The Computation of the Non-Commutative Generalization of the a-Polynomial of the Figure-Eight Knot

Abstract

Abstract. The paper computes the noncommutative A-ideal of the figureeight knot, a noncommutative generalization of the A-polynomial. It is shown that if a knot has the same A-ideal as the figure-eight knot, then all colored Kauffman brackets are the same as those of the figure eight knot.

Confluence theory for graphs

“…For other oriented surfaces, A 1 (F, ∅, R) is isomorphic to the Kauffman bracket skein module of F × I, [HP2,P5,P6,PS]. (For more on Kauffman bracket skein modules see [Bu1,Bu2,BFK1,BFK2,BHMV,BP,FG,FGL,FK,GS1,GS2,GH,HP3,HP4,Le,Sa1,Sa2,S1,S3,Tu].) Consequently, Corollary 4.1 immediately implies the result of Przytycki, [P6, Theorem 3.1]: Theorem 9.4.…”

Kauffman bracket versus Jones polynomial skein modules

“…This should be contrasted with t −6 (2, 3) T − t 6 (2, −1) T + t 3 (1, 7) T − t(1, 5) T + (−t 11 + t 3 − t −1 − t −5 )(1, 3) T +(t 9 − t 5 − t −7 )(1, 1) T + (−t 11 + 2t 7 + t 3 − t −1 + t −9 )(1, −1) T +(t 13 + t)(1, −3) T − t −1 (1, −5) T + t 8 (0, 7) T + (−2t 8 + t 4 − t −4 )(0, 5) T +(−t 12 + t 8 − t 4 − 1 + t −4 )(0, 3) T + (t 12 − t 8 + 1 + t −4 )(0, 1) T which was obtained in [11] as an element of the peripheral ideal defined using the Kauffman bracket. The former gives rise to the following recursive relation for colored Jones polynomials y n = J(K 8 , n) of the figure-eight knot (t 6n+6 − t −2n+2 )y n+2 + (−t 14n+24 + t 10n+16 + t 6n+20 − t 6n+12 + t 6n+8 +t 6n+4 − t 2n+12 + t 2n+8 + t 2n−4 + t −2n+8 − 2t −2n+4 − t −2n + t −2n−4 −t −2n−12 − t −6n+4 − t −6n−8 + t −10n−16 )y n+1 + (t 14n+22 − 2t 10n+18 +t 10n+14 − t 10n+6 − t 6n+18 + t 6n+14 − t 6n+10 − t 6n+6 + t 6n+2 + t 2n+14 −t 2n+10 + t 2n+2 + t 2n−2 + t −2n+10 − t −2n+6 + t −2n−2 + t −2n−6 − t −6n+6 +t −6n+2 − t −6n−2 − t −6n−6 + t −6n−10 − 2t −10n−2 + t −10n−6 − t −10n−14 +t −14n−6 )y n + (t 10n+4 − t 6n+16 − t 6n+4 + t 2n+12 − 2t 2n+8 − t 2n+4 + t 2n −t 2n−8 − t −2n+8 + t −2n+4 + t −2n−8 + t −6n+8 − t −6n + t −6n−4 + t −6n−8 +t −10−4 − t −14n−4 )y n−1 + (t 2n+6 + t −6n−6 )y n−2 = 0.…”

“…For the same reason as in the case of the torus knots discussed above, RT t (S 3 \N (K 8 )) is also free, with the same basis. From the work in [11] one can infer that the action of the algebra K t (T 2 ) on K t (S 2 \N (K 8 )) is determined by the following…”

Some Results About the Kauffman Bracket Skein Module of the Twist Knot Exterior

“…In this perspective, Kauffman bracket skein modules were used to give an alternate description of the A-polynomial of Cooper, Culler, Gillet, Long, and Shalen, and to generalize it to a noncommutative setting [FGL]. The computation of the noncommutative generalization of the A-polynomial of a knot relies heavily on the good understanding of the skein module of the knot complement.The noncommutative generalization of the A-polynomial was computed for the unknot in [FGL], trefoil knot in [G1], partially for (2, 2p + 1)-torus knots in [GS1], and for the figure-eight knot in [GS2]. In those papers, and also in [G2] this knot invariant was linked to the Jones polynomial.…”

“…The noncommutative generalization of the A-polynomial was computed for the unknot in [FGL], trefoil knot in [G1], partially for (2, 2p + 1)-torus knots in [GS1], and for the figure-eight knot in [GS2]. In those papers, and also in [G2] this knot invariant was linked to the Jones polynomial.…”