The Jones polynomial through linear algebra Iain Moffatt University - PowerPoint PPT Presentation

The Jones polynomial through linear algebra Iain Moffatt University of South Alabama Workshop in Knot Theory Waterloo, 24 th September 2011 I. Moffatt (South Alabama) UW 2011 1 / 39

What and why What we’ll see The construction of link invariants through R-matrices. (c.f. Reshetikhin-Turaev invariants, quantum invariants) Why this? Can do some serious math using material from Linear Algebra 1. Illustrates how math works in the wild: start with a problem you want to solve; figure out an easier problem that you can solve; build up from this to solve your original problem. See the interplay between algebra, combinatorics and topology! It’s my favourite bit of math! I. Moffatt (South Alabama) UW 2011 2 / 39

What we’re trying to do A knot is a circle, S 1 , sitting in 3-space R 3 . A link is a number of disjoint circles in 3-space R 3 . Knots and links are considered up to isotopy. This means you can “move then round in space, but you can’t cut or glue them”. I. Moffatt (South Alabama) UW 2011 3 / 39

What we’re trying to do A knot is a circle, S 1 , sitting in 3-space R 3 . A link is a number of disjoint circles in 3-space R 3 . Knots and links are considered up to isotopy. This means you can “move then round in space, but you can’t cut or glue them”. The fundamental problem in knot theory is to determine whether or not two links are isotopic. ? ? = = I. Moffatt (South Alabama) UW 2011 3 / 39 Links To do this we need knot invariants: F : → ( a set ) such

What we’re trying to do Knots and links are considered up to isotopy. This means you can “move then round in space, but you can’t cut or glue them”. The fundamental problem in knot theory is to determine whether or not two links are isotopic. ? ? = = Links To do this we need knot invariants: F : ( Isotopy ) → ( a set ) such that F ( L ) � = F ( L ′ ) = ⇒ L � = L ′ , Aim: construct link invariants using linear algebra. I. Moffatt (South Alabama) UW 2011 3 / 39

Our toolkit: Linear algebra 1 The basics V a vector space over C with basis { v 1 , . . . , v n } . ⇒ v = � n v ∈ V ⇐ i = 1 a i v i . ⇒ v = � n i = 1 f j f : V → V ⇐ i v i .  f 1 f n  1 · · · 1 f linear ↔ matrix . .    . .   . .    f 1 f n . . . n n The direct product V × V = { ( u , v ) | u , bv ∈ V} λ ( u , v ) + ( u ′ , v ′ ) = ( λ u + u ′ , λ v + v ′ ) The dual V ∗ V ∗ = Hom ( V , C ) = { f : V → C | f linear } basis { v i | v i ( v j ) = δ i , j , i = 1 , . . . , n } I. Moffatt (South Alabama) UW 2011 4 / 39

Making life easier: link diagrams Working with 3-D objects is tricky. To make life easy we draw knots on the plane. A link diagram is a drawing of a link on the plane. Link diagrams are considered up to the Reidemeister moves Knot diagrams Reidemeister moves project onto plane = = RI RII = RIII Theorem Links Diagrams ( Isotopy ) = ( R − moves ) I. Moffatt (South Alabama) UW 2011 5 / 39

An example [link] I. Moffatt (South Alabama) UW 2011 6 / 39

The first idea: let’s form a map Our goal ( Isotopy ) → C [ t , t − 1 ] . Links We want to construct a knot polynomial J : How do we get started? Diagrams ( R − moves ) → C [ t , t − 1 ] . Work with link diagrams � look for J : Motivated by algebra: define the map on the “generators” of a diagram. I. Moffatt (South Alabama) UW 2011 7 / 39

The first idea: let’s form a map Our goal ( Isotopy ) → C [ t , t − 1 ] . Links We want to construct a knot polynomial J : How do we get started? Diagrams ( R − moves ) → C [ t , t − 1 ] . Work with link diagrams � look for J : Motivated by algebra: define the map on the “generators” of a diagram. Generators Figure-eight knot I. Moffatt (South Alabama) UW 2011 7 / 39

Cutting down the number of generators The large number of generators is making life difficult. Can we reduce their number? Use the following redrawing of the figure eight knot. Generators Made out of Figure-eight knot A standard position • Fewer generators needed called a braid closure I. Moffatt (South Alabama) UW 2011 8 / 39

Braids and braid closures All of the ‘interesting’ structure is contained in a part of a diagram called a braid: g interesting n i r o b Braid Closure element Closure of a braid A braid: is an intertwining strings attached to top and bottom "bars" such that each string never "turns back up": I. Moffatt (South Alabama) UW 2011 9 / 39

Putting link diagrams in the standard form Alexander’s Theorem: obtaining a braid from a link Choose a point X . Whenever an arc travels counter-clockwise pull it over the base point. Cut open the link to get a braid. X X X Theorem Every link diagram can be written as a braid closure. I. Moffatt (South Alabama) UW 2011 10 / 39

