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Tabulation of Prime Knots in Lens Spaces

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Tabulation of Prime Knots in Lens Spaces Tabulation of Prime Knots in Lens Spaces Bostjanˇ Gabrovsekˇ Abstract We tabulate prime knots up to four crossings and propose a tabulation up to five crossings in the infinite family of lens spaces L(p; q). As a side product we tabulate knots in the solid torus up to four crossings and propose a tabulation up to five cross- ings. For the solid torus we also establish which prime knots are amphichiral. Keywords. Knot, tabulation, skein module, solid torus, lens space 1 Introduction and preliminaries So far knots have been tabulated up to a certain number of crossings only for a handful 3 of spaces: the 3-dimensional Euclidean space R3 [1], the projective space RP [6], and the solid torus T [3], the latter being tabulated only up to a so-called flip. In the following sections we append the infinite family of lens spaces L(p; q), 0 < q < p, GCD(p; q) = 1 to this modest list. By the standard inclusion i : T,! L(p; q) (see Section 2) each knot k : K,! T defines a knot i ◦ k : K,! L(p; q). Thus, for each L(p; q) a subset of prime knots in T yields the knot table in the lens space. The method of providing the knot tables is straightforward: for the given space we generate all knot diagrams up to n crossings and classify them by ambient isotopy. The minimal diagram of each class represents an entry in the knot table. The classification itself has been made by computer [2], the algorithm is presented in Section 4, the final results are presented in Section 5. We start of by defining knot diagrams used in the knot tables and overview knot in- variants used to detect inequivalent knots, namely the Kauffman bracket skein module and the HOMFLYPT skein module. B. Gabrovsek:ˇ FME, University of Ljubljana, Askerˇ cevaˇ 6, 1000 Ljubljana, Slovenia; e-mail: [email protected] Mathematics Subject Classification (2010): Primary 57M27; Secondary 57Q35 1 2 Bostjanˇ Gabrovsekˇ Let K be a knot in the solid torus T = A × I, with A being an annulus (Figure 1(a)). A punctured disk diagram of a knot K is the regular projection of K on A, keeping the information of over- and undercrossings (Figure 1(b)). We resolve the inconvenience of drawing the annulus by making a dot (a puncture) in the region of R2 ⊃ A that bounds the inner component of @A and assume that the outer component of @A lies in the unbounded region of R2 (Figure 1(c)). We call the dotted region the 0-region and the unbounded region the 1-region. (a) (b) (c) Figure 1: Construction of a punctured disk diagram of a link in the solid torus. The Reidemeister moves of a punctured disk diagram correspond to the classical Rei- demeister moves Ω1, Ω2, and Ω3 (Figure 2), except that we cannot perform any move through the puncture. ! ! ! (a) Ω1 (b) Ω2 (c) Ω3 Figure 2: Classical Reidemeister moves. 2 Knots in L(p; q) The lens space L(p; q), 0 < q < p, GCD(p; q) = 1 is the glueing of two solid tori T1 and T2 together by their boundary via the homeomorphism hp;q : @T1 ! @T2 that takes the meridian of @T1 to the (p; q)-curve on @T2 (see Figure 3 as an example). h −−−!3;1 m h3;1(m) = (3; 1) Figure 3: The boundary homeomorphism h3;1 : @T1 ! @T2. Tabulation of Prime Knots in Lens Spaces 3 To construct a diagram of a knot K in L(p; q) we first isotope K into the first compo- nent T1 and project it to the annulus A of T1 = A × I. Such a diagram corresponds to the 3 punctured disk diagram of a knot in T1 . We equip these diagrams with an additional Reidemeister move SLp;q also known as the slide move [4] or the band move [5, 10]. This move arises from the gluing of hp;q and is presented in Figure 4. One can visualize the move by sliding an arc of the knot over the meridional disk of T1 glued to T2. −−! SLp;q q p 9 > = > ; Figure 4: The slide move for L(p; q). Proposition 2.1 (Hoste, Przytycki [9]). Two punctured disk diagrams represent the same link in L(p; q) if and only if one can be transformed into the other by a finite sequence of Reidemeister moves Ω1, Ω2, Ω3, and SLp;q. In the classification skein modules are used to distinguish inequivalent knots. Our pri- mary invariant is the Kauffman bracket skein module (KBSM). For pairs of knots where the KBSM fails to detect inequivalences, we use a stronger invariant, the HOMFLYPT skein module (HSM). Let M be an oriented 3-manifold and let Lfr be the set of isotopy classes of unoriented ±1 framed links in M. Let R = Z[A ] be the ring of Laurent polynomials in A and let RLfr 3A recent yet interesting approach to knots in lens spaces can be found in [15]. 