On Computation of HOMFLY-PT Polynomials of 2–Bridge Diagrams

On computation of HOMFLY-PT polynomials of 2–bridge diagrams .

.. Masahiko Murakami . Joint work with Fumio Takeshita and Seiichi Tani

Nihon University .

December 20th, 2010

1 Masahiko Murakami (Nihon University) On computation of HOMFLY-PT polynomials December 20th, 2010 1 / 28 Contents

Motivation and Results Preliminaries Computation Conclusion

1 Masahiko Murakami (Nihon University) On computation of HOMFLY-PT polynomials December 20th, 2010 2 / 28 Contents

Motivation and Results Preliminaries Computation Conclusion

1 Masahiko Murakami (Nihon University) On computation of HOMFLY-PT polynomials December 20th, 2010 3 / 28 There exist polynomial time algorithms for computing Jones polynomials and HOMFLY-PT polynomials under reasonable restrictions.

Computational Complexities of Knot Polynomials

Alexander polynomial [Alexander](1928) Generally, polynomial time Jones polynomial [Jones](1985) Generally, #P–hard [Jaeger, Vertigan and Welsh](1993) HOMFLY-PT polynomial [Freyd, Yetter, Hoste, Lickorish, Millett, Ocneanu](1985) [Przytycki, Traczyk](1987) Generally, #P–hard [Jaeger, Vertigan and Welsh](1993)

1 Masahiko Murakami (Nihon University) On computation of HOMFLY-PT polynomials December 20th, 2010 4 / 28 Computational Complexities of Knot Polynomials

1 Masahiko Murakami (Nihon University) On computation of HOMFLY-PT polynomials December 20th, 2010 4 / 28 2–bridge diagrams O(n3) time our result implementation

Computational Complexities of Knot Polynomials Jones polynomials, Restricting knots and link types pretzel diagrams O(n2) time Utsumi, Imai(2002) 2–bridge diagrams O(n2) time Diao et al.(2009) Murakami et al.(2007, 2009) Montesinos diagrams O(n2) time Diao et al.(2009) Hara et al.(2009) arborescent diagrams O(n4 log n) time Hara et al.(2009) closed 3–braid diagrams O(n2) time Murakami et al.(2007, 2009) Jones polynomials, Bounded treewidths a constant polynomial time Makowsky(2001) at most two O(n5 log n) time Mighton(1999) HOMFLY-PT polynomials, Restricting knots and link types closed k braid diagrams for ﬁxed k polynomial time Mighton(1999) k–algebraic diagrams for ﬁxed k polynomial time Makowsky et al.(2003)

1 Masahiko Murakami (Nihon University) On computation of HOMFLY-PT polynomials December 20th, 2010 5 / 28 Computational Complexities of Knot Polynomials Jones polynomials, Restricting knots and link types pretzel diagrams O(n2) time Utsumi, Imai(2002) 2–bridge diagrams O(n2) time Diao et al.(2009) Murakami et al.(2007, 2009) Montesinos diagrams O(n2) time Diao et al.(2009) Hara et al.(2009) arborescent diagrams O(n4 log n) time Hara et al.(2009) closed 3–braid diagrams O(n2) time Murakami et al.(2007, 2009) Jones polynomials, Bounded treewidths a constant polynomial time Makowsky(2001) at most two O(n5 log n) time Mighton(1999) HOMFLY-PT polynomials, Restricting knots and link types closed k braid diagrams for ﬁxed k polynomial time Mighton(1999) k–algebraic diagrams for ﬁxed k polynomial time Makowsky et al.(2003) 2–bridge diagrams O(n3) time our result implementation

1 Masahiko Murakami (Nihon University) On computation of HOMFLY-PT polynomials December 20th, 2010 5 / 28 Contents

Motivation and Results Preliminaries Computation Conclusion

1 Masahiko Murakami (Nihon University) On computation of HOMFLY-PT polynomials December 20th, 2010 6 / 28 . .

[Deﬁnition] The Jones polynomial V : −3w(L) e V (L) = (−A) e hLi . t1/2=A−2 L: an oriented link, Le: an oriented link diagram of L, w(Le): the writhe of Le, hLei: the Kauﬀman bracket polynomial of Le with no orientations.

