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What Gauss Knew About Knots & Braids

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What Gauss Knew About Knots & Braids What Gauss Knew about Knots & Braids Minh-Tâm Quang Trinh Massachusetts Institute of Technology This talk is inspired by the paper: In modern language, we’ll be interested in KNOTS and their more general cousins, LINKS. M. Epple. Orbits of Asteroids, a Braid, and the First Link Invariant. Math. Intelligencer, 20(1) (1998), 45–52. Some knot (diagram)s: It discusses: 1 Gauss’s integral to compute the linking number of two closed curves. 2 Gauss’s sketch of a 4-strand braid. Some link (diagram)s: We’ll go further by discussing how these ideas evolved in the 20th and 21st centuries. We’ll also use many pictures from Introduction to Vassiliev (What mathematicians call a knot is what sailors and Knot Invariants by Chmutov–Duzhin–Mostovoy. climbers would call a grommet.) 1 Perhaps the simplest kind of knot is an unknot: A knot is the image of a (smooth) injective map from the circle S1 into 3-space R3. Two such maps u, v : S1 R3 are isotopic iff there exists a ! (smooth) map Below are left- and right-handed trefoils, respectively: ) : S1 0, 1 R3 × » ¼ ! such that: 1 3 )t = ) , t : S R defines a knot for all t 0, 1 . • (− º ! 2 » ¼ )0 = u and ) = v. • 1 Are the trefoils isotopic to the unknot? To each other? Then we say that the associated knots are isotopic as well. Thm (Reidemeister) Two knots are isotopic iff they differ by some sequence of the following local “moves”: 2 Let’s say that a knot is tricolor iff, in some diagram, each A link of n components is the image of an injective map arc can be colored so that Globally, at least two colors are used. S1 S1 R3. • | t{z · · · t } ! At each crossing, either the three arcs are all the same n copies • color, or they each have a different color. Again, we’d like a way to show that various links are not isotopic. An unknot is not tricolor, while a trefoil is. Let’s restrict attention to links of 2 components. If the components are oriented, then we have an intuitive idea of the number of times they intertwine: Tricolorability is preserved by Reidemeister’s moves. should have linking numbers 0, 1, 2, and 1, respectively. This proves that the trefoils are not isotopic to the unknot. − 3 Gauss (1777-1855) lived a century before Reidemeister Apparently, Gauss was led to this integral by his study of (1893-1971). celestial mechanics (and later, electromagnetic induction). Back then, people didn’t bother to define isotopy But his key insight is purely mathematical. We’ll describe rigorously. Topology was still called geometria situs. it with modern machinery. Yet in 1833, Gauss found a precise definition for the As setup, define (ordered) 2-point configuration space to be linking number K1, K2 of an oriented link K1 K2. 3 3 h i t Conf = v , v2 R R : v < v2 . 2 f¹ 1 º 2 × 1 g 2 3 Def-Thm (Gauss) If K1, K2 admit parametrizations Define the Gauss map Γ : Conf 2 S R by 1 3 ! ⊆ v1, v2 : S R , then ! v2 v1 Γ v1, v2 = − . 1 ¼ ¼ v v ¹ º v2 v1 K , K = 2 1 dv dv . j − j 1 2 − 3 1 2 h i 4 v2 v1 · ¹ × º K1 K2 That is, Γ v , v2 is the unit vector pointing from v to v2. j − j ¹ 1 º 1 4 To explain: Gauss’s integrand is the Jacobian of Γ , so Now, a 2-component link v1,v2 1 º 1 1 3 Γ v1 v2 : S S R Jac v1,v2 t t ! 4 S1 S1 ¹ º × gives rise to a composite map is a ratio of signed surface areas—roughly, the number of 1 1 2 v v Γ times that S S wraps over its image in S . 1 1 1 × 2 2 × Γv ,v : S S Conf S 1 2 × −−−−−! 2 −! The preimage of a generic small circle in S2 is a disjoint union of circles in S1 S1: × Both the torus S1 S1 and the unit sphere S2 are closed × oriented surfaces. Any map between such surfaces has an integer degree. What Gauss’s integral really computes is deg Γv ,v Z. Each has a signed local degree. 1 2 2 We’ll only give a vague sketch. Γ deg v1,v2 is their sum, which matches the number above. 