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Topological Robotics Groups in Brunnian and Our Work

Braids and Links in Mathematical Sciences

Jie Wu

National University of Singapore

July 27, 2010. Summer School in Applied Mathematics Dalian University of Technology Topological Robotics Braid Groups in Mathematics Brunnian Braids and Our Work Braids and Links in Mathematical Sciences

Topological Robotics

Braid Groups in Mathematics

Brunnian Braids and Our Work Topological Robotics Braid Groups in Mathematics Brunnian Braids and Our Work A New Area in Mathematics

• From the coming Oberwolfach meeting on Topological Robotics organized by Michael Farber et al (10-16 Oct. 2010). • Topological robotics is a new mathematical discipline studying topological problems inspired by robotics and engineering as well as problems of practical robotics requiring topological tools. • It is a part of a broader newly created research area called “computational topology”. The latter studies topological problems appearing in computer science and algorithmic problems in topology. Topological Robotics Braid Groups in Mathematics Brunnian Braids and Our Work

Problems of topological robotics can roughly be split into three main categories: (A) studying special topological spaces, configuration spaces of important mechanical systems; (B) studying new topological invariants of general topological spaces, invariants which are motivated and inspired by applications in robotics and engineering; (C) studying of random topological spaces which arise in applications as configuration spaces of large systems of various nature. Topological Robotics Braid Groups in Mathematics Brunnian Braids and Our Work Some topics in their meeting

• Configuration Spaces of graphs. The configuration space F(Γ, n) of n distinct particles moving on a graph Γ can be used to model many important processes in engineering. (a) Discrete models of the configuration space. (b) Nonpositive curvature of the discrete configuration space. (c) The asphericity of F(Γ, n). (d) The upper bound on the homological dimension of the fundamental of F(Γ, n). In (a) one constructs a finite cell complex (in fact, a cubical complex) having the type of the configuration space F(Γ, n) (which is not compact). In part (b) one observes that the discrete model is nonpositively curved in the sense of Gromov which implies its asphericity (c), i.e. vanishing of all homotopy groups πn(F(Γ, n)) with n > 1. Topological Robotics Braid Groups in Mathematics Brunnian Braids and Our Work

• theory of the robot arm. • Classical studies subsets K ⊂ Rn, called , viewed the equivalence relation of . The precise meaning of the word knot depends on the context; most common knots are formed by embeddings of spheres (e.g. circles) or disks, subject to requirements of being smooth, piecewise linear, or locally flat. Unknotting theorems of knot theory state that under specific assumptions various knots are all equivalent to each other. • A robot arm is a mechanism with hinges at its vertices and rigid bars at its edges. The hinges can be folded but the bars must maintain their length and cannot cross. Mechanisms of this kind appear widely in robotics. Related mathematical problems were studied in discrete and computational geometry, in knot theory, in molecular biology and polymer physics. Topological Robotics Braid Groups in Mathematics Brunnian Braids and Our Work

• Motion planning algorithms, TC(X) and Schwartz genus. One may start by describing the motion planning problem of robotics and its interpretation as the problem of finding a section of a fibration. • random graphs. The theory of random graphs is the best developed part of stochastic algebraic topology. Many beautiful and deep results about random graphs are described in the literature. • Random 2-dimensional complexes. generalizations of random graphs. • Random polygon spaces. Topological Robotics Braid Groups in Mathematics Brunnian Braids and Our Work Algebraic Definition of The Artin

• The set of all n-strand braids forms a group, denoted by Bn, generated by σi for 1 ≤ i ≤ n − 1 with the defining relations given by

1. σi σj = σj σi for |i − j| ≥ 2. 2. σi σi+1σi = σi+1σi σi+1.

q • B2 = Z, all 2-strand braids are given by σ1 for q ∈ Z. • B3 is a group generated by σ1 and σ2 subject to the relation σ1σ2σ1 = σ2σ1σ2. Topological Robotics Braid Groups in Mathematics Brunnian Braids and Our Work Geometric Definition of Braid Groups

• Let M be a (topological) space. The ordered configuration space n F(M, n) = {(x1, x2,..., xn) ∈ M | xi 6= xj for i 6= j}. • The unordered configuration space B(M, n) = F(M, n)/Sn the orbit space, where Sn acts on F(M, n) by permuting coordinates.

• The braid group over M: Bn(M) = π1(B(M, n)), the of B(M, n). • Intuitive description: a loop in B(M, n) lifts to a path λ(t) = (λ1(t), . . . , λn(t)) in F(M, n) with the ending λ(1) = (λ1(1), . . . , λn(1)) is a coordinate of the fixed basepoint λ(0) = (q1, ··· , qn).A braid over M can be described by the graph of λi (t) for 1 ≤ i ≤ n. ∼ 2 ∼ 2 • The Artin braid group: Bn = Bn(D ) = Bn(R ). Topological Robotics Braid Groups in Mathematics Brunnian Braids and Our Work Braids and Links A is the union of mutually disjoint simple closed (polygonal) curves in R3. Markov Theorem. ANY link can be obtained by closing up a braid. Topological Robotics Braid Groups in Mathematics Brunnian Braids and Our Work Braids and Polynomials • Given an n-strand braid, by taking the intersection of the strands with the planes parallel to the XY -plane, one obtains a path

(z1(t), z2(t),..., zn(t))

with zi (t) 6= zj (t) for each 0 ≤ t ≤ 1 and any i 6= j, and so a polynomial

f (x)(t) = (x − z1(t))(x − z2(t)) ··· (x − zn(t))

with parameter 0 ≤ t ≤ 1.

