On Kauffman polynomial of alternating knot and HOMFLY polynomial of - PowerPoint PPT Presentation

On Kauffman polynomial of alternating knot and HOMFLY polynomial of its Whitehead double 三浦 嵩広 ( 神戸大学大学院 理学研究科 ) 結び目の数理 II 1/24

Kauffman polynomial The Kauffman polynomial F ( L ) = F ( L ; a, x ) ∈ Z [ a ± 1 , x ± 1 ] is a link invariant defined as follows: F ( L ) := a − w ( D ) [ D ] , where w ( D ) is the writhe of a diagram D of a link L . In this talk, we study the highest degree of x in F ( L ) . 2/24

HOMFLY polynomial The HOMFLY polynomial P ( L ) = P ( L ; v, z ) ∈ Z [ v ± 1 , z ± 1 ] is a link invariant defined as follows: W ( K ) : a Whitehead double of a knot K ➔ ➔ In this talk, we study the highest degree of z in P ( W ( K )) . 3/24

✛ ✘ Conj.1 [Kidwell-Stoimenow 2003] K : a nontrivial knot. ⇒ 2( x - maxdeg F ( K )) + 2 = z - maxdeg P ( W ( K )) . ✚ ✙ In this talk, we discuss the following conjecture. ✛ ✘ Conj.2 K : an alternating prime knot. ⇒ 2( x - maxdeg F ( K )) + 2 = z - maxdeg P ( W ( K )) . ✚ ✙ Prop.1 [Thistlethwaite 1988] K : an alternating prime knot ⇒ x - maxdeg F ( K ) = c ( K ) − 1 , where c ( K ) is the crossing number of K . 4/24

From this Fact, Conj.2 ⇔ Conj.2’. ✛ ✘ Conj.2 K : an alternating prime knot. ⇒ 2( x - maxdeg F ( K )) + 2 = z - maxdeg P ( W ( K )) . ✚ ✙ ✛ ✘ Conj.2’ K : an alternating prime knot. ⇒ z - maxdeg P ( W ( K )) = 2 c ( K ) . ✚ ✙ Remark K : a nontrivial knot ⇒ z - maxdeg P ( W ( K )) ≤ 2 c ( K ) , by Morton’s inequality z - maxdeg P ( D ) ≤ 1 − s ( D ) + c ( D ) , where s ( D ) is the number of Seifert circles of a diagram D 5/24

the coefficient of x c − 1 of F ( K ) the coefficient of z 2 c of P ( W ( K )) K a − 3 ( a − 1 + a ) v α ( v − 1 − v ) 3 1 ( a − 1 + a ) v α ( v − 1 − v ) 4 1 a − 5 ( a − 1 + a ) v α ( v − 1 − v ) 5 1 a − 5 ( a − 1 + a ) v α ( v − 1 − v ) 5 2 a − 2 ( a − 1 + a ) v α ( v − 1 − v ) 6 1 a − 2 ( a − 1 + a ) v α ( v − 1 − v ) 6 2 ( a − 1 + a ) v α ( v − 1 − v ) 6 3 a − 7 ( a − 1 + a ) v α ( v − 1 − v ) 7 1 a − 7 ( a − 1 + a ) v α ( v − 1 − v ) 7 2 a − 7 ( a − 1 + a ) v α ( v − 1 − v ) 7 3 a − 7 ( a − 1 + a ) v α ( v − 1 − v ) 7 4 a − 7 ( a − 1 + a ) v α ( v − 1 − v ) 7 5 a − 3 ( a − 1 + a ) v α ( v − 1 − v ) 7 6 a ( a − 1 + a ) v α ( v − 1 − v ) 7 7 —————————————————— α : a certain integer. 6/24

the coefficient of x c − 1 of F ( K ) the coefficient of z 2 c of P ( W ( K )) K a − 3 ( a − 1 + a ) v α ( v − 1 − v ) 8 1 . . . . . . . . . 2 a − 3 ( a − 1 + a ) 2 v α ( v − 1 − v ) 8 16 2 a − 3 ( a − 1 + a ) 2 v α ( v − 1 − v ) 8 17 3 a − 3 ( a − 1 + a ) 3 v α ( v − 1 − v ) 8 18 a − 3 ( a − 1 + a ) v α ( v − 1 − v ) 9 1 . . . . . . . . . 2 a − 3 ( a − 1 + a ) 2 v α ( v − 1 − v ) 9 39 4 a − 3 ( a − 1 + a ) 4 v α ( v − 1 − v ) 9 40 2 a − 3 ( a − 1 + a ) 2 v α ( v − 1 − v ) 9 41 7/24

the coefficient of x c − 1 of F ( K ) the coefficient of z 2 c of P ( W ( K )) K 1 a − 3 ( a − 1 + a ) 1 v α ( v − 1 − v ) 8 1 . . . . . . . . . 2 a − 3 ( a − 1 + a ) 2 v α ( v − 1 − v ) 8 16 2 a − 3 ( a − 1 + a ) 2 v α ( v − 1 − v ) 8 17 3 a − 3 ( a − 1 + a ) 3 v α ( v − 1 − v ) 8 18 1 a − 3 ( a − 1 + a ) 1 v α ( v − 1 − v ) 9 1 . . . . . . . . . 2 a − 3 ( a − 1 + a ) 2 v α ( v − 1 − v ) 9 39 4 a − 3 ( a − 1 + a ) 4 v α ( v − 1 − v ) 9 40 2 a − 3 ( a − 1 + a ) 2 v α ( v − 1 − v ) 9 41 F ( K ; a ) : the coefficient of x c − 1 of F ( K ) . Def. P ( W ( K ); v ) : the coefficient of z 2 c of P ( W ( K )) . ϕ ( K ) := 1 π ( K ) := 1 2 F ( K ; 1) . 2 | P ( W ( K ); i ) | . 8/24

