S 2 , → R 3 has no “knotting” behavior, because there are two few dimensions to move around in, just like S 1 , → R 2 . But S 2 , → R 4 works, and this is a special case of higher- dimensional knot theory : S n , → R n +2 . Think of R 4 as 3-dimensional space moving through time, and think of S 2 as a collection of circles. Now consider embeddings S 2 ⊂ R 4 as a “movie” of embeddings of S 1 ⊂ R 3 where unknots turn into knots and back to unknots. 20.4 Domain X More generally, there is knotting of objects X , → R n for appropriate n based on the dimension of X . For example, tori can be knotted in R 3 , since tori are nothing more than the boundaries of thickened knots! Exercise: Find two 2-holed tori that are not isotopic to each other in R 3 . 32
21 4/12: Jones’ original paper Backtrack to braid theory and the Jones polynomial. Let’s connect them, and in particular, see how Jones originally defined his polynomial (before Kauffman did)! Definition 18. The Temperley-Lieb algebra A n for n > 1 is the free additive algebra on the (multiplicative) generators e 1 , . . . , e n - 1 viewed as a C [ τ, τ - 1 ]-module. Here τ is a variable that commutes with all generators, and the generators satisfy the relations 1) e 2 i = e i 2) e i e i ± 1 e i = τe i 3) e i e j = e j e i when | i - j | > 2 Jones noticed that A n looks like the Braid group B n . He constructed a particular map ρ : B n → A n . He also constructed a particular “trace” map tr : A n → C [ τ, τ - 1 ], which means tr ( ab ) = tr ( ba ). So we get a trace map tr ◦ ρ : B n → C [ τ, τ - 1 ] and this is the Jones polynomial after a particular normalization! Let’s analyze this more closely. By Alexander’s theorem, any oriented link is given by the closure ˆ σ of some braid σ . The non-uniqueness is handled by Markov’s theorem, so we have to make sure that tr ◦ ρ is independent of the “Markov moves”, in order to get an invariant of the link (closure of the braid). Let’s look at the Markov moves: 1) Vacuously, two braids are identified up to isotopy-equivalence in the braid group. 2) Conjugate a braid σ ∈ B n by another braid b ∈ B n . Exercise: \ bσb - 1 = ˆ σ 3) An actual Markov move : Given σ ∈ B n , view it inside B n +1 by attaching an isolated strand at the end, take the generator σ n ∈ B n +1 , and form σσ ± 1 n . Exercise: [ σσ ± 1 n = ˆ σ 4) The “inverse” Markov move , undoing (3), where a braid of the form σσ ± 1 n becomes σ . The definition of “trace” already handles conjugation of braids. To handle the actual Markov moves, Jones’ trace map tr for A n is chosen to satisfy the additional property tr ( we i ) = τ · tr ( w ) whenever w ∈ A i - 1 . To get our desired link invariant, we need to normalize the trace map to account for these τ ’s that appear under Jones’ additional property when a Markov move occurs.
- Spring '08