[Remark] A link diagram with n crossings. HOMFLY-PT Jones The absolute values of the degrees O(n) O(n) The number of the terms O(n2) O(n) The absolute values of the coeﬃcients O(2n) O(2n)

. The HOMFLY-PT Polynomial . [Deﬁnition] The HOMFLY-PT polynomial H: e e 1 H(K( ) = 1)for each trivial( knot) diagram( K. ) . . I − I I 2 lH + l 1H + mH = 0.

1 Masahiko Murakami (Nihon University) On computation of HOMFLY-PT polynomials December 20th, 2010 7 / 28 . .

[Remark] A link diagram with n crossings. HOMFLY-PT Jones The absolute values of the degrees O(n) O(n) The number of the terms O(n2) O(n) The absolute values of the coeﬃcients O(2n) O(2n)

. The HOMFLY-PT Polynomial . [Deﬁnition] The HOMFLY-PT polynomial H: e e 1 H(K( ) = 1)for each trivial( knot) diagram( K. ) . . I − I I 2 lH + l 1H + mH = 0.

1 Masahiko Murakami (Nihon University) On computation of HOMFLY-PT polynomials December 20th, 2010 7 / 28 . .

. The HOMFLY-PT Polynomial . [Deﬁnition] The HOMFLY-PT polynomial H: e e 1 H(K( ) = 1)for each trivial( knot) diagram( K. ) . . I − I I 2 lH + l 1H + mH = 0.

[Remark] A link diagram with n crossings. HOMFLY-PT Jones The absolute values of the degrees O(n) O(n) The number of the terms O(n2) O(n) The absolute values of the coeﬃcients O(2n) O(2n) 1 Masahiko Murakami (Nihon University) On computation of HOMFLY-PT polynomials December 20th, 2010 7 / 28 Integer Tangles

[Deﬁnition]

The 0-tangle twisted k times is called k-tangle and denoted by Ik. U ? 6 6 6K

0–tangle 3–tangle (−2)–tangle ∞–tangle −1−1 +1+1 −1+1 +1−1 I0 I3 I−2 I∞ Integer tangles.

1 Masahiko Murakami (Nihon University) On computation of HOMFLY-PT polynomials December 20th, 2010 8 / 28 2–bride Diagrams

eol or [Deﬁnition] a1, . . . , am: integers, R 1 1 (a1, . . . , am) (2–bridge diagram)

m is an odd number m is an even number e+1+1 e+1−1 R (a1, . . . , am) R (a1, . . . , am) 1 Masahiko Murakami (Nihon University) On computation of HOMFLY-PT polynomials December 20th, 2010 9 / 28 Contents

Motivation and Results Preliminaries Computation Conclusion

1 Masahiko Murakami (Nihon University) On computation of HOMFLY-PT polynomials December 20th, 2010 10 / 28 Our Algorithm

Link diagrams with n crossings

O(n2) time [Murakami et al.](2007) ⇓ O(n2) time [Murakami et al.](2007)

Integer sequences and orientations

O(n3) time Implementation ⇓ O(n3) time Implementation

HOMFLY-PT polynomials

1 Masahiko Murakami (Nihon University) On computation of HOMFLY-PT polynomials December 20th, 2010 11 / 28 Computation of HOMFLY-PT Polynomials of Integer Tangles

[Claim] For any integer k, the following holds. { −2 olor −1 olor l r l r −l H(I − ) − l mH(I − ) if o = o , H(I o o ) = k 2 k 1 k 2 olor olor l r −l H(Ik−2 ) − lmH(I∞ ) if o =6 o . Here, the formula refers to four link diagrams that are exactly the same except near an integer tangle where they diﬀer in the way indicated.

[Sketch of proof] ? ? ? 6 6 6 6 6 6 6 6 6

......

olor olor olor olor olor olor Ik Ik−2 Ik−1 Ik Ik−2 I∞ k > 0 and ol = or k > 0 and ol =6 or ( ) ( ) ( ) lH I + l−1H I + mH I = 0. 1 Masahiko Murakami (Nihon University) On computation of HOMFLY-PT polynomials December 20th, 2010 12 / 28 Computation of HOMFLY-PT Polynomials of Integer Tangles

[Sketch of proof] ? ? ? 6 6 6 6 6 6 6 6 6

......

olor olor olor olor olor olor Ik−2 Ik Ik−1 Ik−2 Ik I∞ k ≤ 0 and ol = or k ≤ 0 and ol =6 or ( ) ( ) ( ) lH I + l−1H I + mH I = 0. 1 Masahiko Murakami (Nihon University) On computation of HOMFLY-PT polynomials December 20th, 2010 13 / 28 Computation of HOMFLY-PT Polynomials of 2–bridge Diagrams

