Introduction M : an oriented closed 3-manifold : 1 ( M ) PSL(2 , C - PowerPoint PPT Presentation

蒲谷祐一 （大阪市立大学 数学研究所） （井上 歩氏（東京工業大学大学院理工学研究科）との共同研究） Quandle による shadow coloring と PSL(2,C) 表現の体積と Chern-Simons 不変量 早稲田大学 2009 年 12 月 24 日 1

Introduction M : an oriented closed 3-manifold ρ : π 1 ( M ) → PSL(2 , C ) : a rep. of the fund. group of M Vol( M, ρ ) ∈ R and CS( M, ρ ) ∈ R / π 2 Z are invariants of the representation ρ . When ρ is a discrete faithful rep. of a hyperbolic mfd M , then Vol and CS are the volume and the Chern-Simons invariant of the hyperbolic metric. 2

The definition of Vol and CS are generalized to the case of manifolds with torus boundary e.g. knot complements. A formula of i (Vol + i CS) ∈ C / π 2 Z was given by Neumann in terms of triangulations of 3-manifolds. We give a formula in terms of knot diagrams by using the quandle formed by parabolic elements of PSL(2 , C ). 3

Quandle str. on C 2 \ { 0 } Define a binary operation ∗ on C 2 \ { 0 } by ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ − x 2 ⎝ x 1 ⎝ x 2 ⎝ 1 − x 2 y 2 ⎝ x 1 ⎠ := ⎠ ∗ 2 ⎠ ⎠ y 2 y 1 y 2 y 1 1 + x 2 y 2 2 This satisfies the quandle axioms: 1. x ∗ x = x for x ∈ C 2 \ { 0 } 2. The inverse of ∗ y : C 2 \ { 0 } → C 2 \ { 0 } is given by ⎛ ⎞ x 2 ⎝ 1 + x 2 y 2 ∗ − 1 y : 2 ⎠ − y 2 1 − x 2 y 2 2 3. ( x ∗ y ) ∗ z = ( x ∗ z ) ∗ ( y ∗ z ) 4

P : the set of parabolic elements of PSL(2 , C ) ( ∼ = the set of parabolic elements of SL(2 , C ) with trace 2) P has a quandle str. by x ∗ y = y − 1 xy . Define a map C 2 \ { 0 } 2:1 − − → P by ⎛ ⎞ ⎛ ⎞ − x 2 ⎝ x ⎝ 1 − xy ⎠ �→ ⎠ y 2 y 1 + xy This map induces a quandle isomorphism P ∼ = ( C 2 \ { 0 } ) / ± 5

Arc coloring by ( C 2 \ { 0 } ) / ± Let D be a diagram of a knot. A map A : { arcs of D } → ( C 2 \ { 0 } ) / ± is called an arc coloring if it satisfies the following relation at each crossing. x ∗ y x, y and x ∗ y ∈ ( C 2 \ { 0 } ) / ± y x 6

Arc coloring of the figure eight knot � � � � 1 0 0 t � � 1 − t 2 This is the figure eight knot. � � − t t (1 + t 2 ) 8

Arc coloring of the figure eight knot � � � � 1 0 0 t � � 1 − t 2 Color two arcs. � � − t t (1 + t 2 ) 9

Arc coloring of the figure eight knot � � � � 1 0 0 t � � 1 Consider the relation at a − t 2 crossing. � � − t t (1 + t 2 ) 10

Arc coloring of the figure eight knot � � � � 1 0 0 t � � 1 ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ − t 2 ⎝ 1 ⎝ 0 ⎝ 1 ⎠ ∗ − 1 ⎠ = ⎠ − t 2 0 t � � − t t (1 + t 2 ) 11

Arc coloring of the figure eight knot � � � � 1 0 0 t � � 1 Consider the relation at an- − t 2 other crossing. � � − t t (1 + t 2 ) 12

Arc coloring of the figure eight knot � � � � 1 0 0 t � � 1 ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ − t 2 ⎝ 0 ⎝ 1 − t ⎠ = ⎠ ∗ ⎝ ⎠ − t 2 t (1 + t 2 ) t � � − t t (1 + t 2 ) 13

