[PPT] - On computation of HOMFLY polynomials of Montesinos links Masahiko PowerPoint Presentation

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On computation of HOMFLY polynomials of Montesinos links

Masahiko Murakami

Nihon University

December 18th, 2013

Masahiko Murakami (Nihon University) On computation of HOMFLY polynomials of Montesinos links December 18th, 2013 1 / 37

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Contents

Motivation and Results Preliminaries Computation

1

2–bridge Diagrams

2

Pretzel Diagrams

3

Montesinos Diagrams

Conclusion

Masahiko Murakami (Nihon University) On computation of HOMFLY polynomials of Montesinos links December 18th, 2013 2 / 37

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Contents

Motivation and Results Preliminaries Computation

1

2–bridge Diagrams

2

Pretzel Diagrams

3

Montesinos Diagrams

Conclusion

Masahiko Murakami (Nihon University) On computation of HOMFLY polynomials of Montesinos links December 18th, 2013 3 / 37

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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 polynomial (HOMFLY-PT polynomial) [Freyd, Yetter, Hoste, Lickorish, Millett, Ocneanu](1985) [Przytycki, Traczyk](1987) Generally, #P–hard [Jaeger, Vertigan and Welsh](1993) There exist polynomial time algorithms for computing Jones polynomials and HOMFLY polynomials under reasonable restrictions.

Masahiko Murakami (Nihon University) On computation of HOMFLY polynomials of Montesinos links December 18th, 2013 4 / 37

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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 polynomials, Restricting knots and link types closed k braid diagrams for fixed k

poly. time

Mighton(1999) k–algebraic diagrams for fixed k

poly. time

Makowsky et al.(2003) 2–bridge diagrams O(n3) time Murakami et al.(2014) pretzel diagrams, Montesinos diagrams O(n3) time result

Masahiko Murakami (Nihon University) On computation of HOMFLY polynomials of Montesinos links December 18th, 2013 5 / 37

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Contents

Motivation and Results Preliminaries Computation

1

2–bridge Diagrams

2

Pretzel Diagrams

3

Montesinos Diagrams

Conclusion

Masahiko Murakami (Nihon University) On computation of HOMFLY polynomials of Montesinos links December 18th, 2013 6 / 37

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The HOMFLY Polynomial

[Definition] The HOMFLY polynomial H:

1 H(

K) = 1 for each trivial knot K.

2 lH

(

✒ ■

) + l−1H (

■ ✒ )

+ mH (

■ ✒ )

= 0. [Definition] The Jones polynomial V : V (L) = (−A)−3w(

L)⟨

L⟩

t1/2=A−2 .

L: an oriented link,

L: an oriented link diagram of L,

w( L): the writhe of L, ⟨ L⟩: the Kauffman bracket polynomial of L with no orientations. [Remark] A link diagram with n crossings. HOMFLY poly. Jones poly. 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)

Masahiko Murakami (Nihon University) On computation of HOMFLY polynomials of Montesinos links December 18th, 2013 7 / 37

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Integer Tangles

[Definition] The 0-tangle twisted k times is called k-tangle and denoted by Ik.

❯ ☛ ✻ ✻ ❄ ✻❑ ☛

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

Masahiko Murakami (Nihon University) On computation of HOMFLY polynomials of Montesinos links December 18th, 2013 8 / 37

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2–bride Diagrams

[Definition] a1, . . . , ak: integers, Rol

1or 1(a1, . . . , ak) (2–bridge diagram)

k is an odd number k is an even number

R+1+1(a1, . . . , ak)
R+1−1(a1, . . . , ak)

Masahiko Murakami (Nihon University) On computation of HOMFLY polynomials of Montesinos links December 18th, 2013 9 / 37

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Pretzel Diagrams

[Definition] a1, . . . , ak: integers, P o1,...,ok(a1, . . . , ak) (pretzel diagram)

P +1,−1,...,+1(a1, . . . , am)

[Definition] a1, . . . , ak: integers, Qo1,...,ok+1(a1, . . . , ak)

Q+1,−1,...,+1(a1, . . . , am)