[Claim] For any integer k, the following holds. { −2 olor −1 olor l r l r −l H(I − ) − l mH(I − ) if o = o , H(I o o ) = k 2 k 1 k 2 olor olor l r −l H(Ik−2 ) − lmH(I∞ ) if o =6 o . [Lemma] l r eo1o1 m = 1 {H(R (a1)) − l r − l r − 2 eo1o1 − − 1 eo1o1 − l r l H(R (a1 2)) l mH(R (a1 1)) if o1 = o1, = l r − 2 eo1o1 − − l 6 r l H(R (a1 2)) lm if o1 = o1. l r eo1o1 m = 2 H(R (a1, . . . , am)) − l r  − 2 eo1o1 −  l H(R (a1, . . . , am−1, am 2))  − l r − 1 eo1o1 − l r l mH(R (a1, . . . , am−1, am 1)) if om = om, = l r  − 2 eo1o1 −  l H(R (a1, . . . , am−1, am 2))  l r − eo1o1 l 6 r lmH(R (a1, . . . , am−1)) if om = om.

1 Masahiko Murakami (Nihon University) On computation of HOMFLY-PT polynomials December 20th, 2010 14 / 28 Computation of HOMFLY-PT Polynomials of 2–bridge Diagrams

[Lemma]

eol or H(R 1 1 (a1, . . . , am)) −1 −1 −1  −lm − l m if m = 1 and a1 = 0,   1 if m = 1 and a1 = ±1 or  if m = 2 and a2 = 0, = eol or  H(R 1 1 (a1 ∓ 1)) if m = 2 and a2 = ±1,  l r  eo1o1 ≥  H(R (a1, . . . , am−2)) if m 3 and am = 0,  eol or H(R 1 1 (a1, . . . , am−2, am−1 ∓ 1)) if m ≥ 3 and am = ±1.

1 Masahiko Murakami (Nihon University) On computation of HOMFLY-PT polynomials December 20th, 2010 15 / 28 Order of Computation of HOMFLY-PT Polynomials of 2–bridge Diagrams

1st tangle 1st and 2nd tangles 1st, . . . , m–th tangles

eol or eol or eol or H(R 1 1 (0)) H(R 1 1 (a1, 0)) H(R 1 1 (a1, . . . , am−1, 0)) ⇓ ⇓ ⇓ eol or eol or eol or H(R 1 1 (1)) or H(R 1 1 (a1, 1)) or H(R 1 1 (a1, . . . , am−1, 1)) or eol or eol or eol or H(R 1 1 (−1)) H(R 1 1 (a1, −1)) H(R 1 1 (a1, . . . , am−1, −1)) ⇓ ⇓ ⇓ eol or eol or eol or H(R 1 1 (2)) or H(R 1 1 (a1, 2)) or H(R 1 1 (a1, . . . , am−1, 2)) or eol or eol or eol or H(R 1 1 (−2)) H(R 1 1 (a1, −2)) H(R 1 1 (a1, . . . , am−1, −2)) ⇓ ⇓ ⇓ . . . . ⇒ . ⇒ · · · ⇒ . ⇓ ⇓ ⇓ eol or eol or eol or H(R 1 1 (a1 − 1)) or H(R 1 1 (a1, a2 − 1)) or H(R 1 1 (a1, . . . , am−1, am − 1)) or eol or eol or eol or H(R 1 1 (a1 + 1)) H(R 1 1 (a1, a2 + 1)) H(R 1 1 (a1, . . . , am−1, am + 1)) ⇓ ⇓ ⇓ eol or eol or eol or H(R 1 1 (a1)) H(R 1 1 (a1, a2)) H(R 1 1 (a1, . . . , am−1, am)) ⇓ ⇓ eol or eol or H(R 1 1 (a1 + 1)) or H(R 1 1 (a1, a2 + 1)) or eol or eol or H(R 1 1 (a1 − 1)) H(R 1 1 (a1, a2 − 1))

1 Masahiko Murakami (Nihon University) On computation of HOMFLY-PT polynomials December 20th, 2010 16 / 28 eol or H(R 1 1 (0)) constant time ⇓ eol or H(R 1 1 (1)) eol or or H(R 1 1 (−1)) constant time ⇓ eol or H(R 1 1 (2)) eol or or H(R 1 1 (−2)) O(n2) time ⇓ . . ⇓ eol or H(R 1 1 (a1)) O(n2) time 2 Total: O(|a1|n ) time