• From this, the braid group Bn is the fundamental group of the space of all monic polynomials of degree n with n distinct roots. • The braid groups are used to study complexity of approximating roots of polynomials. Topological Robotics Braid Groups in Mathematics Brunnian Braids and Our Work Braids and Mapping Classes

• Consider the disk D2 with n distinct punctured points 2 2 2 Qn = {q1,..., qn} in D r ∂D . Let Mn(D ) be the space 2 of all self diffeomorphisms f of D such that f |∂D2 = id∂D2 (i.e f fixes the boundary points pointwise) and f (Qn) = Qn (i.e. f permutes the punctured points q1, q2,..., qn. ∼ 2 • The Artin braid group: Bn = π0(Mn(D )), the 2 path-connected components of Mn(D ) with multiplication induced the composition operation. • In general, for any 2- (surface) S, the group of the path-connected components of the self-diffeomorphisms of S that fix boundary points pointwise and permute the punctured points is called , which is related to braids on S. • Mapping Class Groups ! Heegaard splittings. Dalian is strong on Heegaard splitting (Professor Lei etc). Topological Robotics Braid Groups in Mathematics Brunnian Braids and Our Work The Braids and the Automorphisms of Free Groups

• Let Fn be the of rank n and let Aut(Fn) be the group of the automorphisms of Fn. • The Artin Representation means the φ: Bn → Aut(Fn) given by the action:  xi+1 if j = i  −1 σi · xj = xi+1xi xi+1 if j = i + 1  xj otherwise.

• Artin’s Theorem: The homomorphism φ is a monomorphism (i.e. Artin’s representation is faithful) and φ(Bn) is given by the automorphisms β of Fn with the −1 property that for each i β(xi ) = w xj w for some word w and some j, and β(x1x2 ··· xn) = x1x2 ··· xn. Topological Robotics Braid Groups in Mathematics Brunnian Braids and Our Work Scientific Meaning of Braids

A braid is a locus of points ( or particles or robots or birds... ) moving through time without colliding. • air traffic control problem. • satellites in sky. • many robots working a place. Applications: • robotics. • Medical surgery planning. • molecular biology. • cryptanalysis. • solar activity. Topological Robotics Braid Groups in Mathematics Brunnian Braids and Our Work A robot moving between m obstacles gives an m+1 stranded braid Robot Motion Planning t i m e Topological Robotics Braid Groups in Mathematics Brunnian Braids and Our Work Pure Braids A pure braid means a braid in which each strand has the same ending point as the initial point. The set of n-strand pure braids forms a of Bn, denoted by Pn. The group Pn is generated by 2 −1 −1 −1 Ai,j = σj−1σj−2 ··· σi+1σi σi+1 ··· σj−2σj−1 for 1 ≤ i < j ≤ n.

i j Topological Robotics Braid Groups in Mathematics Brunnian Braids and Our Work Brunnian Braids A braid is called Brunnian if it becomes a trivial braid after removing ANY of its strands.

-1 3 (σ1σ2 ) is Brunnian Topological Robotics Braid Groups in Mathematics Brunnian Braids and Our Work Example of n-strand Brunnian braids −2 −1 2 Given any (n − 1)-strand Brunnian braid β, σn−1β σn−1β is an n-strand Brunnian braid.

-2 σn-1

β-1 2 σn-1 β Topological Robotics Braid Groups in Mathematics Brunnian Braids and Our Work Homotopy Groups

• For any (topological) space, the n πn(X) = [S , X] the set of the (pointed) homotopy classes of (pointed) continuous maps from the n-sphere Sn to X. • Some Breakthroughs on homotopy groups: 1. Hurewicz (Hurewicz Theorem). n 2. Serre : πq(S ) is finite except q = n, or q = 2n − 1 for n even. 3. EHP-sequence (Whitehead, James) and Toda’s computation. 4. Adams Spectral Sequences, Adams-Novikov Spectral Sequences. 5. Cohen-Moore-Neisendorfer Theorem: n 2n+1 p · Torp(π∗(S )) = 0 for p odd prime. Topological Robotics Braid Groups in Mathematics Brunnian Braids and Our Work One of Our Main Results

• The n-th homotopy group of the sphere is given by the quotient of the (n + 1)-strand Brunnian braid group over the sphere modulo the (n + 1)-strand Brunnian braid group over the disk for n ≥ 4. • This gives a connection between (general) homotopy groups and the braid groups, which received 2007 National Science Award of Singapore. • The determination of the general homotopy groups is one of the central problems in the area of algebraic topology, and one of most challenging problems in mathematics. • In our subsequent works, we (joint with Prof. Fuquan Fang, Prof. Fengchun Lei and Dr. Jingyan Li) also established some fundamental connections between (general) homotopy groups and link groups. Topological Robotics Braid Groups in Mathematics Brunnian Braids and Our Work

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