Masahiko Murakami (Nihon University) On computation of HOMFLY polynomials of Montesinos links December 18th, 2013 10 / 37

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Montesinos Diagrams

[Definition] a1,1, . . . , a1,v1, . . . , au,1, . . . , au,vu, a: integers,

Mot

1,ob 1,...,ot u,ob u(a1,1, . . . , a1,v1| · · · |au,1, . . . , au,vu||a) (Montesinos diagram)

Ia1,1 Ia1,v1 Ia1,2 Ia1,v1-1 Ia1,3 Ia2,1 Ia2,v2 Ia2,2 Ia2,v2-1 Iau,1 Iau,vu Iau,2 Iau,vu-1 Iau,3 Ia22 Ia

Masahiko Murakami (Nihon University) On computation of HOMFLY polynomials of Montesinos links December 18th, 2013 11 / 37

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Contents

Motivation and Results Preliminaries Computation

1

2–bridge Diagrams

2

Pretzel Diagrams

3

Montesinos Diagrams

Conclusion

Masahiko Murakami (Nihon University) On computation of HOMFLY polynomials of Montesinos links December 18th, 2013 12 / 37

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Algorithm

Link diagrams with n crossings O(n) time ⇓ O(n) time Integer sequences and orientations O(n3) time ⇓ O(n3) time HOMFLY polynomials

Masahiko Murakami (Nihon University) On computation of HOMFLY polynomials of Montesinos links December 18th, 2013 13 / 37

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Computation of HOMFLY Polynomials of Integer Tangles

[Claim] For any integer k, the following holds. H(Ikolor) = { −l−2H(Ik−2olor) − l−1mH(Ik−1olor) if ol = or, −l2H(Ik−2olor) − lmH(I∞olor) if ol ̸= or. 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] . . .

✻ ✻

. . .

✻ ✻

. . .

✻ ✻

Ikolor Ik−2olor Ik−1olor k > 0 and ol = or . . .

✻ ❄

. . .

✻ ❄

. . .

✻ ❄

Ikolor Ik−2olor I∞olor k > 0 and ol ̸= or lH (

✒ ■

) + l−1H (

■ ✒ )

+ mH (

■ ✒ )

= 0.

Masahiko Murakami (Nihon University) On computation of HOMFLY polynomials of Montesinos links December 18th, 2013 14 / 37

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Computation of HOMFLY Polynomials of Integer Tangles

[Claim] For any integer k, the following holds. H(Ik−2olor) = { −l2H(Ikolor) − lmH(Ik−1olor) if ol = or, −l−2H(Ikolor) − l−1mH(I∞olor) if ol ̸= or. 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] . . .

✻ ✻

. . .

✻ ✻

. . .

✻ ✻

Ik−2olor Ikolor Ik−1olor k ≤ 0 and ol = or . . .

✻ ❄

. . .

✻ ❄

. . .

✻ ❄

Ik−2olor Ikolor I∞olor k ≤ 0 and ol ̸= or lH (

✒ ■

) + l−1H (

■ ✒ )

+ mH (

■ ✒ )

= 0.

Masahiko Murakami (Nihon University) On computation of HOMFLY polynomials of Montesinos links December 18th, 2013 15 / 37

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Contents

Motivation and Results Preliminaries Computation

1

2–bridge Diagrams

2

Pretzel Diagrams

3

Montesinos Diagrams

Conclusion

Masahiko Murakami (Nihon University) On computation of HOMFLY polynomials of Montesinos links December 18th, 2013 16 / 37

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Computation of HOMFLY Polynomials of 2–bridge Diagrams

[Claim] For any integer k, the following holds. H(Ikolor) = { −l−2H(Ik−2olor) − l−1mH(Ik−1olor) if ol = or, −l2H(Ik−2olor) − lmH(I∞olor) if ol ̸= or.