4 Bostjanˇ Gabrovsekˇ be the free R-module generated by Lfr. Let S2;1 be generated by the expressions: − A − A−1 ; (Kauffman relator) L t − (−A2 − A−2)L; (framing relator) where , , and are classes of links with representatives that are identical out- side a small 3-ball but look like the indicated diagrams inside it, here blackboard framing is assumed. The Kauffman bracket skein module S2;1(M) is RLfr modulo S2;1. Proposition 2.2 (Turaev [20]). S2;1(T ) is freely generated by an infinite set of generators n 1 n 0 fx gi=0, where x ; n ≤ q is a parallel copy of n longitudes of T and x is the affine unknot. n bp=2c Proposition 2.3 (Hoste, Przytycki [9]). S2;1(L(p; q)) is freely generated by fx gn=0 , where xn, n > 0 is a parallel copy of n longitudes of T ⊂ L(p; q) and x0 is the affine unknot. If, for a given manifold M, the basis of the KBSM is known, we denote by KBSM M (K) the expression of [K] 2 S2;1(M) written in terms of a fixed basis. Let K be the mirror of the knot K. It follows directly from the Kauffman relator that −1 KBSM M (K) and KBSM M (K) differ by a substitution A ! A . If K(1) is the knot obtained from K by twisting the framing of K by a full positive (1) 3 twist, it follows from the framing relation that KBSM M (K ) = (−A )·KBSM M (K) [16, 17]. The HSM is a much stronger invariant than the KBSM and much more difficult to compute. The HSM has been calculated for the solid torus and recently for the family of lens spaces L(p; 1) [4]4. Let M again be an oriented 3-manifold and let Lor be the set of isotopy classes of oriented links in M to which we also add the empty knot ;. Let R = Z[v±1; z±1] be the ring of Laurent polynomials in variables v and z, and let RLor be the free R-module generated by Lor. Let S3 be generated by the expression: v−1 − v − z : (HOMFLYPT relator) We also add to S3 the expression involving the empty knot: v−1; − v; − z : (HOMFLYPT relator) The HOMFLYPT skein module S3(M) is RLor modulo S3. 4In contrast to the HSM, the KBSM has been a widely studied knot invariant, see for example [17, 13, 14]. Tabulation of Prime Knots in Lens Spaces 5 Proposition 2.4 (Turaev [20]). S3(T ) is freely generated by an infinite set of generators i1 is B = ftk : : : tk : s 2 N; k1 < ··· < ks 2 Z n f0g; i1; : : : ; is 2 Ng [ f;g, where ; is the 1 s ∼ empty knot. For k > 0, tk is the oriented knot in T representing k in π1(T ) = Z that has an ascending diagram with k − 1 positive crossings. For k < 0, tk is tjkj with reversed orientation. For example, t3 and t3t−1 are presented in Figure 5. (a) t3 (b) t3t−1 Figure 5: Two generators of S3(T ). Proposition 2.5 (Gabrovsek,ˇ Mroczkowski [4]). S3(L(p; 1)) is freely generated by an infinite set of generators B = fti1 : : : tis : s 2 ; k < ··· < k 2 n f0g; − p < k < p k1 ks N 1 s Z 2 1 p ··· < ks ≤ 2 ; i1; : : : ; is 2 Ng [ f;g, where ; is the empty knot and tk are knots with diagrams equal to those in Proposition 2.4. 5 Empirical evidence suggests that S3(L(p; 2)) is also free with the same basis as L(p; 1) . We state the following conjecture. Conjecture 2.1. S3(L(p; q)) is freely generated by an infinite set of generators Bp = fti1 : : : tis : s 2 ; k < ··· < k 2 n f0g; − p < k < ··· < k ≤ p ; i ; : : : ; i 2 k1 ks N 1 s Z 2 1 s 2 1 s Ng [ f;g, where ; is the empty knot and tk are knots with diagrams equal to those in Proposition 2.4. Having fixed a basis of S3(M) and having a knot K ⊂ M we denote by HSM M (K) the expression of [K] 2 S3(M) written terms of the basis. 3 Gauss codes This section will prepare us for Section 4: we describe the computer data structure for storing knot diagrams and take a closer on how these diagrams are manipulated.
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