Previous research ✛ ✘ Thm. [Gruber 2009] K : an alternating prime knot. ⇒ ϕ ( K ) ≡ π ( K ) (mod 2 ). ✚ ✙ Main results Main result 1 K : an alternating prime knot with c ( K ) ≤ 12 ⇒ ϕ ( K ) = π ( K ) . Cor. K : an alternating prime knot with c ( K ) ≤ 12 ⇒ 2( x - maxdeg F ( K ))+2 = z - maxdeg P ( W ( K )) . 9/24

Main results Def. The standard projection of the (3 , n ) -torus link ( n = 2 , 3 , . . . ) this tangle of this projection 10/24

Main result 2 D : a link projection . D ′ : a knot projection obtained by repeating the following operations from D . ➔ ➔ ➔ ➔ K : the knot presented by an alternating diagram with D ′ ⇒ ϕ ( K ) = π ( K ) = 2 | β | 2 | γ | ∏ ( n − 1) | δ n | , where | β | , | γ | , n ≥ 2 | δ n | is the number of times of ( β ) , ( γ ) , ( δ n ) to obtain D ′ . 11/24

Example 12/24

Example | β | = 1 13/24

Example | β | = 1 , | γ | = 2 14/24

Example | β | = 1 , | γ | = 2 , | δ 4 | = 1 15/24

Example ϕ ( K ) = π ( K ) = 2 1 2 2 3 1 = 24 | β | = 1 , | γ | = 2 , | δ 4 | = 1 , 16/24

Outline of the proof of Main results D : a connected, link projection. D alt : an alternating diagram with D. ϕ ( D ) := 1 Def. of ϕ ( D ) 2 F ( D alt ; 1) , where F ( D alt ; a ) the coefficient of x c ( D ) − 1 in F ( D alt ; a, x ) . π ( D ) := 1 Def. of π ( D ) 2 | P ( W ( D alt ); i ) | , where P ( W ( D alt ); v ) the coefficient of z 2 c ( D ) in P ( W ( D alt ); v, z ) . e.g. W ( D alt ) D 17/24

Outline of the proof of Main results Prop.2 [Thistlethwaite 1988] (ii) D alt : a reducible diagram ⇒ ϕ ( D ) = 0 (iii) For a non-nugatory crossing, nugatory crossing ( ∵ ) (iii) By Kidwell’s inequality x - maxdeg F ( D ) ≤ c ( D ) − b ( D ) , where b ( D ) is the bridge length of a diagram D. and, the equation 18/24

Outline of the proof of Main results Prop.2 [Thistlethwaite 1988] (ii) D alt : a reducible diagram ⇒ ϕ ( D ) = 0 (iii) For a non-nugatory crossing, Main lemma (ii) D alt : a reducible diagram of nontrivial knot ⇒ π ( D ) = 0 , for non-nugatory 19/24

Outline of the proof of Main lemma (iii) At the relevant crossing P ( D 0 ) = z ( P ( D 5 ) + P ( D 7 )) , by Morton’s inequality. 20/24

Topic related to ϕ ( K ) The chromatic invariant κ ( G ) is the graph invariant defined as follows: (1) (2) If G has a cut-edge or a loop, then κ ( G ) = 0 (3) For non cut-edge, Prop.3 [Thistlethwaite 1988] K : an alternating prime knot. G : a plane graph associated with a reduced diagram of K. ➔ G ⇒ ϕ ( K ) = κ ( G ) . 21/24

Topic related to ϕ ( K ) The chromatic invariant κ ( G ) is the graph invariant defined as follows: (1) (2) If G has a cut-edge or a loop, then κ ( G ) = 0 (3) For non cut-edge, Remind (ii) D alt : a reducible diagram ⇒ ϕ ( D ) = 0 (iii) For a non-nugatory crossing, 22/24

Topic related to π ( K ) g c ( K ) : the canonical genus of a knot K ☛ ✟ := min { g ( S ) | S is a Seifert surface obtained by Seifert’s algorithm } Conj.3 [Tripp 2002] K : a prime knot ⇒ g c ( W ( K )) = c ( K ) . ✡ ✠ Fact z - maxdeg P ( W ( K )) ≤ 2 g c ( W ( K )) ≤ 2 c ( K ) . ✛ ✘ Conj.4 [Tripp 2002, Nakamura 2006, Brittenham-Jensen 2006] K : an alternating prime knot ⇒ π ( K ) ̸ = 0 , therefore g c ( W ( K )) = c ( K ) . ✚ ✙ Cor. K : alternating prime knot s.t. c ( K ) ≤ 12 or satisfying the condition of Main result 2. ⇒ ϕ ( K ) = π ( K ) ̸ = 0 , therefore g c ( W ( K )) = c ( K ) . 23/24

Further research • We can not obtain the following knot K by Main results. ϕ ( K ) = 11 . π ( K ) = ? ✓ ✏ • We propose the following conjectures. Main Conj.1 K : an alternating prime knot ⇒ ϕ ( K ) = π ( K ) . ✒ ✑ ✛ ✘ Main Conj.2 For any non-nugatory crossing, ✚ ✙ 24/24