Gap between the alternation number and the dealternating number Mar - PowerPoint PPT Presentation

Gap between the alternation number and the dealternating number Mar´ ıa de los Angeles Guevara Hern´ andez Osaka City University Advanced Mathematical Institute (CONACYT-Mexico fellow) guevarahernandez.angeles@gmail.com Waseda University, Japan December 25, 2018 Guevara Hern´ andez (OCAMI) alt ( K ) and dalt ( K ) December 25, 2018 1 / 24

Introduction Definition A link is a disjoint union of circles embedded in S 3 , a knot is a link with one component. Guevara Hern´ andez (OCAMI) alt ( K ) and dalt ( K ) December 25, 2018 2 / 24

Introduction Definition A knot that possesses an alternating diagram is called an alternating knot , otherwise it is called a non-alternating knot. Alternating diagram Non-alternating diagram Guevara Hern´ andez (OCAMI) alt ( K ) and dalt ( K ) December 25, 2018 3 / 24

Introduction Definition A knot that possesses an alternating diagram is called an alternating knot , otherwise it is called a non-alternating knot. Alternating diagram Non-alternating diagram In 2015 Greene and Howie,independently, gave a characterization of alternating links. Guevara Hern´ andez (OCAMI) alt ( K ) and dalt ( K ) December 25, 2018 3 / 24

Definition (Adams et al., 1992) The dealternating number of a link diagram D is the minimum number of crossing changes necessary to transform D into an alternating diagram. The dealternating number of a link L, denoted dalt ( L ) , is the minimum dealternating number of any diagram of L. A link with dealternating number k is also called k-almost alternating . We say that a link is almost alternating if it is 1-almost alternating. Guevara Hern´ andez (OCAMI) alt ( K ) and dalt ( K ) December 25, 2018 4 / 24

Definition (Kawauchi, 2010) The alternation number of a link diagram D is the minimum number of crossing changes necessary to transform D into some (possibly non-alternating) diagram of an alternating link. The alternation number of a link L, denoted alt ( L ) , is the minimum alternation number of any diagram of L. Guevara Hern´ andez (OCAMI) alt ( K ) and dalt ( K ) December 25, 2018 5 / 24

alt ( L ) = 1 dalt ( L ) = 2 alt ( L ) ≤ dalt ( L ) Guevara Hern´ andez (OCAMI) alt ( K ) and dalt ( K ) December 25, 2018 6 / 24

alt ( L ) = 1 dalt ( L ) = 2 alt ( L ) ≤ dalt ( L ) Adams et al. showed that an almost alternating knot is either a torus knot or a hyperbolic knot. Guevara Hern´ andez (OCAMI) alt ( K ) and dalt ( K ) December 25, 2018 6 / 24

Turaev genus To a link diagram D , Turaev associated a closed orientable surface embedded in S 3 , called the Turaev surface . Definition (Turaev, 1987) The Turaev genus, g T ( L ) , of a link L is the minimal number of the genera of the Turaev surfaces of diagrams of L. [Dasbach et al., 2008] g T ( L ) = 0 if and only if L is alternating. Guevara Hern´ andez (OCAMI) alt ( K ) and dalt ( K ) December 25, 2018 7 / 24

Khovanov homology [Khovanov, 2000] Let L ∈ S 3 be an oriented link. The Khovanov homology of L , denoted Kh ( L ), is a bigraded Z -module with homological grading i and polynomial i , j Kh i , j ( L ). (or Jones) grading j so that Kh ( L ) = � j � i − 4 − 3 − 2 − 1 0 1 2 7 1 5 3 1 1 1 1 1 − 1 1 1 − 3 1 1 − 5 − 7 1 The coefficients of the monomials t i q j are shown. j − 2 i = s + 1 or j − 2 i = s − 1, where s = 2 is the signature of 9 42 . Guevara Hern´ andez (OCAMI) alt ( K ) and dalt ( K ) December 25, 2018 8 / 24