Computational Complexity for Computing HOMFLY-PT polynomials of 2–bridge Diagrams

1st tangle

eol or −1 −1 −1 H(R 1 1 (0)) = −lm − l m

eol or H(R 1 1 (±1)) = 1

eol or H(R 1 1 (a1)) − l r  − 2 eo1o1 −  l H(R (a1 2))  −1 eol or  −l mH(R 1 1 (a1 − 1)) = l r  if o1 = o1,  2 eol or  −l H(R 1 1 (a − 2)) + lm  1 l 6 r if o1 = o1.

n = |a1| + ··· + |am|

1 Masahiko Murakami (Nihon University) On computation of HOMFLY-PT polynomials December 20th, 2010 17 / 28 ⇓ eol or H(R 1 1 (1)) eol or or H(R 1 1 (−1)) constant time ⇓ eol or H(R 1 1 (2)) eol or or H(R 1 1 (−2)) O(n2) time ⇓ . . ⇓ eol or H(R 1 1 (a1)) O(n2) time 2 Total: O(|a1|n ) time

Computational Complexity for Computing HOMFLY-PT polynomials of 2–bridge Diagrams

eol or H(R 1 1 (0)) 1st tangle constant time

eol or −1 −1 −1 H(R 1 1 (0)) = −lm − l m

eol or H(R 1 1 (±1)) = 1

n = |a1| + ··· + |am|

1 Masahiko Murakami (Nihon University) On computation of HOMFLY-PT polynomials December 20th, 2010 17 / 28 ⇓ eol or H(R 1 1 (2)) eol or or H(R 1 1 (−2)) O(n2) time ⇓ . . ⇓ eol or H(R 1 1 (a1)) O(n2) time 2 Total: O(|a1|n ) time

Computational Complexity for Computing HOMFLY-PT polynomials of 2–bridge Diagrams

eol or H(R 1 1 (0)) 1st tangle constant time ⇓ eol or −1 −1 −1 eol or H(R 1 1 (0)) = −lm − l m H(R 1 1 (1)) eol or eol or or H(R 1 1 (−1)) H(R 1 1 (±1)) = 1 constant time eol or H(R 1 1 (a1)) − l r  − 2 eo1o1 −  l H(R (a1 2))  −1 eol or  −l mH(R 1 1 (a1 − 1)) = l r  if o1 = o1,  2 eol or  −l H(R 1 1 (a − 2)) + lm  1 l 6 r if o1 = o1.

n = |a1| + ··· + |am|

1 Masahiko Murakami (Nihon University) On computation of HOMFLY-PT polynomials December 20th, 2010 17 / 28 ⇓ . . ⇓ eol or H(R 1 1 (a1)) O(n2) time 2 Total: O(|a1|n ) time

Computational Complexity for Computing HOMFLY-PT polynomials of 2–bridge Diagrams

n = |a1| + ··· + |am|

1 Masahiko Murakami (Nihon University) On computation of HOMFLY-PT polynomials December 20th, 2010 17 / 28 2 Total: O(|a1|n ) time

Computational Complexity for Computing HOMFLY-PT polynomials of 2–bridge Diagrams

1 Masahiko Murakami (Nihon University) On computation of HOMFLY-PT polynomials December 20th, 2010 17 / 28 Computational Complexity for Computing HOMFLY-PT polynomials of 2–bridge Diagrams

eol or H(R 1 1 (0)) 1st tangle constant time ⇓ eol or −1 −1 −1 eol or H(R 1 1 (0)) = −lm − l m H(R 1 1 (1)) eol or eol or or H(R 1 1 (−1)) H(R 1 1 (±1)) = 1 constant time eol or ⇓ H(R 1 1 (a1))  eol or −2 eol or H(R 1 1 (2))  −l H(R 1 1 (a − 2))  1 eol or  −1 eol or or H(R 1 1 (−2))  −l mH(R 1 1 (a − 1)) 1 O(n2) time = if ol = or,  1 1 ⇓  2 eol or  −l H(R 1 1 (a − 2)) + lm .  1 . l 6 r if o1 = o1. ⇓ eol or H(R 1 1 (a )) | | ··· | | 1 n = a1 + + am O(n2) time 2 Total: O(|a1|n ) time 1 Masahiko Murakami (Nihon University) On computation of HOMFLY-PT polynomials December 20th, 2010 17 / 28 eol or H(R 1 1 (a1, 0)) constant time ⇓ eol or H(R 1 1 (a1, 1)) eol or or H(R 1 1 (a1, −1)) constant time ⇓ eol or H(R 1 1 (a1, 2)) eol or or H(R 1 1 (a1, −2)) O(n2) time ⇓ . . ⇓ eol or H(R 1 1 (a1, a2)) O(n2) time 2 Total: O(|a2|n ) time

