INTRODUCTION TO ALGEBRAIC THEORY OF QUANDLES Lecture 1
Valeriy Bardakov
Sobolev Institute of Mathematics, Novosibirsk
August 24-28, 2020 ICTS Bengaluru, India
V. Bardakov (Sobolev Institute of Math.) INTRODUCTION TO ALGEBRAIC THEORY OF QUANDLESAugust 27-30, 2019 1 / 53 Green-board in ICTS
V. Bardakov (Sobolev Institute of Math.) INTRODUCTION TO ALGEBRAIC THEORY OF QUANDLESAugust 27-30, 2019 2 / 53 Quandles
A quandle is a non-empty set with one binary operation that satisfies three axioms.
These axioms motivated by the three Reidemeister moves of diagrams 3 of knots in the Euclidean space R .
Quandles were introduced independently by S. Matveev and D. Joyce in 1982.
V. Bardakov (Sobolev Institute of Math.) INTRODUCTION TO ALGEBRAIC THEORY OF QUANDLESAugust 27-30, 2019 3 / 53 Definition of rack and quandle
Definition A rack is a non-empty set X with a binary algebraic operation
(a, b) 7→ a ∗ b
satisfying the following conditions: (R1) For any a, b ∈ X there is a unique c ∈ X such that a = c ∗ b; (R2) Self-distributivity: (a ∗ b) ∗ c = (a ∗ c) ∗ (b ∗ c) for all a, b, c ∈ X.
A quandle X is a rack which satisfies the following condition: (Q1) a ∗ a = a for all a ∈ X.
V. Bardakov (Sobolev Institute of Math.) INTRODUCTION TO ALGEBRAIC THEORY OF QUANDLESAugust 27-30, 2019 4 / 53 Symmetries
Let X be a rack. For any a ∈ X define a map Sa : X −→ X, which sends any element x ∈ X to element x ∗ a, i. e. xSa = x ∗ a.
We shall call Sa the symmetry at a. Proposition
For every a ∈ X the symmetry Sa is an automorphism of X.
V. Bardakov (Sobolev Institute of Math.) INTRODUCTION TO ALGEBRAIC THEORY OF QUANDLESAugust 27-30, 2019 5 / 53 Proof
Rewrite the condition (R2) in the form (a ∗ b)Sc = aSc ∗ bSc. It means that every symmetry is a homomorphism of X.
Suppose that xSa = ySa for some x, y ∈ X. It means that x ∗ a = y ∗ a and from condition (R1) follows that x = y, i.e. Sa is injective. Also, from (R1) follows that Sa is an epimorphism. Hence, every symmetry is an automorphism.
V. Bardakov (Sobolev Institute of Math.) INTRODUCTION TO ALGEBRAIC THEORY OF QUANDLESAugust 27-30, 2019 6 / 53 Consequences of axioms
From (R1) follows that we can define an operation ∗−1 : X × X → X by the rule a = c ∗ b ⇔ c = a ∗−1 b. This equivalent to
(a ∗ b) ∗−1 b = a = (a ∗−1 b) ∗ b.
From this identity and (R2) follow identities
(a ∗ b) ∗−1 c = (a ∗−1 c) ∗ (b ∗−1 c),
(a ∗−1 b) ∗ c = (a ∗ c) ∗−1 (b ∗ c), (a ∗−1 b) ∗−1 c = (a ∗−1 c) ∗−1 (b ∗−1 c).
V. Bardakov (Sobolev Institute of Math.) INTRODUCTION TO ALGEBRAIC THEORY OF QUANDLESAugust 27-30, 2019 7 / 53 Another definition of rack
Definition A rack is a set X equipped with two binary operations
(a, b) 7→ a ∗ b and (a, b) 7→ a ∗−1 b which satisfy the axioms (R1)0 (a ∗ b) ∗−1 b = (a ∗−1 b) ∗ b = a for all a, b, c ∈ Q, (R2) (a ∗ b) ∗ c = (a ∗ c) ∗ (b ∗ c) for all a, b, c ∈ Q.
