A polynomial invariant and the forbidden move of virtual knots . . - PowerPoint PPT Presentation

研究集会「結び目の数学 VI 」 @ Nihon University . . A polynomial invariant and the forbidden move of virtual knots . . . . . Migiwa Sakurai migiwa@cis.twcu.ac.jp December 19, 2013 Graduate School of Science Tokyo Woman’s Christian University 1 / 21

Contents 1 Preliminaries 2 forbidden moves and q t ( K ) 3 Examples 2 / 21

Preliminaries § 1. Preliminaries D : a virtual knot diagram def ⇔ D ; a knot diagram with , and virtual +1 - 1 � ��������������������� �� ��������������������� � real virtual knot diag. knot diag. 3 / 21

Preliminaries K : a virtual knot def ⇔ K ; an eq. class of virtual knot diagrams under GRM 4 / 21

Preliminaries A Gauss daigram : a preimage of a virtual knot diagram with info. of real crossings 1 2 + + + + 1 2 5 / 21

Preliminaries GRM of Gauss diagrams ε ε ε -ε ε -ε _ _ + + _ _ _ _ + + + + { All eq. classes of Gauss } one-to-one { All virtual knots } ←→ diagrams by GRM 6 / 21

Preliminaries Forbidden moves ( F ) ε ε ε ε ε′ ε′ ε′ ε′ { } F or GRM F or GRM u F ( K ) = min the number of F s.t K · · · ⇋ ⇋ 7 / 21

Preliminaries Invariants for virtual knots The arc of γ K : a virtual knot Q ε(γ) G : a Gauss diagram of K P , Q : points on the circle S 1 of G γ = − − → PQ : a chord oriented from P into Q ε ( γ ) : the sign of γ γ ε(γ) ε ( P ) = − ε ( γ ) ε ( Q ) = ε ( γ ) The arc of γ = − − → PQ : the arc in S 1 with the ori. form P to Q P -ε(γ) 8 / 21

Preliminaries . Definition 1 . . . 1 i ( γ ) = ”the sum of the sings of all points on the arc of γ ” : the index of γ 2 ([Satoh-Taniguchi]) J n ( K ) = ∑ i ( γ ) = n ε ( γ ) : n -writhe γ ε ( γ )( t | i ( γ ) | − 1) : index polynomial 3 ([Henrich]) p t ( K ) = ∑ 4 ([Cheng]) When we walk on S 1 from a point in S 1 , we denote the labels of arcs as the following: x + y-1 w+1 - z x+1 γ y w γ z-1 { y − x ( ε ( γ ) = 1) N ( γ ) : = z − w ( ε ( γ ) = − 1) . ∑ ε ( γ ) t N ( γ ) : odd writhe poly. f K ( t ) = . i ( γ ): odd . . . . 9 / 21

Preliminaries . Proposition 2 (Satoh-Taniguchi) . . . n > 0 { J n ( K ) + J − n ( K ) } ( t n − 1) . 1 p t ( K ) = ∑ 2 f K ( t ) = ∑ n ∈ Z J 1 − 2 n ( K ) t 2 n . . . . . . . Definition 3 . . . ∑ J n ( K )( t n − 1) q t ( K ) = n ∈ Z ∑ ε ( γ )( t i ( γ ) − 1) . = γ . . . . . p t ( K ) is induced from q t ( K ). f K ( t ) is induced from q t ( K ). 10 / 21

forbidden moves and q t ( K ) § 2. forbidden moves and q t ( K ) . Theorem 4 . . . F K, K ′ s.t. K ⇋ K ′ q t ( K ) − q t ( K ′ ) = ( t − 1)( ± t ℓ ± t m ) ( ℓ, m ∈ Z ) . . . . . . . Corollary 5 . . . q t ( K ) : = ( t − 1) ∑ n ∈ Z a n t n ∑ n ∈ Z | a n | u F ( K ) ≥ . . 2 . . . . From Thm. 4, ∑ n � 0 | J n ( K ) | u F ( K ) ≥ . 4 11 / 21

