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

Contents Motivation and Results Preliminaries Computation 2–bridge Diagrams 1 Pretzel Diagrams 2 Montesinos Diagrams 3 Conclusion 1 Masahiko Murakami (Nihon University) On computation of HOMFLY polynomials of Montesinos links December 18th, 2013 2 / 37

Contents Motivation and Results Preliminaries Computation 2–bridge Diagrams 1 Pretzel Diagrams 2 Montesinos Diagrams 3 Conclusion 1 Masahiko Murakami (Nihon University) On computation of HOMFLY polynomials of Montesinos links December 18th, 2013 3 / 37

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

Computational Complexities of Knot Polynomials Jones polynomials, Restricting knots and link types O ( n 2 ) time pretzel diagrams Utsumi, Imai(2002) O ( n 2 ) time 2–bridge diagrams Diao et al .(2009) Murakami et al .(2007, 2009) O ( n 2 ) time Montesinos diagrams Diao et al .(2009) Hara et al .(2009) O ( n 4 log n ) time arborescent diagrams Hara et al .(2009) O ( n 2 ) time closed 3–braid diagrams Murakami et al .(2007, 2009) Jones polynomials, Bounded treewidths a constant polynomial time Makowsky(2001) O ( n 5 log n ) time at most two 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) O ( n 3 ) time Murakami et al .(2014) 2–bridge diagrams pretzel diagrams, Montesinos diagrams O ( n 3 ) time result 1 Masahiko Murakami (Nihon University) On computation of HOMFLY polynomials of Montesinos links December 18th, 2013 5 / 37

Contents Motivation and Results Preliminaries Computation 2–bridge Diagrams 1 Pretzel Diagrams 2 Montesinos Diagrams 3 Conclusion 1 Masahiko Murakami (Nihon University) On computation of HOMFLY polynomials of Montesinos links December 18th, 2013 6 / 37

The HOMFLY Polynomial [Definition] The HOMFLY polynomial H : 1 H ( � K ) = 1 for each trivial knot � K . ( ) ( ■ ✒ ) ( ■ ✒ ) ■ ✒ + l − 1 H 2 lH + mH = 0 . [Definition] The Jones polynomial V : � � V ( L ) = ( − A ) − 3 w ( � L ) ⟨ � L ⟩ � t 1 / 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. O ( n ) O ( n ) The absolute values of the degrees O ( n 2 ) O ( n ) The number of the terms O (2 n ) O (2 n ) The absolute values of the coefficients 1 Masahiko Murakami (Nihon University) On computation of HOMFLY polynomials of Montesinos links December 18th, 2013 7 / 37

Integer Tangles [Definition] The 0 -tangle twisted k times is called k -tangle and denoted by I k . ❄ ❯ ☛ ✻ ☛ ✻ ✻❑ 0 –tangle 3 –tangle ( − 2) –tangle ∞ –tangle I 0 − 1 − 1 I 3+1+1 I − 2 − 1+1 I ∞ +1 − 1 Integer tangles. 1 Masahiko Murakami (Nihon University) On computation of HOMFLY polynomials of Montesinos links December 18th, 2013 8 / 37

2–bride Diagrams [Definition] a 1 , . . . , a k : integers, � R o l 1 o r 1 ( a 1 , . . . , a k ) (2–bridge diagram) k is an odd number k is an even number � � R +1+1 ( a 1 , . . . , a k ) R +1 − 1 ( a 1 , . . . , a k ) 1 Masahiko Murakami (Nihon University) On computation of HOMFLY polynomials of Montesinos links December 18th, 2013 9 / 37

Pretzel Diagrams [Definition] a 1 , . . . , a k : integers, � P o 1 ,...,o k ( a 1 , . . . , a k ) (pretzel diagram) � P +1 , − 1 ,..., +1 ( a 1 , . . . , a m ) [Definition] a 1 , . . . , a k : integers, � Q o 1 ,...,o k +1 ( a 1 , . . . , a k ) � Q +1 , − 1 ,..., +1 ( a 1 , . . . , a m ) 1 Masahiko Murakami (Nihon University) On computation of HOMFLY polynomials of Montesinos links December 18th, 2013 10 / 37

Montesinos Diagrams [Definition] a 1 , 1 , . . . , a 1 ,v 1 , . . . , a u, 1 , . . . , a u,v u , a : integers, M o t � 1 ,o b 1 ,...,o t u ,o b u ( a 1 , 1 , . . . , a 1 ,v 1 | · · · | a u, 1 , . . . , a u,v u || a ) (Montesinos diagram) I a 22 I a I a u, 1 I a 1,1 I a 2,1 I a 1,2 I a u, 2 I a 2,2 I a u, 3 I a 1,3 I a 2, v 2 -1 I a u,v u -1 I a 1, v 1 -1 I a 2, v 2 I a u,v u I a 1, v 1 1 Masahiko Murakami (Nihon University) On computation of HOMFLY polynomials of Montesinos links December 18th, 2013 11 / 37