V. Bardakov (Sobolev Institute of Math.) INTRODUCTION TO ALGEBRAIC THEORY OF QUANDLESAugust 27-30, 2019 8 / 53 Presentation of rack elements
Using the second definition of rack we can prove Proposition Let X be a rack with a set of generators A. Then any element u ∈ X can be presented in a form
ε1 ε2 ε3 εm−1 u = (... ((ai1 ∗ ai2 ) ∗ ai3 ) ∗ ...) ∗ aim , aij ∈ A, εk ∈ {−1, 1}.
V. Bardakov (Sobolev Institute of Math.) INTRODUCTION TO ALGEBRAIC THEORY OF QUANDLESAugust 27-30, 2019 9 / 53 Trivial quandles
The simplest example of quandle is the so called trivial quandle. A quandle X is called trivial if a ∗ b = a for all a, b ∈ X, i. e. any symmetry Sb is the trivial automorphism. We see that a trivial quandle can contains arbitrary number of elements.
We shall denote the trivial quandle with n elements by Tn.
V. Bardakov (Sobolev Institute of Math.) INTRODUCTION TO ALGEBRAIC THEORY OF QUANDLESAugust 27-30, 2019 10 / 53 Comparing of group axioms and quandle axioms
It is interesting to understand: how far quandles from groups, i.e. what group axioms are true for quandles?
Proposition 1) A quandle is associative if and only if it is a trivial quandle. 2) If a quandle X contains a unit element e, then X = {e} is the trivial 1-element quandle T1.
V. Bardakov (Sobolev Institute of Math.) INTRODUCTION TO ALGEBRAIC THEORY OF QUANDLESAugust 27-30, 2019 11 / 53 Proof
1) Suppose that (x ∗ y) ∗ z = x ∗ (y ∗ z). If we put z = y, then (x ∗ y) ∗ y = x ∗ y. Since elements x, y are arbitrary, we get trivial quandle. 2) Since e is a unit element, then for any x ∈ X we have
e ∗ x = x ∗ e = x.
On the other side x ∗ x = x. Then from (R1) follows that x = e.
V. Bardakov (Sobolev Institute of Math.) INTRODUCTION TO ALGEBRAIC THEORY OF QUANDLESAugust 27-30, 2019 12 / 53 A rack that is not a quandle
Hence, only trivial quandle is associative, i.e. is a semi-group. Properties of trivial quandles are different from properties of other quandles.
Example
If we define on the additive group (Zn, +), n > 1, the operation x ∗ y ≡ x + 1 (mod n) for all x, y ∈ Zn, then (Zn, ∗) is a rack which is not a quandle.
V. Bardakov (Sobolev Institute of Math.) INTRODUCTION TO ALGEBRAIC THEORY OF QUANDLESAugust 27-30, 2019 13 / 53 Examples of quandles: conjugation quandle
Many examples of quandles comes from groups. Example If G is a group and n is a natural number, then the set G equipped with the binary operations
a ∗ b = b−nabn, a ∗−1 b = bnab−n,
gives a quandle structure on G called the n-conjugation quandle, denoted by Conjn(G). If n = 1, then we shall call this quandle the conjugation quandle and write Conj(G).
If G is abelian group, then Conj(G) is a trivial quandle.
V. Bardakov (Sobolev Institute of Math.) INTRODUCTION TO ALGEBRAIC THEORY OF QUANDLESAugust 27-30, 2019 14 / 53 Examples of quandles: core quandle
Example If G is a group, then the set G equipped with the binary operations
a ∗ b = ba−1b, a ∗−1 b = ba−1b
gives a quandle structure on G called the core quandle, denoted by Core(G).
On the core quandles the operation ∗ is the same as the operation ∗−1 we shall call these quandles by involutory quandles.
V. Bardakov (Sobolev Institute of Math.) INTRODUCTION TO ALGEBRAIC THEORY OF QUANDLESAugust 27-30, 2019 15 / 53 Examples of quandles: Takasaki quandle
Example If G is an abelian group, then the core quandle is the Takasaki quandle that is the quandle with the binary operation a ∗ b = 2b − a, where we use additive operation in G. The Takasaki quandle is denoted by T ak(G).
