Braids Without Twists Jim Belk and Francesco Matucci

Braids Without Twists Jim Belk* and Francesco Matucci

The Braid Group A braid is any picture similar to the one below. This braid has four strands, which twist around each other in a certain pattern. We can regard the strands as the paths of motion for four points moving in the plane. So a braid is really a loop in the configuration space of four points in 2.

The Braid Group A braid is any picture similar to the one below. We can multiply braids using vertical concatenation. Under this product, the set of all braids with four strands forms a group. This is the braid group B 4.

Allowed Moves = =

Thompson’s Group F A strand diagram is any picture similar to the one below. A strand diagram is like a braid, but it has splits and merges: split merge

Thompson’s Group F A strand diagram is any picture similar to the one below. A strand diagram is like a braid, but it has splits and merges: split merge

Allowed Moves = These two moves are called reductions. They do not change the underlying strand diagram. =

Multiplication We multiply strand diagrams in the same way as braids:

Multiplication Usually the result will not be reduced.

Multiplication Usually the result will not be reduced.

Multiplication Usually the result will not be reduced.

Multiplication Usually the result will not be reduced.

Multiplication Usually the result will not be reduced.

Multiplication Usually the result will not be reduced.

Multiplication Usually the result will not be reduced

Multiplication The group of all strand diagrams is called Thompson’s group F.

Multiplication The group of all strand diagrams is called Thompson’s group F.

Planar Knots

Planar Knots If you join together the top and bottom of a braid, you get a knot.

Planar Knots If we join together the top and bottom of a strand diagram, we get a trivalent directed graph drawn on an annulus. This is called an annular strand diagram.

Planar Knots We can reduce annular strand diagrams using the two allowed moves.

Planar Knots We can reduce annular strand diagrams using the two allowed moves.

Planar Knots We can reduce annular strand diagrams using the two allowed moves.

Planar Knots We can reduce annular strand diagrams using the two allowed moves.

Planar Knots We can reduce annular strand diagrams using the two allowed moves.

Planar Knots We can reduce annular strand diagrams using the two allowed moves.

Planar Knots We can reduce annular strand diagrams using the two allowed moves.

Theorem (Guba, Sapir, B, and Matucci). Two elements of are conjugate if and only if they have the same reduced annular strand diagram.

Theorem (Guba, Sapir, B, and Matucci). Two elements of are conjugate if and only if they have the same reduced annular strand diagram.

Thompson’s Group Theorem (B and Matucci). Two elements of are conjugate if and only if they have the same reduced toral strand diagram.

Thompson’s Group Theorem (B and Matucci). Two elements of V are conjugate if and only if they have the same reduced closed diagram with crossings.

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