w Kh ( K ) δ Kh δ ( L ). δ = j − 2 i so that Kh ( L ) = � Let δ min be the minimum δ -grading where Kh ( L ) is nontrivial and δ max be the maximum δ -grading where Kh ( L ) is nontrivial. Kh ( L ) is said to be [ δ min , δ max ]-thick, and the Khovanov width of L is defined as w Kh ( L ) = 1 2( δ max − δ min ) + 1 . Guevara Hern´ andez (OCAMI) alt ( K ) and dalt ( K ) December 25, 2018 9 / 24

alt ( K ) ≤ dalt ( K ) . (1) g T ( K ) ≤ dalt ( K ) . (2) w Kh ( K ) − 2 ≤ g T ( K ) . (3) � w HF ( K ) − 1 ≤ g T ( K ) . (4) (2) [Abe and Kishimoto, 2010]; (3)[Champanerkar et al., 2007] and [Champanerkar and Kofman, 2009]; (4)[M. Lowrance, 2008]. Guevara Hern´ andez (OCAMI) alt ( K ) and dalt ( K ) December 25, 2018 10 / 24

| σ ( K ) − s ( K ) | ≤ alt ( K ) . (5) 2 | σ ( K ) − s ( K ) | ≤ g T ( K ) . (6) 2 Skein relation 0 ≤ σ ( K + ) − σ ( K − ) ≤ 2 . (7) 0 ≤ s ( K + ) − s ( K − ) ≤ 2 . (8) where σ ( K ) and s ( K ) are the signature and Rasmussen s -invariant of a knot K , respectively, and both invariants are equal to 2 for the positive trefoil knot. (5) [Abe, 2009]; (6)[Dasbach and Lowrance, 2011]; (7) [Cochran and Lickorish, 1986]; (8) [Rasmussen, 2010]. Guevara Hern´ andez (OCAMI) alt ( K ) and dalt ( K ) December 25, 2018 11 / 24

alt ( K ) and dalt ( K ) [Abe and Kishimoto, 2010] Examples where the alternation number equals the dealternating number. [Lowrance, 2015] For all n ∈ N there exists a knot K , which is the iteration of Whitehead doubles of eight figure-eight knot, such that alt ( K ) = 1 and n ≤ dalt ( K ). [Guevara-Hern´ andez, 2017] For all n ∈ N there exist a knot family DS n such that if K ∈ DS n then alt ( K ) = 1 and dalt ( K ) = n . Guevara Hern´ andez (OCAMI) alt ( K ) and dalt ( K ) December 25, 2018 12 / 24

Families of knots 3 ( σ 1 σ 2 ) 3 n · c ) 2 σ − 1 N (( σ 2 σ 3 ) 3( m +1) σ l where l , m , n ∈ N . Guevara Hern´ andez (OCAMI) alt ( K ) and dalt ( K ) December 25, 2018 13 / 24

Families of knots 3 ( σ 1 σ 2 ) 3 n · c ) 2 σ − 1 N (( σ 2 σ 3 ) 3( m +1) σ l where l , m , n ∈ N . Guevara Hern´ andez (OCAMI) alt ( K ) and dalt ( K ) December 25, 2018 13 / 24

Families of knots 3 ( σ 1 σ 2 ) 3 n · c ) 2 σ − 1 N (( σ 2 σ 3 ) 3( m +1) σ l where l , m , n ∈ N . Guevara Hern´ andez (OCAMI) alt ( K ) and dalt ( K ) December 25, 2018 13 / 24

Families of knots 3 ( σ 1 σ 2 ) 3 n · c ) 2 σ − 1 N (( σ 2 σ 3 ) 3( m +1) σ l where l , m , n ∈ N . Guevara Hern´ andez (OCAMI) alt ( K ) and dalt ( K ) December 25, 2018 13 / 24

Families of knots 3 ( σ 1 σ 2 ) 3 n · c ) 2 σ − 1 N (( σ 2 σ 3 ) 3( m +1) σ l where l , m , n ∈ N . Guevara Hern´ andez (OCAMI) alt ( K ) and dalt ( K ) December 25, 2018 13 / 24