For G = Z/nZ, T ak(G) is called a dihedral quandle and is denoted by Rn.
V. Bardakov (Sobolev Institute of Math.) INTRODUCTION TO ALGEBRAIC THEORY OF QUANDLESAugust 27-30, 2019 16 / 53 Geometric interpretation of the dihedral quandle
Take the regular n-gon and denote its vertices by a0, a1, ..., an−1, going in the counter-clockwise direction.
The operation on the quandle Rn we can present as reflection of this n-gon, i.e. ai ∗ aj is the vertex which one get from ai using reflection of n-gon under the line which goes across aj and is the axis of symmetry of n-gon.
V. Bardakov (Sobolev Institute of Math.) INTRODUCTION TO ALGEBRAIC THEORY OF QUANDLESAugust 27-30, 2019 17 / 53 Geometric interpretation of dihedral quandles
Figure: ai ∗ aj = aiSaj
V. Bardakov (Sobolev Institute of Math.) INTRODUCTION TO ALGEBRAIC THEORY OF QUANDLESAugust 27-30, 2019 18 / 53 Geometric interpretations for the Takasaki quandle on the circle S1
Figure: Quandle operation on S1
V. Bardakov (Sobolev Institute of Math.) INTRODUCTION TO ALGEBRAIC THEORY OF QUANDLESAugust 27-30, 2019 19 / 53 Geometric interpretations for the Takasaki quandle on the set of real numbers R
Figure: Quandle operation on R
V. Bardakov (Sobolev Institute of Math.) INTRODUCTION TO ALGEBRAIC THEORY OF QUANDLESAugust 27-30, 2019 20 / 53 Symmetric space
The following definition is used in differential geometry Definition A symmetric space in the sense of Loos is a manifold M endowed with a multiplication ∗ : M × M → M, satisfying the following four conditions: 1) x ∗ x = x; 2) (y ∗ x) ∗ x = y; 3) (y ∗ z) ∗ x = (y ∗ x) ∗ (z ∗ x); 4) every point x ∈ M has a neighbourhood U such that y ∗ x = y implies y = x for all y ∈ U.
O. Loos, ”Symmetric spaces”, 1–2, Benjamin (1969).
V. Bardakov (Sobolev Institute of Math.) INTRODUCTION TO ALGEBRAIC THEORY OF QUANDLESAugust 27-30, 2019 21 / 53 Examples of quandles: Alexander quandle
Example −1 Let M be a module over the ring of Laurent polynomials Z t, t . Then the Alexander quandle At(M) is the set M with the quandle operation given by a ∗ b = ta + (1 − t)b. If we put t = −1, then we get the Takasaki quandle.
V. Bardakov (Sobolev Institute of Math.) INTRODUCTION TO ALGEBRAIC THEORY OF QUANDLESAugust 27-30, 2019 22 / 53 Examples of quandles: Joyce-Matveev quandle
Example If G is a group and ϕ ∈ Aut(G), then the set G with binary operation a ∗ b = ϕ(ab−1)b gives a quandle structure on G, which we denote by Jϕ(G) and call the Joice-Matveev quandle.
V. Bardakov (Sobolev Institute of Math.) INTRODUCTION TO ALGEBRAIC THEORY OF QUANDLESAugust 27-30, 2019 23 / 53 Connections between some quandles
If ϕ = bbn is the conjugation by bn, then
a ∗ b = ϕ(ab−1)b = b−nabn,
and we get the n-conjugation quandle, i.e. in this case Jϕ(G) = Conjn(G) . Let G be an abelian group. If ϕ = −I, where I is the identity automorphism of G, then a ∗ b = 2b − a. Thus we see that Jϕ(G) = T ak(G), the Takasaki quandle of G. On the other hand, for arbitrary α ∈ Aut(G), a ∗ b = α(a) + (I − α)(b). Thus, in this case, Jϕ(G) = Aϕ(G) is the Alexander quandle.