Computational Complexity for Computing HOMFLY-PT polynomials of 2–bridge Diagrams

2nd tangle

eol or H(R 1 1 (a1, 0)) = 1 eol or eol or H(R 1 1 (a1, ±1)) = H(R 1 1 (a1 ∓ 1)) eol or H(R 1 1 (a , a ))  1 2 − l r  − 2 eo1o1 −  l H(R (a1, a2 2))  −1 eol or  −l mH(R 1 1 (a1, a2 − 1))  l r if o2 = o2, = l r  − 2 eo1o1 −  l H(R (a1, a2 2))  l r  − eo1o1  lmH(R (a1)) l 6 r if o2 = o2

n = |a1| + ··· + |am|

1 Masahiko Murakami (Nihon University) On computation of HOMFLY-PT polynomials December 20th, 2010 18 / 28 ⇓ eol or H(R 1 1 (a1, 1)) eol or or H(R 1 1 (a1, −1)) constant time ⇓ eol or H(R 1 1 (a1, 2)) eol or or H(R 1 1 (a1, −2)) O(n2) time ⇓ . . ⇓ eol or H(R 1 1 (a1, a2)) O(n2) time 2 Total: O(|a2|n ) time

Computational Complexity for Computing HOMFLY-PT polynomials of 2–bridge Diagrams

eol or H(R 1 1 (a , 0)) 2nd tangle 1 constant time eol or H(R 1 1 (a1, 0)) = 1 eol or eol or H(R 1 1 (a1, ±1)) = H(R 1 1 (a1 ∓ 1)) eol or H(R 1 1 (a , a ))  1 2 − l r  − 2 eo1o1 −  l H(R (a1, a2 2))  −1 eol or  −l mH(R 1 1 (a1, a2 − 1))  l r if o2 = o2, = l r  − 2 eo1o1 −  l H(R (a1, a2 2))  l r  − eo1o1  lmH(R (a1)) l 6 r if o2 = o2

n = |a1| + ··· + |am|

1 Masahiko Murakami (Nihon University) On computation of HOMFLY-PT polynomials December 20th, 2010 18 / 28 ⇓ eol or H(R 1 1 (a1, 2)) eol or or H(R 1 1 (a1, −2)) O(n2) time ⇓ . . ⇓ eol or H(R 1 1 (a1, a2)) O(n2) time 2 Total: O(|a2|n ) time

Computational Complexity for Computing HOMFLY-PT polynomials of 2–bridge Diagrams

eol or H(R 1 1 (a , 0)) 2nd tangle 1 constant time l r ⇓ eo1o1 H(R (a1, 0)) = 1 eol or H(R 1 1 (a1, 1)) l r l r l r eo1o1 ± eo1o1 ∓ eo o H(R (a1, 1)) = H(R (a1 1)) or H(R 1 1 (a1, −1)) eol or constant time H(R 1 1 (a , a ))  1 2 − l r  − 2 eo1o1 −  l H(R (a1, a2 2))  −1 eol or  −l mH(R 1 1 (a1, a2 − 1))  l r if o2 = o2, = l r  − 2 eo1o1 −  l H(R (a1, a2 2))  l r  − eo1o1  lmH(R (a1)) l 6 r if o2 = o2

n = |a1| + ··· + |am|

1 Masahiko Murakami (Nihon University) On computation of HOMFLY-PT polynomials December 20th, 2010 18 / 28 ⇓ . . ⇓ eol or H(R 1 1 (a1, a2)) O(n2) time 2 Total: O(|a2|n ) time

Computational Complexity for Computing HOMFLY-PT polynomials of 2–bridge Diagrams

n = |a1| + ··· + |am|

1 Masahiko Murakami (Nihon University) On computation of HOMFLY-PT polynomials December 20th, 2010 18 / 28 2 Total: O(|a2|n ) time

