1 BRAIDS AND THEIR SEIFERT SURFACES Andrew Ranicki (Edinburgh) http://www.maths.ed.ac.uk/ � aar Drawings by Carmen Rovi NUI, Maynooth 7th May, 2014
2 A braid in the Book of Kells
3 The mathematical definition of a braid ◮ Fix n � 2 and n distinct points z 1 , z 2 , . . . , z n ∈ D 2 . ◮ An n -strand braid β is an embedding � I = { 1 , 2 , . . . , n } × I ⊂ D 2 × I ; ( k , t ) �→ β ( k , t ) β : n such that each of the composites β ( k , − ) � D 2 × I projection � I (1 � k � n ) I is a homeomorphism, and β ( k , 0) = ( z k , 0) ∈ D 2 ×{ 0 } , β ( k , 1) = ( z σ ( k ) , 1) ∈ D 2 ×{ 1 } for a permutation σ ∈ Σ n of { 1 , 2 , . . . , n } . ◮ β defines n disjoint forward paths t �→ β ( k , t ) in D 2 × I ⊂ R 3 from ( z k , 0) to ( z σ ( k ) , 1), such that each section β ( { 1 , 2 , . . . , n } × I ) ∩ ( D 2 × { t } ) ( t ∈ I ) consists of n points.
4 An example of a 3-strand braid with σ = (132) Z Z Z 1 2 3 2 D X {0} β 2 D X I 2 D X {1} Z Z Z 2 3 1
5 A braid drawn by Gauss (1833) ◮ ◮ Further 19th century developments: Listing, Tait, Hurwitz. ◮ See Moritz Epple’s history paper Orbits of asteroids, a braid, and the first link invariant, Mathematical Intelligencer, 20, 45-52 (1996)
6 Concatenation of braids ◮ The concatenation of n -strand braids β, β ′ is the n -strand braid � I ⊂ D 2 × I ⊂ R 3 β ′ β : n defined by � β ( k , 2 t ) if 0 � t � 1 / 2 ( β ′ β )( k , t ) = β ′ ( k , 2 t − 1) if 1 / 2 � t � 1 for 1 � k � n , t ∈ I , with permutation the composite σ ′ σ .
7 An example of concatenation Z Z Z 1 2 3 β β ’ β β ’ Z Z Z 1 2 3
8 Isotopy of braids ◮ Two n -strand braids � I ⊂ D 2 × I β 0 , β 1 : n are isotopic if there exist braids � I ⊂ D 2 × I ( s ∈ I ) β s : n such that the function � I → D 2 × I ; ( s , k , t ) �→ β s ( k , t ) I × n is continuous. ◮ Same permutations σ 0 = σ s = σ 1 ∈ Σ n . ◮ Braid applet
Theorie der Z6pfe. 49 aus den Definitionen hervorgeht, daft bei diesem Prozess der ite Faden yon Z~ nicht notwendig mit dem i t~ Faden yon Z~ zu verkntipfen ist. 1st vielmehr #~ die Verbindung yon A~ und B,.,, so hat man ja B,., mit dem Punkt A', yon Z2 zusammenfallen zu lassen, so daft ~i mit dem Faden #', yon Z~ verkniipft wird. In Fig. 2 ist z. B. der erste Faden 9 yon Z~ mit dem ch'itten Faden yon Z~ verbunden. Das assoziative Gesetz Artin ◮ Emil Artin founded the modern theory of braids in Theorie der (1) (z, z,) -- (z, z,) Z¨ opfe (1925), defining the n -strand braid group B n : the set of isotopy classes of n -strand braids under concatenation. ffir unsere Komposition leuchtet unmittelbar ein. Denn offenbar erscheint ◮ A trivial braid, a braid with an overcrossing and a braid with derselbe Zopf, wenn man an Z~ den bereits verknfipften Z~Z..~ anh~tngt an undercrossing oder abet an ZL den Zopf Z2 und an das Kompositionsresultat Z..~. Dagegen ist im allgemeinen die Reihenfolge von Z~ �9 undZ~ wesentlich' d'h" es gilt nicht das k~ l m Gesetz. u t a t i v e l l l Die einfachsten Typen von Z0pfen T/ter Ordnung sind in Fig. 3 dargestellt. Wit haben: I X ] " 1. Den Zopf E, bei dem der Punkt Ai mit Bi verbunde~ ist und die F~den t~ miteinander nicht verschlungen sind. (Bei passender Deformation schneiden sich dann dieProjektionen unsererKurven l X ] ~ nicht.) Ersichtlich gilt, wenn Z ein beliebiger Zopf ist: Fig. 3. zE = EZ Z. Unser Zorf E spielt also die Rolle der Einheit und werde deshalb auch einfach mit 1 bezeichnet. 2. Der Zopf (ri, bei dem A~ mit B~+I und Ai+l mit Bi verbunden ist. wobei der z ~e Faden einmal /tber dem (i+ 1) ten Faden li~uft, die iibrigen Faden aber wie bei E laufen. (Also unverschlungen yon A,. nach Br.) 3. Der Zopf %-1, b ei dem derselbe Sachverhalt wie bei a/ vorliegt, nur dal3 der ite Faden einmal unter dem (i+ 1) t~n lauft. Komponiert man den Zopf ~ mit ai -1, so kann man den ~en Faden v0m (i~-1) ten herunterheben, erhiilt also den Zopf E. Ebenso wenn a-i mit % komponiert wird. Es gilt also: (3) a i. a. -1 ----- a: a. a. = 1. Aus diesem Grunde wurde der dritte Typus a/-~ genannt.