V. Bardakov (Sobolev Institute of Math.) INTRODUCTION TO ALGEBRAIC THEORY OF QUANDLESAugust 27-30, 2019 24 / 53 3-element quandles
On 3-element set {1, 2, 3} there exist three non-isomorphic quandles with multiplication tables:
T3 1 2 3 R3 1 2 3 Cz(4) 1 2 3 1 1 1 1 1 1 3 2 1 1 1 1 , , , 2 2 2 2 2 3 2 1 2 3 2 2 3 3 3 3 3 2 1 3 3 2 3 3
where T3 is the trivial quandle, R3 is the dihedral quandle. Note that there exist 7 non-isomorphic 4-element quandles, 22 non-isomorphic 5-element quandles, 73 non-isomorphic 6-element quandles, 298 non-isomorphic 7-element quandles.
V. Bardakov (Sobolev Institute of Math.) INTRODUCTION TO ALGEBRAIC THEORY OF QUANDLESAugust 27-30, 2019 25 / 53 What operations define rack (quandles) in arbitrary group?
We have seen that the conjugation quandle and the core quandle are defined by the words
−1 −1 w1(a, b) = b ab and w2(a, b) = ba b,
respectively, in arbitrary group G. We can formulate
Question Let w = w(x, y) be a word, which is a reduced word in the free group F2 = F2(x, y). Under what conditions for arbitrary group G the algebraic system (G, ∗w) with binary operation
g ∗w h = w(g, h)
is a rack (quandle)?
V. Bardakov (Sobolev Institute of Math.) INTRODUCTION TO ALGEBRAIC THEORY OF QUANDLESAugust 27-30, 2019 26 / 53 Verbal quandles
If the algebraic system (G, ∗w) is a rack (respectively, quandle), then we call this rack (respectively, quandle) a verbal rack (respectively, verbal quandle) defined by the word w.
Proposition (V. B. – T. Nasybullov – M. Singh)
Let w = w(x, y) ∈ F (x, y) be such that Q = (G, ∗w) is a rack for every group G. Then, in fact, Q is a quandle, and
−1 −n n w(x, y) = yx y or w(x, y) = y xy for some n ∈ Z.
V. Bardakov (Sobolev Institute of Math.) INTRODUCTION TO ALGEBRAIC THEORY OF QUANDLESAugust 27-30, 2019 27 / 53 Verbal quandles
As we seen in examples of quandles, there exist quandles which can be defined by words with parameters. For example, the Joyce-Matveev quandle is defined by the word w = w(x, y, z) = z−1(xy−1)zy. Research problem
Find reduced words w = w(x, y, z1, z2, . . . , zm) in the free product F2 ∗ Fm such that (F2 ∗ Fm, ∗w) is a rack (quandle).
Let G be a class of groups (abelian, nilpotent, solvable etc.) Research problem
Find reduced group words w = w(x, y) such that (G, ∗w) is a rack (quandle) for any G ∈ G.
V. Bardakov (Sobolev Institute of Math.) INTRODUCTION TO ALGEBRAIC THEORY OF QUANDLESAugust 27-30, 2019 28 / 53 Subquandles and quandle homomorphisms
Definition A subset Y of a quandle (X, ∗) is called a subquandle of X if Y also forms a quandle under the operation ∗.
Definition. Let (X, ∗) and (Y, ·) be two quandles. A map f : X → Y is called a quandle homomorphism if f(x ∗ y) = f(x) · f(y) for all x, y ∈ X.
We denote by Homqnd(X,Y ) the set of quandle homomorphisms from X to Y .
V. Bardakov (Sobolev Institute of Math.) INTRODUCTION TO ALGEBRAIC THEORY OF QUANDLESAugust 27-30, 2019 29 / 53 Application of quandle homomorphisms
Using homomorphism of quandles it is possible to prove that some knots are non-trivial. Example The fundamental quandle of the trefoil knot has the presentation
Qtr = ha, b, c | a ∗ b = c, b ∗ c = a, c ∗ a = bi.
To prove that the trefoil knot is not equivalent to the trivial knot, we can see that the map
a 7→ a0, b 7→ a1, c 7→ a2
define the epimorphism Qtr → R3 to the dihedral quandle R3 = {a0, a1, a2}, which contains more than one element. As we know the fundamental quandle of the trivial knot is T1, the trivial 1-element quandle.