forbidden moves and q t ( K ) Sketch Proof ′ ) ′ ) -ε(γ 1 -ε(γ 2 -ε(γ 1 ) -ε(γ 2 ) γ 1 ′ γ 2 ′ γ 1 γ 2 ε(γ 1 ′ ) ε(γ 2 ′ ) ε(γ 2 ) ε(γ 1 ) G 1 G 2 { i ( γ ′ 1 ) = i ( γ 1 ) − ε ( γ 2 ) i ( γ ′ 2 ) = i ( γ 2 ) + ε ( γ 1 ) . ε ( γ i ) = ε ( γ ′ i ) ( i = 1 , 2) . 12 / 21

forbidden moves and q t ( K ) q t ( K 1 ) − q t ( K 2 ) = ε ( γ 1 )( t i ( γ 1 ) − 1) + ε ( γ 2 )( t i ( γ 2 ) − 1) 1 ) − 1) − ε ( γ ′ 2 ) − 1) 1 )( t i ( γ ′ 2 )( t i ( γ ′ − ε ( γ ′  ( t − 1)( t i ( γ 1 ) − 1 − t i ( γ 2 ) ) ( ε ( γ i ) = 1)    ( t − 1)( − t i ( γ 1 ) + t i ( γ 2 ) )   ( ε ( γ 1 ) = 1 , ε ( γ 2 ) = − 1)   = ( t − 1)( − t i ( γ 1 ) − 1 + t i ( γ 2 ) − 1 )  ( ε ( γ 1 ) = − 1 , ε ( γ 2 ) = 1)     ( t − 1)( t i ( γ 1 ) − t i ( γ 2 ) − 1 )  ( ε ( γ i ) = − 1)  □ 13 / 21

forbidden moves and q t ( K ) . Theorem 6 . . . 1 ([S]) p t ( K ) : = ( t − 1) ∑ n > 0 b n t n ∑ n > 0 | b n | u F ( K ) ≥ . 2 2 ([Crans, Ganzell, Mellor]) f K ( t ) : = ∑ n � 0 c n t n ∑ n � 0 | c n | u F ( K ) ≥ . 2 . . . . . . Proposition 7 . . . ∑ ∑ ∑ ∑ n � 0 | a n | n > 0 | b n | n � 0 | c n | n � 0 | J n ( K ) | u F ( K ) ≥ ≥ , , . 2 2 2 4 . . . . . 14 / 21

Examples § 3. Examples . Example 8 . . . ∀ n ∈ N , ∃ K n s.t. ” u F ( K n ) can not be determined by p t ( K n ), f K n ( t )”, ”it can be determined by q t ( K n )”. . . . . . m : even ( > 0) 1 ... m +2 m +3 2 m +1 2 m +2 ... m +1 3 2 K m 15 / 21

Examples 1 2 - + 3 + m +1 + ... ... + 2 m +2 + 2 m +1 + + m +2 m +3 16 / 21

Examples i (1) = − m 1 2 - + 3 + m +1 + ... ... + 2 m +2 + 2 m +1 + + m +2 m +3 16 / 21

Examples i (1) = − m 1 i (2) = − 1 2 - + 3 . + . . i ( m + 1) = − 1 m +1 + ... ... 2 m +2 + + 2 m +1 + + m +2 m +3 16 / 21

Examples i (1) = − m 1 i (2) = − 1 2 - + 3 . + . . i ( m + 1) = − 1 m +1 + i ( m + 2) = − 1 ... ... 2 m +2 + . . . + 2 m +1 i (2 m + 1) = − 1 + + m +2 m +3 16 / 21

Examples i (1) = − m 1 i (2) = − 1 2 - + 3 . + . . i ( m + 1) = − 1 m +1 + i ( m + 2) = − 1 ... ... 2 m +2 + . . . + 2 m +1 i (2 m + 1) = − 1 + + m +2 m +3 i (2 m + 2) = m 16 / 21

Examples p t ( K m ) & f K m ( t ) p t ( K m ) = 2 m ( t − 1) . ∑ → | b n | = 2 m . n > 0 f K m ( t ) = 2 mt 2 ∑ → | c n | = 2 m . n ∈ Z q t ( K m ) q t ( K m ) = ( t − 1) { t m − 1 + · · · + 1 + ( − 2 m + 1) t − 1 + · · · + t − m } ∑ → | a k | = 4 m − 2 . n ∈ Z 17 / 21