Braids Theorem Every link diagram can be written as a braid closure. Rather than working with links, we can work with braids. Braids have only three generators: Generators: A braid � We now have generators! If we want to work with braids, we need to know: How do the generators generate braids? When are braids equivalent? When do braids represent the same links? I. Moffatt (South Alabama) UW 2011 11 / 39

Braids Theorem Every link diagram can be written as a braid closure. Rather than working with links, we can work with braids. Braids have only three generators: Generators: A braid � We now have generators! If we want to work with braids, we need to know: How do the generators generate braids? When are braids equivalent? When do braids represent the same links? I. Moffatt (South Alabama) UW 2011 11 / 39

Operations on braids Composition Tensor product σ ′ = σ ′ = σ σ σ ⊗ σ ′ σ σ ′ (n+m) strings n strings m strings n strings n strings n strings stack up place beside I. Moffatt (South Alabama) UW 2011 12 / 39

Operations on braids Composition Tensor product σ ′ = σ ′ = σ σ σ σ ′ ⊗ σ σ ′ (n+m) strings n strings n strings n strings n strings m strings stack up place beside • With these operations every braid can be built from ors: . ⊗ ⊗ ( ) ( ) ( ) ( ) = ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ � We now have generators and generating operations! I. Moffatt (South Alabama) UW 2011 12 / 39

Braid equivalence Different diagrams can represent the same braid: = .... = = Braids are considered up to the following moves B-moves = = = = These can be written algebraically using “ ◦ ” and “ ⊗ ”. I. Moffatt (South Alabama) UW 2011 13 / 39

The Markov moves We want to study links using braids, we need to know when braids represent the same link. = I. Moffatt (South Alabama) UW 2011 14 / 39

The Markov moves We want to study links using braids, we need to know when braids represent the same link. = The Markov moves (M-moves) σ σ ′ σ σ σ = = = σ σ ′ (n+1) strings n strings (n+1) strings MI-move MII-move I. Moffatt (South Alabama) UW 2011 14 / 39

Markov’s Theorem σ σ ′ σ σ σ = = = = = = = σ σ ′ (n+1) strings n strings (n+1) strings MI-move MII-move Markov’s Theorem Braids describe equal links ⇐ ⇒ related by B-moves and M-moves. Links Diagrams Braids ( Isotopy ) = ( R − moves ) = ( B - moves , M - moves ) Sufficiency is easy, e.g. σ σ ′ σ = σ = σ = ′ σ σ σ ′ Braids related by B-moves � closures related by R-moves. Necessity is hard. I. Moffatt (South Alabama) UW 2011 15 / 39

Summary So far: ( Isotopy ) → C [ q , q − 1 ] . Links We want to construct J : Too hard! Let’s make it easier. We have shown that every link can be represented by a braid: Diagrams Links Braids ( Isotopy ) = ( R − moves ) = We have seen ( B - moves , M - moves ) . ( B - moves , M - moves ) → C [ q , q − 1 ] Braids Thus enough to construct J : Easier as braids are generated by under ⊗ and ◦ . I. Moffatt (South Alabama) UW 2011 16 / 39

Summary So far: ( Isotopy ) → C [ q , q − 1 ] . Links We want to construct J : Too hard! Let’s make it easier. We have shown that every link can be represented by a braid: Diagrams Links Braids ( Isotopy ) = ( R − moves ) = We have seen ( B - moves , M - moves ) . ( B - moves , M - moves ) → C [ q , q − 1 ] Braids Thus enough to construct J : Easier as braids are generated by under ⊗ and ◦ . I. Moffatt (South Alabama) UW 2011 16 / 39

Summary So far: ( Isotopy ) → C [ q , q − 1 ] . Links We want to construct J : Too hard! Let’s make it easier. We have shown that every link can be represented by a braid: X X X Diagrams Links Braids ( Isotopy ) = ( R − moves ) = We have seen ( B - moves , M - moves ) . ( B - moves , M - moves ) → C [ q , q − 1 ] Braids Thus enough to construct J : Easier as braids are generated by under ⊗ and ◦ . I. Moffatt (South Alabama) UW 2011 16 / 39

Summary So far: ( Isotopy ) → C [ q , q − 1 ] . Links We want to construct J : Too hard! Let’s make it easier. We have shown that every link can be represented by a braid: X X X Diagrams Links Braids ( Isotopy ) = ( R − moves ) = We have seen ( B - moves , M - moves ) . ( B - moves , M - moves ) → C [ q , q − 1 ] Braids Thus enough to construct J : Easier as braids are generated by under ⊗ and ◦ . I. Moffatt (South Alabama) UW 2011 16 / 39