Masahiko Murakami (Nihon University) On computation of HOMFLY polynomials of Montesinos links December 18th, 2013 17 / 37

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Computation of HOMFLY Polynomials of 2–bridge Diagrams

H( Rol

1or 1(a1, . . . , ak))

=                  −lm−1 − l−1m−1 if k = 1 and a1 = 0, 1 if k = 1 and a1 = ±1 or if k = 2 and a2 = 0, H( Rol

1or 1(a1 ∓ 1))

if k = 2 and a2 = ±1, H( Rol

1or 1(a1, . . . , ak−2))

if k ≥ 3 and ak = 0, H( Rol

1or 1(a1, . . . , ak−2, ak−1 ∓ 1))

if k ≥ 3 and ak = ±1.

Masahiko Murakami (Nihon University) On computation of HOMFLY polynomials of Montesinos links December 18th, 2013 18 / 37

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Order of Computation of HOMFLY Polynomials of 2–bridge Diagrams

1st tangle 1st and 2nd tangles 1st, . . . , k–th tangles H( Rol

1or 1(0))

H( Rol

1or 1(a1, 0))

H( Rol

1or 1(a1, . . . , ak−1, 0))

⇓ ⇓ ⇓ H( Rol

1or 1(1)) or

H( Rol

1or 1(a1, 1)) or

H( Rol

1or 1(a1, . . . , ak−1, 1)) or

H( Rol

1or 1(−1))

H( Rol

1or 1(a1, −1))

H( Rol

1or 1(a1, . . . , ak−1, −1))

⇓ ⇓ ⇓ H( Rol

1or 1(2)) or

H( Rol

1or 1(a1, 2)) or

H( Rol

1or 1(a1, . . . , ak−1, 2)) or

H( Rol

1or 1(−2))

H( Rol

1or 1(a1, −2))

H( Rol

1or 1(a1, . . . , ak−1, −2))

⇓ ⇓ ⇓ . . . ⇒ . . . ⇒ · · · ⇒ . . . ⇓ ⇓ ⇓ H( Rol

1or 1(a1 − 1)) or

H( Rol

1or 1(a1, a2 − 1)) or

H( Rol

1or 1(a1, . . . , ak−1, ak − 1)) or

H( Rol

1or 1(a1 + 1))

H( Rol

1or 1(a1, a2 + 1))

H( Rol

1or 1(a1, . . . , ak−1, ak + 1))

⇓ ⇓ ⇓ H( Rol

1or 1(a1))

H( Rol

1or 1(a1, a2))

H( Rol

1or 1(a1, . . . , ak−1, ak))

⇓ ⇓ H( Rol

1or 1(a1 + 1)) or

H( Rol

1or 1(a1, a2 + 1)) or

H( Rol

1or 1(a1 − 1))

H( Rol

1or 1(a1, a2 − 1))

Masahiko Murakami (Nihon University) On computation of HOMFLY polynomials of Montesinos links December 18th, 2013 19 / 37

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Computational Complexity for Computing HOMFLY polynomials of 2–bridge Diagrams

The HOMFLY polynomials of O(n) 2–bridge diagrams are computed. Each HOMFLY polynomial is computed in O(n2) time. [Claim] For any integer k, the following holds. H(Ikolor) = { −l−2H(Ik−2olor) − l−1mH(Ik−1olor) if ol = or, −l2H(Ik−2olor) − lmH(I∞olor) if ol ̸= or. [Theorem] The HOMFLY polynomial of Rol

1or 1(a1, . . . , ak) is computable in O(n3)

time, where n is the number of the crossings of the input 2–bridge diagram.

Masahiko Murakami (Nihon University) On computation of HOMFLY polynomials of Montesinos links December 18th, 2013 20 / 37

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Contents

Motivation and Results Preliminaries Computation

1

2–bridge Diagrams

2

Pretzel Diagrams

3

Montesinos Diagrams

Conclusion

Masahiko Murakami (Nihon University) On computation of HOMFLY polynomials of Montesinos links December 18th, 2013 21 / 37

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Computation of HOMFLY Polynomials of Pretzel Diagrams

[Claim] For any integer k, the following holds. H(Ikolor) = { −l−2H(Ik−2olor) − l−1mH(Ik−1olor) if ol = or, −l2H(Ik−2olor) − lmH(I∞olor) if ol ̸= or.

