The state numbers of a knot

Transcription

1 ILDT2014 RIMS, Kyoto University 21 May 2014 The state numbers of a knot Takuji NAKAMURA (Osaka Electro-Communication University) Joint work with Y. NAKANISHI, S. Satoh and Y. Tomiyama (Kobe University) 1

2 Min-type inv. for a knot K The crossing number c(k) :=min{# of crossings in D} The unknotting number u(k) :=min{# of c.c. for D to unknot} and so on Complexity of K 2

3 The n-state number for a knot K (n N) s n (K) :=min{# of n-states for D } Problems : 0. Determine s n (K) for a given K 1. Give upper/lower bounds by the crossing number c(k) of K the Jones and the Miyazawa polynomial 2. Is S n (i) :={K s n (K) =i} finite? 3

4 Contents 1. Preliminaries 2. Bounds for 1-,2-,3-state numbers 3. Knots with given 1-,2-,3-state numbers 4. The 1-state number and polynomial invariants 4

5 What is a virtual knot? 1. Preliminaries D : a virtual knot diagram def D : an immersed circle in R 2 with, real virtual Examples : classical 5

6 K : a virtual knot def K =[D]; an equivalence class of virtual knot diagrams D under generalized Reidemeister moves 6

7 State of a virtual knot diagram D : a virtual knot diagram S : a state of D def a union of circles by splicing all real crossings in D D Example: D : S :,, 7

8 S : an n-state def S consists of n circles (n 1) s n (D) :=#of n-state of D Example: D : 1-state 1-state 1-state 2-state s 1 (D) =3, s 2 (D) =1, s i (D) =0(i 3) 8

9 Example: 1-state 1-state 1-state 2-state 2-state 1-state 1-state 2-state s 1 (D) =5, s 2 (D) =3, s i (D) =0(i 3) 9

10 Example: 2-state 1-state 1-state 1-state 2-state 2-state 2-state 3-state s 1 (D) =3, s 2 (D) =4, s 3 (D) =1, s i (D) =0(i 4) 10

11 The n-state number for a virtual knot K : a virtual knot s n (K) :=min{s n (D) D : a diagram for K} ; the n-state number for K Fact 1.1. K : the trivial knot s 1 (K) =1, s i (K) =0(i 2). 11

12 Gauss diagram D : a virtual knot diagram G(D) : the Gauss diagram of D def preimage ofd; a circle with signed and oriented chords. Achordγ is oriented from the upper to the lower. 12

13 State of a Gauss diagram splice a crossing in D or splice a chord in G(D) or G: underlying Gauss diagram Lemma 1.2. D & D have the same underlying Gauss diagram G. s n (D )=s n (D) Notation: s n (G) :=s n (D) 13

14 2. Bounds for 1-,2-,3-state numbers K : a virtual knot c(k) : the minimal real crossing number for K Theorem 2.1. For any knot K, wehave (1) 1 s 1 (K) 2 2c(K) +( 1) c(k), 3 (2) 0 s 2 (K) 1 2 2c(K), (3) 0 s 3 (K) 3 8 2c(K). 14

15 Example: Gauss diagram F r : 1 2 r s 1 (F r )= 2 2r +( 1) r 3 F r? aknotk with s1 (K) = 2 2c(K) +( 1) c(k) 3 15

16 Example: Gauss diagrams F r : r F r 1 +1: free chord 1 2 r r 1 s 2 (F r )=s 2(F r 1 +1) = 1 2 2r F r? aknotk with s 2 (K) = 1 2 2c(K) 16

17 Example: Gauss diagrams F r 2 +2: F r 3 +3: s 3 (F r 2 +2) =s 3(F r 3 +3) = 3 8 2r 17

18 3. Finiteness problem A set of virtual knots with s n (K) =i; S n (i) :={K s n (K) =i} Proposition 3.1. (1) S 1 (1) = {trivial knot} (2) S 1 (2k) = (k 0) (3) S 2 (0) = {trivial knot} Is S n (i) finite? 18

19 Theorem 3.2. S 2 (i) is finite for any i. For any virtual knot K with c(k) 3, we see that c(k) s 2 (K). 19

20 Theorem 3.2. S 2 (i) is finite for any i. For any virtual knot K with c(k) 3, we see that c(k) s 2 (K). Theorem 3.3. S n (0) is infinite for n 3. S 3 (0) Remark: Any non-trivial K S 3 (0) is non-classical. 19-a