V. Bardakov (Sobolev Institute of Math.) INTRODUCTION TO ALGEBRAIC THEORY OF QUANDLESAugust 27-30, 2019 30 / 53 Fibonacci groups
The Fibonacci groups:
F (2, n) = hx0, x1, . . . , xn−1 | x0x1 = x2, x1x2 = x3,...,
xn−2xn−1 = x0, xn−1x0 = x1i, were introduced by J. H. Conway (Advanced problem 5327, Amer Math. Monthly, 72 (1965), 915), who formulated a question about the order of F (2, 5). For n = 2m and m ≥ 4, these groups are fundamental groups of certain hyperbolic manifolds, F (2, 9) is hyperbolic. The group F (2, 6) is known to contain a free abelian subgroup of rank 2 and so is not hyperbolic. All other groups, when n < 8, are finite.
V. Bardakov (Sobolev Institute of Math.) INTRODUCTION TO ALGEBRAIC THEORY OF QUANDLESAugust 27-30, 2019 31 / 53 Fibonacci quandles
By analogy with Fibonacci groups let us introduce
Definition A Fibonacci quandle F ib(n), n ≥ 2 is a quandle that have a presentation
hf0, f1, . . . , fn−1 | f0 ∗ f1 = f2, . . . fn−2 ∗ fn−1 = f0, fn−1 ∗ f0 = f1i.
It is easy to see that the fundamental quandle of the trefoil knot Qtr is isomorphic to F ib(3).
V. Bardakov (Sobolev Institute of Math.) INTRODUCTION TO ALGEBRAIC THEORY OF QUANDLESAugust 27-30, 2019 32 / 53 Research problems
Research problems 1) What can we say on properties of F ib(n)?
2) For what n the Fibonacci quandle F ib(n) is a fundamental quandle of some link?
V. Bardakov (Sobolev Institute of Math.) INTRODUCTION TO ALGEBRAIC THEORY OF QUANDLESAugust 27-30, 2019 33 / 53 Abelian quandles
Definition An abelian quandle is a quandle Q satisfying the identity
(w ∗ x) ∗ (y ∗ z) = (w ∗ y) ∗ (x ∗ z) for all x, y, z, w ∈ Q.
Example Let T be an invertible linear transformation on a vector space V . V becomes an abelian quandle with the operations
x ∗ y = T (x − y) + y and x ∗−1 y = T −1(x − y) + y.
V. Bardakov (Sobolev Institute of Math.) INTRODUCTION TO ALGEBRAIC THEORY OF QUANDLESAugust 27-30, 2019 34 / 53 Involutory quandles
Definition An involutory quandle is a quandle Q which satisfies the identity
(x ∗ y) ∗ y = x for all x, y ∈ Q.
In involutory quandle
x ∗ y = x ∗−1 y.
V. Bardakov (Sobolev Institute of Math.) INTRODUCTION TO ALGEBRAIC THEORY OF QUANDLESAugust 27-30, 2019 35 / 53 Examples of abelian and involutory quandles
The following proposition is trivial.
Proposition 1) Any Alexander quandle is abelian. 2) For any group G the core quandle Core(G) is involutory.
V. Bardakov (Sobolev Institute of Math.) INTRODUCTION TO ALGEBRAIC THEORY OF QUANDLESAugust 27-30, 2019 36 / 53 n-quandles
A generalization of involutory quandles are n-quandles.
Definition n n For a natural n let x ∗ y = xSy . An n-quandle is a quandle satisfying the identity x ∗n y = x.
Hence, a 1-quandle is a trivial quandle, 2-quandle is an involutory quandle.
V. Bardakov (Sobolev Institute of Math.) INTRODUCTION TO ALGEBRAIC THEORY OF QUANDLESAugust 27-30, 2019 37 / 53 Quandle automorphisms
The set of all automorphisms of a quandle X is denoted by Aut(X).
As we noted, any symmetry Sa, a ∈ X, is an automorphism of X.