Examples p t ( K m ) & f K m ( t ) p t ( K m ) = 2 m ( t − 1) . ∑ → | b n | = 2 m . ∴ u F ( K m ) ≥ m . ✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿ n > 0 f K m ( t ) = 2 mt 2 ∑ → | c n | = 2 m . n ∈ Z q t ( K m ) q t ( K m ) = ( t − 1) { t m − 1 + · · · + 1 + ( − 2 m + 1) t − 1 + · · · + t − m } ∑ → | a k | = 4 m − 2 . n ∈ Z 17 / 21

Examples p t ( K m ) & f K m ( t ) p t ( K m ) = 2 m ( t − 1) . ∑ → | b n | = 2 m . ∴ u F ( K m ) ≥ m . ✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿ n > 0 f K m ( t ) = 2 mt 2 ∑ → | c n | = 2 m . ∴ u F ( K m ) ≥ m . ✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿ n ∈ Z q t ( K m ) q t ( K m ) = ( t − 1) { t m − 1 + · · · + 1 + ( − 2 m + 1) t − 1 + · · · + t − m } ∑ → | a k | = 4 m − 2 . n ∈ Z 17 / 21

Examples p t ( K m ) & f K m ( t ) p t ( K m ) = 2 m ( t − 1) . ∑ → | b n | = 2 m . ∴ u F ( K m ) ≥ m . ✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿ n > 0 f K m ( t ) = 2 mt 2 ∑ → | c n | = 2 m . ∴ u F ( K m ) ≥ m . ✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿ n ∈ Z q t ( K m ) q t ( K m ) = ( t − 1) { t m − 1 + · · · + 1 + ( − 2 m + 1) t − 1 + · · · + t − m } ∑ → | a k | = 4 m − 2 . ∴ u F ( K m ) ≥ 2 m − 1 . ✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿ n ∈ Z 17 / 21

Examples - + + + + ... ... + ++ 18 / 21

Examples - - + + + + + + + ... ... ... ... ++ + ++ ++ 18 / 21

Examples - - - + + + + + + ... + + + + ... ... ... ... ... ... ++ + + + ++ ++ ++ 18 / 21

Examples - - - + + + + + + ... + + + + ... ... ... ... ... ... ++ + + + ++ ++ ++ - + + + ... ... + + ++ 18 / 21

Examples - - - + + + + + + ... + + + + ... ... ... ... ... ... ++ + + + ++ ++ ++ - - + + + + ... + + ... ... ... ... + + + + ++ ++ 18 / 21

Examples - - - + + + + + + ... + + + + ... ... ... ... ... ... ++ + + + ++ ++ ++ - - + + + + ... + + ... ... ... ... + + + + ++ ++ 18 / 21

Examples - - - + + + + + + ... + + + + ... ... ... ... ... ... ++ + + 1 2 m + ++ ++ ++ m +1 - - + + + + ... + + ... ... ... m +2 2 m -1 ... + + + + ++ ++ u F ( K m ) = 2 m − 1 . 18 / 21

Examples Unknotting numbers for virtual knots with with up to 4 real crossing points K u F ( K ) K u F ( K ) K u F ( K ) K u F ( K ) K u F ( K ) 0.1 0 4.6 1-2 4.20 1 4.34 1 4.48 3 2.1 1 4.7 2 4.21 2 4.35 1 4.49 1 3.1 1 4.8 1-2 4.22 1 4.36 2 4.50 1 3.2 1 4.9 1-2 4.23 1 4.37 3 4.51 1 3.3 2 4.10 1-2 4.24 2-3 4.38 1 4.52 1 3.4 1 4.11 2 4.25 2 4.39 1 4.53 2 3.5 2-3 4.12 1-2 4.26 2 4.40 1 4.54 1 3.6 1-4 4.13 1-2 4.27 1-2 4.41 1 4.55 1 3.7 2-3 4.14 1-2 4.28 2 4.42 1 4.56 1 4.1 2 4.15 2 4.29 2 4.43 2 4.57 1 4.2 1-2 4.16 1 4.30 1-2 4.44 1-2 4.58 1 4.3 2 4.17 1 4.31 1 4.45 2 4.59 1 4.4 1 4.18 1 4.32 1 4.46 1-2 4.60 1 4.5 1 4.19 1-2 4.33 1 4.47 2 4.61 1-4 19 / 21