21 Proposition 3.4. S 1 (i) is finite for i 7. In particular, S 1 (3) = {2.1, 3.5, 3.6, 3.7}. 20

22 Proposition 3.4. S 1 (i) is finite for i 7. In particular, S 1 (3) = {2.1, 3.5, 3.6, 3.7}. Theorem 3.5. S 1 (9) is infinite. 20-a

23 Proposition 3.4. S 1 (i) is finite for i 7. In particular, S 1 (3) = {2.1, 3.5, 3.6, 3.7}. Theorem 3.5. S 1 (9) is infinite. K m : G m : }{{} 2m crossings s 1 (G m )=9 K m K m if m m by the Miyazawa polynomial. a virtual knot invariant 20-b

24 4. Lower bounds for s 1 (K) by polynomials Lower bounds for s 1 (K) by Jones polynomial f K (A) =( A 3 ) w(d) S A a(s) b(s) ( A 2 A 2 ) S 1 w(d) : writhe of D, S := #{circles in a state S of D} a(s) :=#{A-splice D S}, b(s) :=#{B-splice D S} A-splice B-splice f K (t 1 4)=V K (t) : Jones polynomial of K Proposition 4.1. s 1 (K) f K (ξ) = V K ( 1),whereξ = e π 4 i 21

25 Example: AAA AAB ABA BAA ABB BAB BBA BBB 22

26 Example: d = A 2 A 2 A 3 0 d 2 1 A 2 1 d 1 1 A 2 1 d 1 1 A 2 1 d 1 1 A 1 2 d 2 1 A 1 2 d 2 1 A 1 2 d 2 1 A 0 3 d 3 1 f K (A) =A 4 + A 12 A 16 A=ξ f K (ξ) =3 s 1 (K) 3 22-a

27 Miyazawa polynomial R K (A, x) R K (A, x) = F 0 (A) + F 1 (A) x 1 + F 01 (A) x 2 + F 2 (A)x F 001 (A) x 3 + F 11 (A)x 1 x 2 + F 3 (A)x : a generalization of the Jones polynomial F 0 } 0 {{} 1(A) is derived from 1-states. k 1 F 0 (A) =F 0 (A)/( A 2 A 2 ) Theorem 4.2. s 1 (K) F 0 (ξ) + k=1 F 0 } 0 {{} 1(ξ). k 1 23

28 Example K: virtual knot 3.3 in Green s table D s.t. s 1 (D) =5 V K (t) = t t t 1 V K ( 1) = 5 R K (A, x) =( A 2 A 2 ) ( A 8 + A 4 ) F 0 (ξ) + k=1 + ( A 10 + A 6 ) x 1 A 10 x ( A12 ) x 2 F Ik (ξ) = F 0 (ξ) + F 1 (ξ) + F 01 (ξ) =5 5 s 1 (K) s 1 (D) =5 24

29 Definition of Miyazawa polynomial R K (A, x) D an oriented virtual diagram S : a poled state of D obtained by Example: A B D poled state S 25

30 Reduction of the number of poles (1) Sliding a pole through virtual crossings. (2) Remove two successive poles on the same side C : the circle obtained from C by reductions C: p(c) := #(poles in C) 2 26

31 a(s) :=#{A-splice D S}, b(s) :=#{B-splice D S} c i (S) :=#{C in S p(c) =i} S D := A a(s) b(s) ( A 2 A 2 ) c 0(S) x c 1(S) 1 x c 2(S) 2 Example B C 2 S C 1 B a(s) =0, b(s) =2, p(c 1 )=0, p(c 2 )=1, c 0 (S) =1, c 1 (S) =1, c i (S) =0(i 2) S D = A 0 2 x 0 x 1 = A 2 ( A 2 A 2 )x 1 R K (A, x) :=( A 3 ) w(d) S S D Z[A, A 1,x 1,x 2,...] 27

32 Example: K : the virtual knot 3.3 in Green s table diagram D AAA AAB ABA BAA w(d) = 3 ABB BAB BBA BBB 28

33 Example: K : the virtual knot 3.3 in Green s table diagram D A 3 0 x 2 A 2 1 x 1 A 2 1 x 1 A 2 1 x 2 1 w(d) = 3 x 0 = A 2 A 2 A 1 2 x 0 x 1 A 1 2 x 0 A 1 2 x 0 A 0 3 x 2 0 R K (A, x) =( A 2 A 2 )( A 8 + A 4 ) +( A 10 + A 6 )x 1 A 10 x 2 1 A12 x 2 28-a