P +1,−1,...,+1(a1, . . . , am)
Q+1,−1,...,+1(a1, . . . , am)

Masahiko Murakami (Nihon University) On computation of HOMFLY polynomials of Montesinos links December 18th, 2013 22 / 37

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Computation of HOMFLY Polynomials of Pretzel Diagrams

H( P o1,...,ok(a1, . . . , ak)) =              1 if k = 1, H( Ro1,−o2(a1 + a2)) if k = 2, H( Qo1,...,ok−1(a1, . . . , ak−2)) if k ≥ 3, ak−1 = 0 and ak = ±1, H( P o1,...,ok−2(a1, . . . , ak−2)) if k ≥ 3, ak−1 = ±1 and ak = ∓1, H( Qo1,...,ok(a1, . . . , ak−1)) if k ≥ 3 and ak = 0.

P +1,−1,...,+1(a1, . . . , am)
Q+1,−1,...,+1(a1, . . . , am)

Masahiko Murakami (Nihon University) On computation of HOMFLY polynomials of Montesinos links December 18th, 2013 23 / 37

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Computation of HOMFLY Polynomials of Pretzel Diagrams

[Claim] For any integer k, the following holds. H(Ikolor) = { −l−2H(Ik−2olor) − l−1mH(Ik−1olor) if ol = or, −l2H(Ik−2olor) − lmH(I∞olor) if ol ̸= or. H( Qo1,...,ok+1(a1, . . . , ak)) =      H( Ro1,−o2(a1)) if k = 1, −m−1(l + l−1)H( Qo1,...,ok(a1, . . . , ak−1)) if k ≥ 2 and ak = 0, H( Qo1,...,ok(a1, . . . , ak−1)) if k ≥ 2 and ak = ±1.

Q+1,−1,...,+1(a1, . . . , am)

Masahiko Murakami (Nihon University) On computation of HOMFLY polynomials of Montesinos links December 18th, 2013 24 / 37

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Computational Complexity for Computing HOMFLY polynomials of pretzel Diagrams

P(a1)
P(a1, a2)

· · ·

P(a1, . . . , ak−1)
P(a1, . . . , ak)
P(a1, 1)
P(a1, a2, 1)

· · ·

P(a1, . . . , ak−1, 1)
P(a1, −1)
P(a1, a2, −1)

· · ·

P(a1, . . . , ak−1, −1)
Q(a1)
Q(a1, a2)

· · ·

Q(a1, . . . , ak−1)

O(|a1|) O(|a2|) · · · O(|ak−1|) O(|ak|) The HOMFLY polynomials of O(n) link diagrams are computed. Each HOMFLY polynomial is computed in O(n2) time. [Theorem] The HOMFLY polynomial of P o1,...,ok(a1, . . . , ak) is computable in O(n3) time, where n is the number of the crossings of the input pretzel diagram.

Masahiko Murakami (Nihon University) On computation of HOMFLY polynomials of Montesinos links December 18th, 2013 25 / 37

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Contents

Motivation and Results Preliminaries Computation

1

2–bridge Diagrams

2

Pretzel Diagrams

3

Montesinos Diagrams

Conclusion

Masahiko Murakami (Nihon University) On computation of HOMFLY polynomials of Montesinos links December 18th, 2013 26 / 37

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Computation of HOMFLY Polynomials of Montesinos Diagrams

Ia1,1 Ia1,v1 Ia1,2 Ia1,v1-1 Ia1,3 Ia2,1 Ia2,v2 Ia2,2 Ia2,v2-1 Iau,1 Iau,vu Iau,2 Iau,vu-1 Iau,3 Ia22 Ia

Mot

1,ob 1,...,ot u,ob u(a1,1, . . . , a1,v1| · · · |au,1, . . . , au,vu||a) Masahiko Murakami (Nihon University) On computation of HOMFLY polynomials of Montesinos links December 18th, 2013 27 / 37

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Computation of HOMFLY Polynomials of Montesinos Diagrams