Families of knots 3 ( σ 1 σ 2 ) 3 n · c ) N (( σ 2 σ 3 ) 3( m +1) σ l 2 σ − 1 where l , m , n ∈ N . Guevara Hern´ andez (OCAMI) alt ( K ) and dalt ( K ) December 25, 2018 13 / 24

w Kh ( K ) − 2 ≤ dalt ( K ) Theorem (Khovanov, 2010) There are long exact sequences · · · Kh i − e − 1 , j − 3 e − 2 ( D h ) → Kh i , j ( D + ) → Kh i , j − 1 ( D v ) → Kh i − 3 , j − 3 e − 2 ( D h ) → · · · and · · · Kh i , j +1 ( D v ) → Kh i , j ( D − ) → Kh i − e +1 , j − 3 e +2 ( D h ) → Kh i +1 , j +1 ( D v ) → · · · When only the δ = j − 2 i grading is considered, the long exact sequence become f δ − e g δ f δ − 1 · · · Kh δ − e ( D h ) + → Kh δ ( D + ) → Kh δ − 1 ( D v ) + + → Kh δ − e − 2 ( D h ) → · · · − − − − − − − and f δ +1 g δ h δ − e · · · Kh δ +1 ( D v ) → Kh δ ( D − ) → Kh δ − e ( D h ) → Kh δ − 1 ( D v ) → · · · − − − − − − − − − − e = neg ( D h ) − neg ( D + ) The crossings D + , D − , D v , D h , respectively. Guevara Hern´ andez (OCAMI) alt ( K ) and dalt ( K ) December 25, 2018 14 / 24

w Kh ( K ) Corollary Let D + , D − , D v and D h be as above. Suppose Kh ( D v ) is [ v min , v max ] -thick and Kh ( D h ) is [ h min , h max ] -thick. Then Kh ( D + ) is [ δ + min , δ + max ] -thick, and Kh ( D − ) is [ δ − min , δ − max ] -thick, where  min { v min + 1 , h min + e } if v min � = h min + e + 1  if v min = h min + e + 1 and h v min δ + min = v min + 1 is surjective + if v min = h min + e + 1 and h v min v min − 1 is not surjective,  +  max { v max + 1 , h max + e } if v max � = h max + e + 1  if v max = h max + e + 1 and h v max δ + max = v max − 1 is injective + if v max = h max + e + 1 and h v max v max + 1 is not injective,  +  min { v min − 1 , h min + e } if v min � = h min + e − 1  if v min = h min + e − 1 and h v min δ − min = v min + 1 is surjective − if v min = h min + e − 1 and h v min v min − 1 is not surjective,  −  max { v max − 1 , h max + e } if v max � = h max + e − 1  if v max = h max + e − 1 and h v max max = v max − 1 is injective δ − − if v max = h max + e − 1 and h v max v max + 1 is not injective.  − Guevara Hern´ andez (OCAMI) alt ( K ) and dalt ( K ) December 25, 2018 15 / 24

Lemma (G.) 3 ( σ 1 σ 2 ) 3 n · c ) , then Kh ( D ) is 2 σ − 1 If D = N (( σ 2 σ 3 ) 3( m +1) σ l [4 m + l + 2 , 6 m + 2 n + l + 4] -thick. Hence, w Kh ( D ) = m + n + 2 . Proof. (outline) D + Guevara Hern´ andez (OCAMI) alt ( K ) and dalt ( K ) December 25, 2018 16 / 24

Lemma (G.) 3 ( σ 1 σ 2 ) 3 n · c ) , then Kh ( D ) is 2 σ − 1 If D = N (( σ 2 σ 3 ) 3( m +1) σ l [4 m + l + 2 , 6 m + 2 n + l + 4] -thick. Hence, w Kh ( D ) = m + n + 2 . Proof. (outline) D + D v D h Guevara Hern´ andez (OCAMI) alt ( K ) and dalt ( K ) December 25, 2018 16 / 24