EnvelGroup R →* G→* G.

Note: Type-valued relations are not common. The fact it is Type-valued is what makes toEnvelGroup.mapAux.well_def have well-founded recursion.

inductive Rack.PreEnvelGroup (R : Type u) :
• unit: {R : Type u} →
• incl: {R : Type u} →
• mul: {R : Type u} →
• inv: {R : Type u} →

Free generators of the enveloping group.

Instances For
Equations
• = { default := Rack.PreEnvelGroup.unit }
inductive Rack.PreEnvelGroupRel' (R : Type u) [inst : Rack R] :

Relations for the enveloping group. This is a type-valued relation because toEnvelGroup.mapAux.well_def inducts on it to show toEnvelGroup.map is well-defined. The relation PreEnvelGroupRel is the Prop-valued version, which is used to define EnvelGroup itself.

Instances For
instance Rack.PreEnvelGroupRel'.inhabited (R : Type u) [inst : Rack R] :
Inhabited (Rack.PreEnvelGroupRel' R Rack.PreEnvelGroup.unit Rack.PreEnvelGroup.unit)
Equations
• = { default := Rack.PreEnvelGroupRel'.refl }
inductive Rack.PreEnvelGroupRel (R : Type u) [inst : Rack R] :
• rel: ∀ {R : Type u} [inst : Rack R] {a b : },

The PreEnvelGroupRel relation as a Prop. Used as the relation for PreEnvelGroup.setoid.

Instances For
theorem Rack.PreEnvelGroupRel'.rel {R : Type u} [inst : Rack R] {a : } {b : } :

A quick way to convert a PreEnvelGroupRel' to a PreEnvelGroupRel.

theorem Rack.PreEnvelGroupRel.refl {R : Type u} [inst : Rack R] {a : } :
theorem Rack.PreEnvelGroupRel.symm {R : Type u} [inst : Rack R] {a : } {b : } :
theorem Rack.PreEnvelGroupRel.trans {R : Type u} [inst : Rack R] {a : } {b : } {c : } :
instance Rack.PreEnvelGroup.setoid (R : Type u_1) [inst : Rack R] :
Equations
• = { r := , iseqv := (_ : ) }
def Rack.EnvelGroup (R : Type u_1) [inst : Rack R] :
Type u_1

The universal enveloping group for the rack R.

Equations
instance Rack.instDivInvMonoidEnvelGroup (R : Type u_1) [inst : Rack R] :
Equations
instance Rack.instGroupEnvelGroup (R : Type u_1) [inst : Rack R] :
Equations
instance Rack.EnvelGroup.inhabited (R : Type u_1) [inst : Rack R] :
Equations
• = { default := 1 }
def Rack.toEnvelGroup (R : Type u_1) [inst : Rack R] :

The canonical homomorphism from a rack to its enveloping group. Satisfies universal properties given by toEnvelGroup.map and toEnvelGroup.univ.

Equations
• One or more equations did not get rendered due to their size.
def Rack.toEnvelGroup.mapAux {R : Type u_1} [inst : Rack R] {G : Type u_2} [inst : ] (f : ShelfHom R ()) :

The preliminary definition of the induced map from the enveloping group. See toEnvelGroup.map.

Equations
theorem Rack.toEnvelGroup.mapAux.well_def {R : Type u_1} [inst : Rack R] {G : Type u_2} [inst : ] (f : ShelfHom R ()) {a : } {b : } :

Show that toEnvelGroup.mapAux sends equivalent expressions to equal terms.

def Rack.toEnvelGroup.map {R : Type u_1} [inst : Rack R] {G : Type u_2} [inst : ] :
ShelfHom R () ()

Given a map from a rack to a group, lift it to being a map from the enveloping group. More precisely, the EnvelGroup functor is left adjoint to Quandle.Conj.

Equations
• One or more equations did not get rendered due to their size.
theorem Rack.toEnvelGroup.univ (R : Type u_1) [inst : Rack R] (G : Type u_2) [inst : ] (f : ShelfHom R ()) :
ShelfHom.comp (Quandle.Conj.map (Rack.toEnvelGroup.map f)) () = f

Given a homomorphism from a rack to a group, it factors through the enveloping group.

theorem Rack.toEnvelGroup.univ_uniq (R : Type u_1) [inst : Rack R] (G : Type u_2) [inst : ] (f : ShelfHom R ()) (g : ) (h : ) :
g = Rack.toEnvelGroup.map f

The homomorphism toEnvelGroup.map f is the unique map that fits into the commutative triangle in toEnvelGroup.univ.

def Rack.envelAction {R : Type u_1} [inst : Rack R] :
→* R R

The induced group homomorphism from the enveloping group into bijections of the rack, using Rack.toConj. Satisfies the property envelAction_prop.

This gives the rack R the structure of an augmented rack over EnvelGroup R.

Equations
• Rack.envelAction = Rack.toEnvelGroup.map ()
@[simp]
theorem Rack.envelAction_prop {R : Type u_1} [inst : Rack R] (x : R) (y : R) :
↑(Rack.envelAction ()) y =