Definition The subgroup of Aut(X), which is generated by all symmetries is called the subgroup of inner automorphisms and is denoted by Inn(X). −1 The subgroup of Inn(X) generated by automorphisms of the form SxSy for x, y in X, is called the transvection group of X and is denoted by Trans(X).
V. Bardakov (Sobolev Institute of Math.) INTRODUCTION TO ALGEBRAIC THEORY OF QUANDLESAugust 27-30, 2019 38 / 53 Connections between Aut(X), Inn(X), and Trans(X)
Inn(X) is a normal subgroup of Aut(X), and Trans(X) is normal in both Inn(X) and Aut(X), i. e. we have the normal series
Trans(X) ¢ Inn(X) ¢ Aut(X).
The quotient group Inn(X)/Trans(X) is a cyclic group. The elements of Trans(X) are the automorphisms of the form
ε1 εn Sx1 ...Sxn such that e1 + ... + en = 0.
V. Bardakov (Sobolev Institute of Math.) INTRODUCTION TO ALGEBRAIC THEORY OF QUANDLESAugust 27-30, 2019 39 / 53 Symmetries and Inn(X)
The following example shows that the set Inn(X) is bigger than the set of all symmetries {Sb | b ∈ X}.
Example
Consider the dihedral quandle R3 = {a0, a1, a2}. The set of symmetrises contains only three elements, but the group Inn(R3) is equal to Aut(R3) and is the full permutation group of 3 symbols a0, a1, a2.
V. Bardakov (Sobolev Institute of Math.) INTRODUCTION TO ALGEBRAIC THEORY OF QUANDLESAugust 27-30, 2019 40 / 53 Automorphisms of racks and quandles
Question Let X be a rack (quandle). Find the groups Aut(X) and Inn(X).
If X = Rn is the dihedral quandle, then these groups were found in the paper: M. Elhamdadi, J. Macquarrie, and R. Restrepo, Automorphism groups of quandle, Journal of Algebra and Its Applications, 11, No. 1 (2012).
V. Bardakov (Sobolev Institute of Math.) INTRODUCTION TO ALGEBRAIC THEORY OF QUANDLESAugust 27-30, 2019 41 / 53 Adjoint group
Definition Given a quandle X, the adjoint group Ad(X) is a group that is generated −1 by ex, x ∈ X and is defined by the relations ey exey = ex∗y with x, y ∈ X. In other words, Ad(X) has the presentation
−1 Ad(X) = hex, x ∈ X | ey exey = ex∗y, x, y ∈ Xi.
The name “adjoint group” is not generally accepted. Some authors call this group by “associated group” or “enveloping group”.
V. Bardakov (Sobolev Institute of Math.) INTRODUCTION TO ALGEBRAIC THEORY OF QUANDLESAugust 27-30, 2019 42 / 53 Examples of adjoint groups
Example |T | 1) If T is a trivial quandle, then Ad(T ) = Z .
2) If K is a knot and QK is its fundamental quandle, then Ad(QK ) is 3 the knot group GK = π1(S \ K), that is the fundamental group of the 3 compliment of K in 3-sphere S .
V. Bardakov (Sobolev Institute of Math.) INTRODUCTION TO ALGEBRAIC THEORY OF QUANDLESAugust 27-30, 2019 43 / 53 Adjoint functor
Note that every quandle homomorphism X → Y canonically induces a group homomorphism Ad(X) → Ad(Y ). Hence, we can regard Ad as a functor from the category of quandles to the category of groups:
Ad : Quand → Group.
The functor Ad is the left adjoint to the functor
Conj : Group → Quand.
It means that for any quandle X and any group G, there is a natural bijection Homgr(Ad(X),G) ' Homqnd(X, Conj(G)).
V. Bardakov (Sobolev Institute of Math.) INTRODUCTION TO ALGEBRAIC THEORY OF QUANDLESAugust 27-30, 2019 44 / 53 Universal property
The function η : Q → Conj(Ad(Q)) sending x to ex is a quandle homomorphism whose image is a union of conjugacy classes of Ad(Q). Also, η has the universal property that for any quandle homomorphism h : Q → Conj(G), G a group, there exists a unique group homomorphism H : Ad(Q) → G such that h = H ◦ η.
