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Introduction Warping polynomial Span of warping polynomial Span and dealternating number Ayaka Shimizu (Osaka City University) The span of the warping polynomial of a knot diagram

Introduction Warping polynomial Span of warping polynomial Span and dealternating number § 0. Introduction D : an oriented knot diagram d ( D ) , d ( D ) , a b c ( D ): the crossing number of D c ( D ) d ( D ) d ( D ): the warping degree of D cd ( D ) cd ( D ) = ( c ( D ) , d ( D )): the complexity of D Ayaka Shimizu (Osaka City University) The span of the warping polynomial of a knot diagram

Introduction Warping polynomial Span of warping polynomial Span and dealternating number § 0. Introduction d ( D ) , d ( D ) , a b D D' c ( D ) d ( D ) cd ( D ) cd ( D ) = ( 8,2 ) cd ( D' ) = ( 8,2 ) Ayaka Shimizu (Osaka City University) The span of the warping polynomial of a knot diagram

Introduction Warping polynomial Span of warping polynomial Span and dealternating number § 0. Introduction d ( D ) , d ( D ) , a b D D' 4 4 3 3 5 7 c ( D ) d ( D ) 6 6 5 5 3 5 4 2 4 2 6 4 2 6 4 4 3 5 5 7 cd ( D ) 5 3 5 3 6 4 cd ( D ) = ( 8,2 ) cd ( D' ) = ( 8,2 ) Ayaka Shimizu (Osaka City University) The span of the warping polynomial of a knot diagram

Introduction Warping polynomial Span of warping polynomial Span and dealternating number § 0. Introduction D D' 4 4 d ( D ) , d ( D ) , a b 3 3 5 7 6 6 5 5 3 5 4 2 2 4 4 6 2 6 4 4 c ( D ) d ( D ) 3 7 5 5 5 3 5 3 6 4 cd ( D ) 2 3 4 5 6 7 W ( t ) =t +3t +5t +4t +t +t D 2 3 4 5 6 7 W ( t ) =2t +3t +3t +4t +3t +t D' W ( t ): the warping polynomial of D W ( t ) D D Ayaka Shimizu (Osaka City University) The span of the warping polynomial of a knot diagram

Introduction Warping polynomial Span of warping polynomial Span and dealternating number Contents § 1. Warping polynomial § 2. Span of the warping polynomial § 3. Span and dealternating number Ayaka Shimizu (Osaka City University) The span of the warping polynomial of a knot diagram

Introduction warping degree of Db Warping polynomial warping degree of D Span of warping polynomial warping polynomial Span and dealternating number properties § 1. Warping polynomial § 1.1. Warping degree of D b § 1.2. Warping degree of D § 1.3. Warping polynomial § 1.4. Properties of the warping polynomial Ayaka Shimizu (Osaka City University) The span of the warping polynomial of a knot diagram

Introduction warping degree of Db Warping polynomial warping degree of D Span of warping polynomial warping polynomial Span and dealternating number properties § 1.1. Warping degree of D b D : an oriented knot diagram b : a base point of D A crossing point p of D is a warping crossing point of D b if we meet the point first at the under-crossing when we go along D by starting from b . D b D p q b warping non-warping Ayaka Shimizu (Osaka City University) The span of the warping polynomial of a knot diagram

Introduction warping degree of Db Warping polynomial warping degree of D Span of warping polynomial warping polynomial Span and dealternating number properties § 1.1. Warping degree of Db the warping degree of D b d ( D b ) = ♯ { warping crossing points of D b } D b d ( D ) =1 b D d ( D ) =0 c c Ayaka Shimizu (Osaka City University) The span of the warping polynomial of a knot diagram

Introduction warping degree of Db Warping polynomial warping degree of D Span of warping polynomial warping polynomial Span and dealternating number properties § 1.2. Warping degree of D the warping degree of D d ( D ) = min b d ( D b ) D -D d ( D ) =0 d (- D ) =1 (Warping degree depends on the orientation.) Ayaka Shimizu (Osaka City University) The span of the warping polynomial of a knot diagram

Introduction warping degree of Db Warping polynomial warping degree of D Span of warping polynomial warping polynomial Span and dealternating number properties § 1.3. Warping polynomial Warping degree labeling for D is a labeling s.t. every edge e has the value d ( D b ), where b ∈ e . F 0 0 2 D E 3 1 1 1 0 1 1 2 1 2 2 Ayaka Shimizu (Osaka City University) The span of the warping polynomial of a knot diagram

Introduction warping degree of Db Warping polynomial warping degree of D Span of warping polynomial warping polynomial Span and dealternating number properties § 1.3. Warping polynomial Lemma. i i+1 i i-1 F 0 0 2 D E 3 1 1 1 0 1 1 2 1 2 2 Ayaka Shimizu (Osaka City University) The span of the warping polynomial of a knot diagram