Computational Complexity for Computing HOMFLY-PT polynomials of 2–bridge Diagrams

eol or H(R 1 1 (a , 0)) 2nd tangle 1 constant time l r ⇓ eo1o1 H(R (a1, 0)) = 1 eol or H(R 1 1 (a1, 1)) l r l r l r eo1o1 ± eo1o1 ∓ eo o H(R (a1, 1)) = H(R (a1 1)) or H(R 1 1 (a1, −1)) eol or constant time H(R 1 1 (a , a ))  1 2 ⇓ − l r l r  − 2 eo1o1 − eo o  l H(R (a1, a2 2)) H(R 1 1 (a1, 2))  −1 eol or l r  −l mH(R 1 1 (a , a − 1)) eo1o1 −  1 2 or H(R (a1, 2)) l r 2 if o2 = o2, O(n ) time = 2 eol or  −l H(R 1 1 (a , a − 2)) ⇓  1 2 .  l r .  − eo1o1 .  lmH(R (a1)) ⇓ l 6 r if o2 = o2 eol or H(R 1 1 (a1, a2)) O(n2) time n = |a1| + ··· + |am|

1 Masahiko Murakami (Nihon University) On computation of HOMFLY-PT polynomials December 20th, 2010 18 / 28 Computational Complexity for Computing HOMFLY-PT polynomials of 2–bridge Diagrams

eol or H(R 1 1 (a , 0)) 2nd tangle 1 constant time l r ⇓ eo1o1 H(R (a1, 0)) = 1 eol or H(R 1 1 (a1, 1)) l r l r l r eo1o1 ± eo1o1 ∓ eo o H(R (a1, 1)) = H(R (a1 1)) or H(R 1 1 (a1, −1)) eol or constant time H(R 1 1 (a , a ))  1 2 ⇓ − l r l r  − 2 eo1o1 − eo o  l H(R (a1, a2 2)) H(R 1 1 (a1, 2))  −1 eol or l r  −l mH(R 1 1 (a , a − 1)) eo1o1 −  1 2 or H(R (a1, 2)) l r 2 if o2 = o2, O(n ) time = 2 eol or  −l H(R 1 1 (a , a − 2)) ⇓  1 2 .  l r .  − eo1o1 .  lmH(R (a1)) ⇓ l 6 r if o2 = o2 eol or H(R 1 1 (a1, a2)) O(n2) time n = |a1| + ··· + |am| 2 Total: O(|a2|n ) time 1 Masahiko Murakami (Nihon University) On computation of HOMFLY-PT polynomials December 20th, 2010 18 / 28 Computational Complexity for Computing HOMFLY-PT polynomials of 2–bridge Diagrams

2 1st tangle ··· O(|a1|n ) time 2 2nd tangle ··· O(|a2|n ) time . . 2 m–th tangle ··· O(|am|n ) time 2 2 2 3 Total: O(|a1|n + |a2|n + ··· + |am|n ) = O(n ) time (Recall n = |a1| + ··· + |am|)

[Theorem] eol or 3 The HOMFLY polynomial of R 1 1 (a1, . . . , am) is computable in O(n ) time, where n is the number of the crossings of the input 2–bridge diagram.

1 Masahiko Murakami (Nihon University) On computation of HOMFLY-PT polynomials December 20th, 2010 19 / 28 Implementation

crossing # # of 2–bridge diagrams time(s) 20 524801 1 21 1049258 3 22 524891 5 Mac mini 23 4195668 12 CPU: Intel Core 2 Duo 24 8390657 25 2.53 GHz, 2.4GHz 25 16779946 80 Memory: 4GB 26 33558529 126 Mac OS X 10.6.5 27 67114324 291 C programming 28 134225921 588 language 29 268447378 1868 30 536887297 2815 31 1073763668 6421 32 2147516417 12567

1 Masahiko Murakami (Nihon University) On computation of HOMFLY-PT polynomials December 20th, 2010 20 / 28 Contents

Motivation and Results Preliminaries Computation Conclusion

1 Masahiko Murakami (Nihon University) On computation of HOMFLY-PT polynomials December 20th, 2010 21 / 28 Conclusion

Future Works Our Algorithm Extension of our algorithm 2–bridge diagrams with n crossings to other links (Ex. pretzel links, O(n2) time ⇓ [Murakami et al.](2007) arborescent links, Montesinos links and Integer sequences and orientations closed n–braid links). Extension of our algorithm O(nm ) time ⇓ Our result n to other invariants HOMFLY-PT polynomials (Ex. the Kauﬀman polynomial).