Ia1,1 Ia1,v1 Ia1,2 Ia1,v1-1 Ia1,3 Ia2,1 Ia2,v2 Ia2,2 Ia2,v2-1 Iau,1 Iau,vu Iau,2 Iau,vu-1 Iau,3

Not

1,ob 1,...,ot u+1,ob u+1(a1,1, . . . , a1,v1| · · · |au,1, . . . , au,vu||a) Masahiko Murakami (Nihon University) On computation of HOMFLY polynomials of Montesinos links December 18th, 2013 28 / 37

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Computational Complexity for Computing HOMFLY polynomials of Montesinos Diagrams

The HOMFLY polynomials of O(n) link diagrams are computed. Each HOMFLY polynomial is computed in O(n2) time. [Theorem] The HOMFLY polynomial of

Mot

1,ob 1,...,ot u,ob u(a1,1, . . . , a1,v1| · · · |au,1, . . . , au,vu||a) is computable in

O(n3) time, where n is the number of the crossings of the input Montesinos diagram.

Masahiko Murakami (Nihon University) On computation of HOMFLY polynomials of Montesinos links December 18th, 2013 29 / 37

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Contents

Motivation and Results Preliminaries Computation

1

2–bridge Diagrams

2

Pretzel Diagrams

3

Montesinos Diagrams

Conclusion

Masahiko Murakami (Nihon University) On computation of HOMFLY polynomials of Montesinos links December 18th, 2013 30 / 37

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Conclusion

Algorithm Montesinos diagrams with n crossings O(n) time ⇓ O(n) time Integer sequences and orientations O(n3) time ⇓ O(n3) time HOMFLY polynomials Future Works Extension of the algorithm to other links (Ex. arborescent links and closed k–braid links). Extension of the algorithm to other invariants (Ex. the Kauffman polynomial). Implementation of the algorithm. Enumeration of the HOMFLY polynomials of Montesinos links.

Masahiko Murakami (Nihon University) On computation of HOMFLY polynomials of Montesinos links December 18th, 2013 31 / 37

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Computational Complexity of Unknotting Problem

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

⇌ ?

Equivalent trivial knot

Masahiko Murakami (Nihon University) On computation of HOMFLY polynomials of Montesinos links December 18th, 2013 32 / 37

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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.

Masahiko Murakami (Nihon University) On computation of HOMFLY polynomials of Montesinos links December 18th, 2013 32 / 37

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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).

Masahiko Murakami (Nihon University) On computation of HOMFLY polynomials of Montesinos links December 18th, 2013 32 / 37

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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?

Masahiko Murakami (Nihon University) On computation of HOMFLY polynomials of Montesinos links December 18th, 2013 32 / 37

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Computational Complexity of Equivalence Problem

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

⇌ ?

Equivalent

Masahiko Murakami (Nihon University) On computation of HOMFLY polynomials of Montesinos links December 18th, 2013 33 / 37

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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.

Masahiko Murakami (Nihon University) On computation of HOMFLY polynomials of Montesinos links December 18th, 2013 33 / 37

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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.

Masahiko Murakami (Nihon University) On computation of HOMFLY polynomials of Montesinos links December 18th, 2013 33 / 37

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Tait Graphs

[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. [Example]

Masahiko Murakami (Nihon University) On computation of HOMFLY polynomials of Montesinos links December 18th, 2013 34 / 37

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Tait Graphs

[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. [Example]

Masahiko Murakami (Nihon University) On computation of HOMFLY polynomials of Montesinos links December 18th, 2013 35 / 37

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Tait Graphs

[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. [Example]

Masahiko Murakami (Nihon University) On computation of HOMFLY polynomials of Montesinos links December 18th, 2013 36 / 37

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Tait Graphs

[Definition] Color the faces black and white. Put a vertex on each black face. Join two vertices by a labeled edge if they share a crossing.

+1 −1

Labels of edges. [Example] [Remark] #crossings = #edges

+1 +1 −1 −1 +1 +1

Masahiko Murakami (Nihon University) On computation of HOMFLY polynomials of Montesinos links December 18th, 2013 37 / 37