Arc coloring of the figure eight knot The relation at this crossing � � � � 1 0 is 0 t ⎛ ⎛ ⎞ ⎛ ⎞ ⎞ − t ⎝ 0 ⎠ ∗ ⎠ = ⎝ ⎝ ⎠ t (1 + t 2 ) t � � ⎛ ⎞ ⎛ ⎞ 1 − t 3 ⎝ 1 − t 2 ⎠ = ⎝ ⎠ t (1 + t 2 + t 4 ) 0 ⎧ ( t + 1)( t 2 − t + 1) = 0 ⎨ t ( t 2 + t + 1)( t 2 − t + 1) = 0 ⎩ � � ∴ t 2 − t + 1 = 0 − t t (1 + t 2 ) 14

Arc coloring of the figure eight knot The relation at this crossing � � � � 1 0 is 0 t ⎛ ⎛ ⎞ ⎛ ⎞ ⎞ ⎝ 1 ⎝ 1 ⎠ ∗ ⎠ = ⎝ ⎠ − t 2 0 � � ⎛ ⎞ ⎛ ⎞ 1 ⎝ 1 + t 2 − t − t 2 ⎠ = ⎝ ⎠ t (1 + t 2 ) − t 2 ⎧ t 2 + t + 1 = 0 ⎨ t ( t 2 + t + 1) = 0 ⎩ � � ∴ t 2 + t + 1 = 0 − t t (1 + t 2 ) 15

Arc coloring of the figure eight knot There are two relations t 2 + t + 1 = 0 , t 2 − t + 1 = 0 which do not have any common solution. But we have a √ √ coloring by ( C 2 \ { 0 } ) / ± ∼ = P ( t = ± 1+ 3 i or ± 1 − 3 i ). 2 2 Because the trace of the longitude is − 2, the coloring by P does not lift to a coloring by C 2 \ { 0 } . But we can color the long knot by C 2 \ { 0 } . 16

Arc coloring of the figure eight knot � � � � 1 0 0 t A parabolic representation � � 1 can be obtained by the map − t 2 ⎛ ⎞ ⎛ ⎞ x 2 ⎝ 1 − xy ⎝ x ⎠ �→ ⎠ − y 2 y 1 + xy � � − t t (1 + t 2 ) 17

Arc coloring of the figure eight knot � � � � 1 1 1 0 0 1 − t 2 1 A parabolic representation � � 1 + t 2 1 can be obtained by − t 4 1 − t 2 ⎛ ⎞ ⎛ ⎞ x 2 ⎝ 1 − xy ⎝ x ⎠ �→ ⎠ − y 2 y 1 + xy � � 1 + t 2 + t 4 t 2 1 − t 2 − t 4 − t 2 (1 + t 2 ) 2 18

Arc coloring of the figure eight knot � � � � 1 1 1 0 √ 0 1 − 1 − 3 i 1 2 ⎛ ⎞ √ 1+ 3 i 1 √ ⎝ ⎠ 2 √ √ When t 2 = − 1+ 3 i − 1 − 3 i 3 − 3 i : 2 2 2 ⎛ ⎞ √ − 1+ 3 i 0 ⎝ ⎠ 2 √ 1+ 3 i 2 2 19

Region coloring Let D be a diagram and A be an arc coloring by ( C 2 \ { 0 } ) / ± . A map D : { regions of D } → ( C 2 \ { 0 } ) / ± is called an region coloring if it satisfies the following relation at each arc of D . x ∗ y x, y and x ∗ y ∈ ( C 2 \ { 0 } ) / ± y x A pair S = ( A , R ) ( A : arc coloring, R : region coloring) is called a shadow coloring . 20

Region coloring of the figure eight knot � � � � 1 0 √ � � 0 � � − 1+ 3 i 2 2 √ 2 � � 2 − 3 i 1 1 � � √ � � 1 2 − 3 i Put a region color at a region 1 ⎛ ⎞ √ 1 1 − 3 i ⎝ 1 √ � � e.g. ⎠ . 2 2 − 3 i 1 √ − 1 − 3 i 2 ⎛ ⎞ √ 3 − 3 3 i ⎝ ⎠ 2 √ − 1 − 3 i 2 ⎛ ⎞ √ − 1 − 3 i ⎝ ⎠ 2 √ − 1+ 3 2 21

Region coloring of the figure eight knot � � � � 1 0 The color of an adjacent re- √ � � 0 � � − 1+ 3 i 2 2 √ 2 � � gion is determined by the re- 2 − 3 i 1 1 � � √ � � 1 2 − 3 i lation. 1 √ 1 1 − 3 i √ ⎛ ⎞ ⎛ ⎞ � � 0 2 ⎝ 1 2 − 3 i ⎠ ∗ − 1 √ √ ⎝ ⎠ − 1 − 3 i − 1+ 3 i 1 2 2 ⎛ ⎞ ⎛ ⎞ √ 1 3 − 3 3 i √ = ⎝ ⎠ ⎝ ⎠ 2 √ 2 − 3 i − 1 − 3 i 2 ⎛ ⎞ √ − 1 − 3 i ⎝ ⎠ 2 √ − 1+ 3 2 22

