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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 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) There exist polynomial time algorithms for computing Jones polynomials and HOMFLY-PT polynomials under reasonable restrictions. 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: [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. 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: [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) 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 1{tangle −1−1 +1+1 −1+1 +1−1 I0 I3 I−2 I1 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(I1 ) 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 I1 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 [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(I1 ) 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−2 Ik Ik−1 Ik−2 Ik I1 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(I1 ) 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 8H(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 8H(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)) + + + .
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