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 2 X there is a unique c 2 X such that a = c ∗ b; (R2) Self-distributivity: (a ∗ b) ∗ c = (a ∗ c) ∗ (b ∗ c) for all a; b; c 2 X. A quandle X is a rack which satisfies the following condition: (Q1) a ∗ a = a for all a 2 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 2 X define a map Sa : X −! X, which sends any element x 2 X to element x ∗ a, i. e. xSa = x ∗ a. We shall call Sa the symmetry at a. Proposition For every a 2 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 2 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 2 Q, (R2) (a ∗ b) ∗ c = (a ∗ c) ∗ (b ∗ c) for all a; b; c 2 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 2 X can be presented in a form "1 "2 "3 "m−1 u = (::: ((ai1 ∗ ai2 ) ∗ ai3 ) ∗ :::) ∗ aim ; aij 2 A; "k 2 {−1; 1g: 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 2 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 = feg 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 2 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 2 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 2 M has a neighbourhood U such that y ∗ x = y implies y = x for all y 2 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 ' 2 Aut(G), then the set G with binary operation a ∗ b = '(ab−1)b gives a quandle structure on G, which we denote by J'(G) and call the Joice-Matveev quandle.