Region coloring of the figure eight knot � � � � 1 0 √ The color of an adjacent re- � � 0 � � − 1+ 3 i 2 2 √ 2 � � gion is determined by the re- 2 − 3 i 1 1 � � √ � � 1 2 − 3 i lation. 1 √ 1 1 − 3 i √ ⎛ ⎞ ⎛ ⎞ � � 2 ⎝ 1 ⎝ 1 2 − 3 i ⎠ ∗ − 1 √ ⎠ − 1 − 3 i 1 0 2 ⎛ ⎞ ⎛ ⎞ √ ⎝ 2 3 − 3 3 i = ⎠ ⎝ ⎠ 2 √ 1 − 1 − 3 i 2 ⎛ ⎞ √ − 1 − 3 i ⎝ ⎠ 2 √ − 1+ 3 2 23

Region coloring of the figure eight knot � � � � 1 0 √ � � 0 � � − 1+ 3 i 2 2 √ 2 � � 2 − 3 i 1 1 � � √ � � The color of an adjacent re- 1 2 − 3 i 1 √ 1 1 − 3 i √ � � gion is determined by the re- 2 2 − 3 i √ − 1 − 3 i lation. 2 ⎛ ⎞ √ 3 − 3 3 i ⎝ ⎠ 2 √ − 1 − 3 i 2 ⎛ ⎞ √ − 1 − 3 i ⎝ ⎠ 2 √ − 1+ 3 2 24

⎛ ⎞ ⎝ 1 Fix an element p 0 of C 2 \ { 0 } e.g. p 0 = ⎠ . 2 x y At a corner colored by r ( x ↔ under arc, y ↔ over arc), we let z =det( p 0 , y ) det( r, x ) det( r, y ) det( p 0 , x ) p π i =Log(det( p 0 , y )) + Log(det( r, x )) − Log(det( r, y )) − Log(det( p 0 , x )) − Log( z ) q π i =Log(det( p 0 , x )) + Log(det( r, y )) 1 − Log(det( p 0 , r )) − Log(det( x, y )) − Log( 1 − z ) where Log( z ) = log | z | + i arg( z ) ( − π < arg( z ) ≤ π ) 25

We remark that p, q ∈ Z . Then define the sign in the following rule: x y y x x y y x r r r r x y y x x y y x and r r r r +[ z ; p, q ] − [ z ; p, q ] (in-out or out-in) (in-in or out-out) 26

Let � � �� − π 2 R ( z ; p, q ) = R ( z ) + π i 1 q Log( z ) − p Log 6 . 2 1 − z where R ( z ) is given by � z Log(1 − t ) dt + 1 R ( z ) = − 2Log( z )Log(1 − z ) 0 t Theorem 1 � ε c R ( z c ; p c , q c ) = i (Vol( S 3 \ K, ρ ) + i CS( S 3 \ K, ρ )) c :corners where ρ is the parabolic representation determined by the arc coloring. 27

Background materials X : a quandle G X = ⟨ x ∈ X | y − 1 xy = x ∗ y ⟩ : the associated group X has a right G X -action defined by x ∗ ( x ε 1 1 x ε 2 1 . . . x ε n n ) = ( . . . (( x ∗ ε 1 x 1 ) ∗ ε 2 x 2 ) . . . ) ∗ x ε n n So Z [ X ] is a right Z [ G X ]-module. 28

Quandle homology Let C R n ( X ) = span Z [ G X ] { ( x 1 , . . . , x n ) | x i ∈ X } . Define the bound- ary operator ∂ : C R n ( X ) → C R n − 1 ( X ) by n � ( − 1) i { ( x 1 , . . . , � ∂ ( x 1 , . . . x n ) = x i , . . . , x n ) i =1 − x i ( x 1 ∗ x i , . . . , x i − 1 ∗ x i , x i +1 , . . . , x n ) } Let M be a right Z [ G X ]-module. The homology of M ⊗ Z [ G X ] C R n ( X ) is the rack homology H R n ( X ; M ). Considering non-degenerate chains, we also define the quandle homology H Q n ( X ; M ). 29