Introduction warping degree of Db Warping polynomial warping degree of D Span of warping polynomial warping polynomial Span and dealternating number properties § 1.3. Warping polynomial D : an oriented knot diagram The warping polynomial W D ( t ) of D is e t i ( e ) , W D ( t ) = ∑ where i ( e ) is the value of an edge e w.r.t. warping degree labeling. Ayaka Shimizu (Osaka City University) The span of the warping polynomial of a knot diagram

Introduction warping degree of Db Warping polynomial warping degree of D Span of warping polynomial warping polynomial Span and dealternating number properties § 1.3. Warping polynomial Example. F 0 0 2 D E 3 1 1 1 0 1 1 2 1 2 2 2 2 3 W (t)=2+2t W (t)=2t+2t W (t)=1+2t+2t +t D E F Ayaka Shimizu (Osaka City University) The span of the warping polynomial of a knot diagram

Introduction warping degree of Db Warping polynomial warping degree of D Span of warping polynomial warping polynomial Span and dealternating number properties § 1.4. Properties of the warping polynomial D : an oriented knot diagram with c ( D ) ≥ 1 − D : D with orientation reversed D ∗ : the mirror image of D • W ⃝ ( t ) = 1 • mindeg W D ( t ) = d ( D ) • W D (1) = 2 c ( D ) • W D ( − 1) = 0 • W − D ( t ) = W D ∗ ( t ) = t c ( D ) W D ( t − 1 ) Ayaka Shimizu (Osaka City University) The span of the warping polynomial of a knot diagram

Introduction span Warping polynomial crossing number Span of warping polynomial span 1–3 Span and dealternating number § 2.1. Span of the warping polynomial the span of f ( t ) span f ( t ) = maxdeg f ( t ) − mindeg f ( t ) Proposition. • span W D ( t ) = c ( D ) − ( d ( D ) + d ( − D )) . • ∀ n ≥ 0, ∃ D s.t. span W D ( t ) = n . Ayaka Shimizu (Osaka City University) The span of the warping polynomial of a knot diagram

Introduction span Warping polynomial crossing number Span of warping polynomial span 1–3 Span and dealternating number § 2.2. Span and crossing number D : a knot diagram with c ( D ) ≥ 1 Proposition. span W D ( t ) ≤ c ( D ). “ = ” ⇔ D is a one-bridge diagram. 1 D 0 2 3 4 Ayaka Shimizu (Osaka City University) The span of the warping polynomial of a knot diagram

Introduction span Warping polynomial crossing number Span of warping polynomial span 1–3 Span and dealternating number § 2.3. Warping polynomials with span 1–3 Theorem. (i) f ( t ) is a warping polynomial with span f ( t ) = 1 ⇔ f ( t ) = ct d + ct d + 1 , where 1 ≤ c , 1 ≤ d ≤ c − 1. (ii) f ( t ) is a warping polynomial with span f ( t ) = 2 ⇔ f ( t ) = at d + ct d + 1 + ( c − a ) t d + 2 , where 2 ≤ c , 1 ≤ a ≤ c − 1, 0 ≤ d ≤ c − 2. (iii) f ( t ) is a warping polynomial with span f ( t ) = 3 ⇔ f ( t ) = at d + bt d + 1 + ( c − a ) t d + 2 + ( c − b ) t d + 3 , where 3 ≤ c , 1 ≤ a < b ≤ c − 1, 0 ≤ d ≤ c − 3. Ayaka Shimizu (Osaka City University) The span of the warping polynomial of a knot diagram

Introduction dealternating number Warping polynomial alternating diagram Span of warping polynomial almost alternating diagram Span and dealternating number § 3.1. Span and dealternating number D : a knot diagram The dealternating number dalt( D ) of D is the minimal number of crossing changes which turn the diagram into an alternating diagram. Proposition. dalt( D ) ≥ span W D ( t ) − 1 . 2 Ayaka Shimizu (Osaka City University) The span of the warping polynomial of a knot diagram

Introduction dealternating number Warping polynomial alternating diagram Span of warping polynomial almost alternating diagram Span and dealternating number § 3.2. Span and alternating diagram D : a knot diagram with c ( D ) ≥ 1 Theorem [S. 2008]. d ( D ) + d ( − D ) + 1 ≤ c ( D ). “ = ” ⇔ D is an alternating diagram. Corollary. span W D ( t ) = 1 ⇔ D is an alternating diagram. Ayaka Shimizu (Osaka City University) The span of the warping polynomial of a knot diagram

Introduction dealternating number Warping polynomial alternating diagram Span of warping polynomial almost alternating diagram Span and dealternating number § 3.3. Span and almost alternating diagram Proposition. If a knot diagram D is an almost alternating diagram, then span W D ( t ) is two or three. Furthermore, (i)if D is obtained from an alternating diagram by a Reidemeister move I, then span W D ( t ) = 2. (ii)Otherwise, span W D ( t ) = 3. (i) (ii) E D 2 3 2 3 W (t)=1+2t+2t +t W (t)=t+4t +3t E D Ayaka Shimizu (Osaka City University) The span of the warping polynomial of a knot diagram