V. Bardakov (Sobolev Institute of Math.) INTRODUCTION TO ALGEBRAIC THEORY OF QUANDLESAugust 27-30, 2019 45 / 53 Properties of η
Let us show that η need not be injective in general. Example Consider the 3-element involutory quandle Cs(4), determined by the relations a ∗ b = c, c ∗ b = a, b ∗ a = b ∗ c = b. −1 Since b ∗ a = b, ea commutes with eb. But a ∗ b = c, so eb eaeb = ec. Therefore, ea = ec, and η is not injective.
V. Bardakov (Sobolev Institute of Math.) INTRODUCTION TO ALGEBRAIC THEORY OF QUANDLESAugust 27-30, 2019 46 / 53 Verbal algebraic systems
We know that there exists some possibilities to construct quandle by group. If w = w(x, y) is a word in the free group F (x, y) on generators x and y, then for any group G, the word w defines a map G × G → G given by (g, h) 7→ w(g, h) for g, h ∈ G. Setting
g ∗w h = w(g, h)
gives an algebraic system (G, ∗w). We know conditions under which this algebraic system is a quandle.
V. Bardakov (Sobolev Institute of Math.) INTRODUCTION TO ALGEBRAIC THEORY OF QUANDLESAugust 27-30, 2019 47 / 53 Some analogous of adjoint groups
Suppose that the algebraic system (G, ∗w) is a quandle X, we define the group
Adw(X) = ex, x ∈ X | ex∗y = w(ex, ey), x, y ∈ X .
−1 We note that if w(x, y) = y xy, then Adw(X) is simply denoted by Ad(X).
V. Bardakov (Sobolev Institute of Math.) INTRODUCTION TO ALGEBRAIC THEORY OF QUANDLESAugust 27-30, 2019 48 / 53 Left adjoint functors
Proposition (V. B – M. Singh) −1 −n n Let w(x, y) = yx y or y xy for some n ∈ Z. Then
Adw : Quand → Group
is a functor that is left adjoint to the functor Qw : Group → Quand.
V. Bardakov (Sobolev Institute of Math.) INTRODUCTION TO ALGEBRAIC THEORY OF QUANDLESAugust 27-30, 2019 49 / 53 Research problems
Research problems Let w(x, y) = yx−1y and X be a quandle (a dihedral quandle or link quandle). What can we say on groups Adw(X)?
The adjoint group Ad(Qtr) of the fundamental quandle for the trefoil knot Qtr is the braid group B3 on 3 strings.
Research problem
What is the adjoint groups Adw(F ib(n)) of the Fibonacci quandle F ib(n), n > 1?
These groups Adw(X) were defined in 2020 and I don’t know results in this direction.
V. Bardakov (Sobolev Institute of Math.) INTRODUCTION TO ALGEBRAIC THEORY OF QUANDLESAugust 27-30, 2019 50 / 53 Structure of Ad(X)
Suppose that X be an arbitrary quandle. Then the adjoint quandle Ad(X) is generated by elements ex, x ∈ X. Using induction on the length of g ∈ Ad(X) it is not difficult to check the equality
−1 ex·g = g exg ∈ Ad(X), x ∈ X, g ∈ Ad(X).
V. Bardakov (Sobolev Institute of Math.) INTRODUCTION TO ALGEBRAIC THEORY OF QUANDLESAugust 27-30, 2019 51 / 53 Ad(X) as a central extension
This action of Ad(X) on X gives a group epimorphism
ψX : Ad(X) → Inn(X), ψX (x) = Sx.
Proposition There exists the short exact sequence
0 → Ker(ψX ) → Ad(X) → Inn(X) → 0,
where Ker(ψX ) lies in the center of Ad(X).
V. Bardakov (Sobolev Institute of Math.) INTRODUCTION TO ALGEBRAIC THEORY OF QUANDLESAugust 27-30, 2019 52 / 53 Thank you!
V. Bardakov (Sobolev Institute of Math.) INTRODUCTION TO ALGEBRAIC THEORY OF QUANDLESAugust 27-30, 2019 53 / 53