5 In knot theory, there’s a tension between: Consider the local crossings “Geometric” definitions that don’t rely on a choice of • diagram, e.g., Gauss’s linking integral. “Diagrammatic” definitions, e.g., tricolorability. • V is defined inductively by the local rule In 1928, Alexander found an invariant 2 2 1 q− V L q V L = q q− V L0 Δ : links isotopy Z q ¹ ¸º − ¹ −º ¹ − º ¹ º f g/ ! » ¼ and the base case V unknot = 1. that had both geometric and diagrammatic definitions. ¹ º In 1985, Jones found a different invariant In 1989, Witten gave a surprising geometric definition of V, in terms of path integrals in a quantum field theory. 1 V : links isotopy Z q± f g/ ! » ¼ that only had a diagrammatic definition. Unlike Δ, it Reshetikhin–Turaev showed his formula to be rigorous, could distinguish left- and right-handed trefoils. but only by returning to diagrammatics. 6 3 Thm (D–G–K–M–N–S–V ) Given v , v2, v3 R , let ∗ 1 2 Here’s a more recent integral in knot theory. 1 = v3 v2, 2 = v1 v3, 3 = v2 v1, How can we detect three components that are linked − − − without being pairwise linked? and let Γ v , v2, v3 be the sum ¡¹ 1 º 2 3 2 2 3 3 1 1 × × × 1 . ¸ 2 ¸ 3 ¸ 2 ¸ 2 3 ¸ 3 j 1 j j j j j j 1 jj j j jj j j jj 1 j Then: 3 3 1 Γ= Γ Γ is a map Conf S2. Let Conf = v1, v2, v3 R : v1 < v2 < v3 < v1 . ¡ ¡ 3 3 f¹ º 2 ¹ º g /j j ! 2 2 The “Pontryagin -invariant” of Γv ,v ,v detects the We want a map Γ : Conf 3 S , so that 1 2 3 ! Borromean rings. 1 1 1 3 v v2 v3 : S S S R 1 t t t t ! The blue part only vanishes when v1, v2, v3 is gives rise to 4¹ º equilateral; the green part only vanishes when v1, v2, v3 v v v Γ are collinear. Moreover, they are orthogonal. 1 1 1 1 × 2 × 3 2 Γv ,v ,v : S S S Conf S . 1 2 3 × × −−−−−−−−! 3 −! ∗ DeTurck, Gluck, Komendarczyk, Melvin, Nuchi, Shonkwiler, Vela-Vick (2013) 7 In 1923, before discovering Δ, Alexander had shown that As we’ll explain, the braids on n strands form a group, every link can be “combed” into the closure of a braid. which Emil Artin introduced in 1947. A braid is like a link, but it connects n inputs at one end of Yet the following sketch appears in Gauss’s notebooks a box to n outputs at the other without trackbacks. between 1815 and 1830. Only the red diagram depicts a braid (for n = 3). A braid can be closed up into a link ˆ: Did Gauss hope to construct an isotopy invariant of Different braids can give the same link! braids? 8 A close-up of the table of numbers: Groups lurk implicitly throughout much of Gauss’s work. Strangely, he never took interest in developing an abstract definition. A group is a set G equipped with a binary operation : G G G (−)◦ (−) × ! How do the changes between columns of numbers and an identity 1 G such that: correspond to crossings in the braid? 2 For all x, y, z G, we have x y z = x y z . • 2 ¹ ◦ º ◦ ◦ ¹ ◦ º Epple’s article offers a precise interpretation, after For all x G, we have x 1 = 1 x = x. • 2 ◦ ◦ remarking: 1 Every x G has an inversex − G such that • 1 21 2 “While Gauss was looking for a notation. to decide x x− = x− x = 1. ◦ ◦ whether or not two braids were equivalent, he came close to defining a nontrivial invariant for braids, The symmetric group G = Sn is the set of self-bijections of namely[,] the last row of the table he set up. ” 1,..., n , where is composition of maps and 1 = id. f g ◦ 9 Once you’ve found a set of generators for a group, it The set Brn of isotopy classes of braids on n strands gives remains to find the relations among the “words” they rise to a group, where: generate. The operation is concatenating braids end-to-end. • ◦ Artin showed that Brn only has two types of relations: The identity 1 is the braid that connects inputs to • 1 i j = j i when i j 2: outputs by straight lines, without any crossings. ◦ ◦ j − j ≥ We can generate Brn by the elements 1,..., n 1, where − i is (the isotopy class of) 2 i i 1 i = i 1 i i 1 for all i: ◦ ¸ ◦ ¸ ◦ ◦ ¸ That is, every braid is formed by repeated concatenation 1 of the i and i− in some order. Note that (2) is an analogue of a Reidemeister move. 10 We conclude this talk by describing P. Recall that a braid gives rise to a link closure ˆ. Using the , fix inclusions Br Br Br ... Notably, if , Brn, then it turns out = . i 1 2 3 2 ◦ ◦ ⊆ ⊆ ⊆ So, to study isotopy invariants of links, we might study Thm (Ocneanu) The following rules uniquely determine Z z 1 -linear functions functions f on Brn such that » ± ¼ 1 1 1 f = f trn : Z z± Brn Z a± , z± ¹ ◦ º ¹ ◦ º » ¼» ¼ ! » ¼ such that trn = trn for all n and , Brn: always holds.
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