10 The n -strand braids σ 0 , σ 1 , . . . , σ n − 1 ◮ The trivial n -strand braid is � I ⊂ D 2 × I ; t i �→ ( z i , t i ) σ 0 : n i i i+1 i+1 ◮ For i = 1 , 2 , . . . , n − 1 the elementary n -strand braid σ i is obtained from σ 0 by introducing an overcrossing of the i th strand and the ( i + 1)th strand, with permutation ( i i + 1) ∈ Σ n . i i+1 i+1 i ◮ The elementary n -strand braid σ − 1 is defined in the same i way but with an under crossing. i i+1 i+1 i
11 The n -strand braid group B n ◮ The concatenation of two n -strand braids β, β ′ is the n -strand braid ββ ′ obtained by identifying β (1 i ) = β ′ (0 i ). ◮ B n is the set of isotopy classes of n -strand braids β , with composition by concatenation, and unit σ 0 . ◮ B n has generators σ 1 , σ 2 , . . . , σ n − 1 and relations � σ i σ j = σ j σ i if | i − j | � 2 σ i σ j σ i = σ j σ i σ j if | i − j | = 1 . ◮ Every n -strand braid β is represented by a word in B n in ℓ generators, corresponding to a sequence of ℓ crossings in a plane projection. ◮ The concatenation βσ i is obtained from β by adding to the sequence a crossing of the i th strand over the ( i + 1)th strand. ◮ The representation theory of the braid groups much studied. Highlight: the Jones polynomial.
12 The closure of a braid ◮ The closure of an n -strand braid β is the c -component link � � � S 1 ⊂ R 3 � β = β ∪ σ 0 : I ∪ σ I = n n c with c = |{ 1 , 2 , . . . , n } /σ | the number of cycles in σ ∈ Σ n . ◮ Alexander proved in A lemma on systems of knotted curves (1923) that every link is the closure � β of a braid β . ◮ Example A braid representation of the figure eight knot, with 3 strands and 4 crossings 3 2 1 −1 −1 σ 1 σ 2 σ 1 σ 2
13 The closure of σ 1 σ 1 is the Hopf link i+1 i i+1 i The 2-strand braid β = σ σ 1 1 The closure β = Hopf link
14 The Seifert surfaces of a link ◮ A Seifert surface for a link � S 1 ⊂ R 3 L : is a surface F 2 ⊂ R 3 with boundary � S 1 ) ⊂ R 3 . ∂ F = L ( ◮ Seifert in ¨ Uber das Geschlecht von Knoten (1935) proved that every link L admits a Seifert surface of the type � � D 2 ∪ D 1 × D 1 ⊂ R 3 F = n ℓ using an algorithm starting with a plane projection. ◮ A link L has many projections, and many Seifert surfaces.
15 The algorithm for a Seifert surface ◮ For any link L : � S 1 ⊂ R 3 there exists a linear map P : R 3 → R 2 (many in fact) such that the image of the composite PL : � S 1 → R 2 is a collection of oriented curves with ℓ transverse double points labelled as over/underpasses. This is a plane projection of L . ◮ Given L and a plane projection traverse the curves, switching each intersection according to over/underpasses, giving n “Seifert circles”. Construct a Seifert surface with n 0-handles and ℓ 1-handles � � D 2 ∪ D 1 × D 1 ⊂ R 3 F = n ℓ with � S 1 ) ⊂ R 3 . ∂ F = L (
16 Examples of Seifert’s algorithm for knots ◮ ◮
17 The canonical Seifert surface F β of a braid ◮ An n -strand braid β with ℓ crossings is represented by a word in B n of length ℓ in the generators σ 1 , σ 2 , . . . , σ n − 1 , so that β = β 1 β 2 . . . β ℓ is the concatenation of ℓ elementary braids. ◮ Stallings in Constructions of fibred knots and links (1978) observed that the closure � β has a canonical projection with n Seifert circles and ℓ intersections, and hence a canonical Seifert surface with n 0-handles and ℓ 1-handles � � D 2 ∪ D 1 × D 1 ⊂ R 3 . F β = n ℓ ◮ Lemma F β is homotopy equivalent to the CW complex � n � ℓ e 0 e 1 X β = i ∪ j i =1 j =1 with ∂ e 1 j = e 0 i ∪ e 0 i +1 if j th crossing is between strands i , i + 1 H 1 ( F β ) = H 1 ( X β ) = ker( d : C 1 ( X β ) → C 0 ( X β )) = ker( d : Z ℓ → Z n ) = Z m .
18 An example of the canonical Seifert surface F β for the closure � β of a braid β 3 3 1 2 1 2
19 SeifertView ◮ Arjeh Cohen and Jack van Wijk wrote a programme SeifertView (2005) and a paper The visualization of Seifert surfaces (2006) for drawing the canonical Seifert surfaces F β of the closures � β of braids β . ◮ A screenshot ◮ Try the SeifertView rollercoaster!
20 More braids β and canonical Seifert surfaces F β I. σ σ 1 0 σ σ 0 1 F F σ σ 0 1
21 More braids β and canonical Seifert surfaces F β II. σ σ σ σ σ 1 1 1 1 1 σ σ σ σ σ 1 1 1 1 1 F F σ σ σ σ 1 σ 1 1 1 1
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