Contents Motivation and Results Preliminaries Computation 2–bridge Diagrams 1 Pretzel Diagrams 2 Montesinos Diagrams 3 Conclusion 1 Masahiko Murakami (Nihon University) On computation of HOMFLY polynomials of Montesinos links December 18th, 2013 12 / 37

Algorithm Link diagrams with n crossings O ( n ) time ⇓ O ( n ) time Integer sequences and orientations O ( n 3 ) time ⇓ O ( n 3 ) time HOMFLY polynomials 1 Masahiko Murakami (Nihon University) On computation of HOMFLY polynomials of Montesinos links December 18th, 2013 13 / 37

Computation of HOMFLY Polynomials of Integer Tangles [Claim] For any integer k , the following holds. { − l − 2 H ( I k − 2 o l o r ) − l − 1 mH ( I k − 1 o l o r ) if o l = o r , H ( I ko l o r ) = if o l ̸ = o r . − l 2 H ( I k − 2 o l o r ) − lmH ( I ∞ o l o r ) 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] ✻ ❄ ✻ ❄ ✻ ❄ ✻ ✻ ✻ ✻ ✻ ✻ . . . . . . . . . . . . . . . . . . I ko l o r I k − 2 o l o r I k − 1 o l o r I ko l o r I k − 2 o l o r I ∞ o l o r k > 0 and o l = o r k > 0 and o l ̸ = o r ( ) ( ■ ✒ ) ( ■ ✒ ) ■ ✒ + l − 1 H + mH = 0 . lH 1 Masahiko Murakami (Nihon University) On computation of HOMFLY polynomials of Montesinos links December 18th, 2013 14 / 37

Computation of HOMFLY Polynomials of Integer Tangles [Claim] For any integer k , the following holds. { − l 2 H ( I ko l o r ) − lmH ( I k − 1 o l o r ) if o l = o r , H ( I k − 2 o l o r ) = if o l ̸ = o r . − l − 2 H ( I ko l o r ) − l − 1 mH ( I ∞ o l o r ) 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] ✻ ❄ ✻ ❄ ✻ ❄ ✻ ✻ ✻ ✻ ✻ ✻ . . . . . . . . . . . . . . . . . . I k − 2 o l o r I ko l o r I k − 1 o l o r I k − 2 o l o r I ko l o r I ∞ o l o r k ≤ 0 and o l = o r k ≤ 0 and o l ̸ = o r ( ) ( ■ ✒ ) ( ■ ✒ ) ■ ✒ + l − 1 H + mH = 0 . lH 1 Masahiko Murakami (Nihon University) On computation of HOMFLY polynomials of Montesinos links December 18th, 2013 15 / 37

Contents Motivation and Results Preliminaries Computation 2–bridge Diagrams 1 Pretzel Diagrams 2 Montesinos Diagrams 3 Conclusion 1 Masahiko Murakami (Nihon University) On computation of HOMFLY polynomials of Montesinos links December 18th, 2013 16 / 37

Computation of HOMFLY Polynomials of 2–bridge Diagrams [Claim] For any integer k , the following holds. { − l − 2 H ( I k − 2 o l o r ) − l − 1 mH ( I k − 1 o l o r ) if o l = o r , H ( I ko l o r ) = if o l ̸ = o r . − l 2 H ( I k − 2 o l o r ) − lmH ( I ∞ o l o r ) 1 Masahiko Murakami (Nihon University) On computation of HOMFLY polynomials of Montesinos links December 18th, 2013 17 / 37

Computation of HOMFLY Polynomials of 2–bridge Diagrams R o l 1 o r H ( � 1 ( a 1 , . . . , a k ))  − lm − 1 − l − 1 m − 1  if k = 1 and a 1 = 0 ,     1 if k = 1 and a 1 = ± 1 or    if k = 2 and a 2 = 0 , = H ( � R o l 1 o r 1 ( a 1 ∓ 1)) if k = 2 and a 2 = ± 1 ,     H ( � R o l 1 o r  if k ≥ 3 and a k = 0 , 1 ( a 1 , . . . , a k − 2 ))    H ( � R o l 1 o r 1 ( a 1 , . . . , a k − 2 , a k − 1 ∓ 1)) if k ≥ 3 and a k = ± 1 . 1 Masahiko Murakami (Nihon University) On computation of HOMFLY polynomials of Montesinos links December 18th, 2013 18 / 37