1 Masahiko Murakami (Nihon University) On computation of HOMFLY-PT polynomials December 20th, 2010 22 / 28 One of the fundamental problems in knot theory. in NP. Algorithm ··· [Haken](1961). Computational complexity ··· [Hass, Lagarias and Pippenger](1999). in co–NP?

Computational Complexity of Unknotting Problem

Input: A knot. Question: Is the knot equivalent to the trivial knot?

? Equivalent

trivial knot

1 Masahiko Murakami (Nihon University) On computation of HOMFLY-PT polynomials December 20th, 2010 23 / 28 in NP. Algorithm ··· [Haken](1961). Computational complexity ··· [Hass, Lagarias and Pippenger](1999). in co–NP?

Computational Complexity of Unknotting Problem

Input: A knot. Question: Is the knot equivalent to the trivial knot?

? Equivalent

trivial knot One of the fundamental problems in knot theory.

1 Masahiko Murakami (Nihon University) On computation of HOMFLY-PT polynomials December 20th, 2010 23 / 28 in co–NP?

Computational Complexity of Unknotting Problem

Input: A knot. Question: Is the knot equivalent to the trivial knot?

? Equivalent

trivial knot One of the fundamental problems in knot theory. in NP. Algorithm ··· [Haken](1961). Computational complexity ··· [Hass, Lagarias and Pippenger](1999).

1 Masahiko Murakami (Nihon University) On computation of HOMFLY-PT polynomials December 20th, 2010 23 / 28 Computational Complexity of Unknotting Problem

Input: A knot. Question: Is the knot equivalent to the trivial knot?

? Equivalent

trivial knot One of the fundamental problems in knot theory. in NP. Algorithm ··· [Haken](1961). Computational complexity ··· [Hass, Lagarias and Pippenger](1999). in co–NP?

1 Masahiko Murakami (Nihon University) On computation of HOMFLY-PT polynomials December 20th, 2010 23 / 28 For knots Algorithm ··· [Haken](1961), [Hemion](1979) and [Matveev](1997). Computational complexity ··· unknown. For links Algorithm ··· unknown.

Computational Complexity of Equivalence Problem

Input: Two links. Question: Are the links equivalent?

? Equivalent

1 Masahiko Murakami (Nihon University) On computation of HOMFLY-PT polynomials December 20th, 2010 24 / 28 For links Algorithm ··· unknown.

Computational Complexity of Equivalence Problem

Input: Two links. Question: Are the links equivalent?

? Equivalent

For knots Algorithm ··· [Haken](1961), [Hemion](1979) and [Matveev](1997). Computational complexity ··· unknown.

1 Masahiko Murakami (Nihon University) On computation of HOMFLY-PT polynomials December 20th, 2010 24 / 28 Computational Complexity of Equivalence Problem

Input: Two links. Question: Are the links equivalent?

? Equivalent

For knots Algorithm ··· [Haken](1961), [Hemion](1979) and [Matveev](1997). Computational complexity ··· unknown. For links Algorithm ··· unknown.

1 Masahiko Murakami (Nihon University) On computation of HOMFLY-PT polynomials December 20th, 2010 24 / 28 Tait Graphs

[Example] [Deﬁnition] Color the faces black and white in such a way that the unique unbounded face is white, and no two faces with a common arc are the same color. Put a vertex on each black face. Join two vertices by a labeled edge if they share a crossing.

1 Masahiko Murakami (Nihon University) On computation of HOMFLY-PT polynomials December 20th, 2010 25 / 28 Tait Graphs

1 Masahiko Murakami (Nihon University) On computation of HOMFLY-PT polynomials December 20th, 2010 26 / 28 Tait Graphs

1 Masahiko Murakami (Nihon University) On computation of HOMFLY-PT polynomials December 20th, 2010 27 / 28 Tait Graphs [Deﬁnition] [Example] [Remark] #crossings = #edges Color the faces black and white. Put a vertex on each +1 +1 black face. Join two vertices by a labeled edge if they share a crossing. −1 −1

+1 −1 Labels of edges. +1 1 Masahiko Murakami (Nihon University) On computation of HOMFLY-PT polynomials December 20th, 2010 28 / 28