Kauffman Bracket Skein Module of a Family of Prism Manifolds

Mroczkowski, Maciej

doi:10.1142/s0218216511008681

Cited by 22 publications

(22 citation statements)

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“…The pair 5 26 [14,21,20]. The knots 5 26 and 5 27 and their corresponding links are presented in Figure 10. Note that we have oriented 5 26 and 5 27 in such a way that the linking numbers match, lk(L(5 26 )) = lk(L(5 27 )).…”

Section: The Resultsmentioning

confidence: 99%

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Tabulation of Prime Knots in Lens Spaces

Gabrovšek

2017

Mediterr. J. Math.

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Using computational techniques we tabulate prime knots up to five crossings in the solid torus and the infinite family of lens spaces L(p, q). For these knots we calculate the second and third skein module and establish which prime knots in the solid torus are amphichiral. Most knots are distinguished by the skein modules. For the handful of cases where the skein modules fail to detect inequivalent knots, we calculate and compare the hyperbolic structures of the knot complements. We were unable to resolve a handful of 5-crossing cases for p ≥ 13.

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Section: The Resultsmentioning

confidence: 99%

“…Appendix B: Equivalences of knots in lens spaces 26 5 93 ∼ 5 80 5 95 ∼ 3 5 5 3 , 5 23 ∼ 4 8 5 74 ∼ 5 6…”

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confidence: 99%

Tabulation of Prime Knots in Lens Spaces

Gabrovšek

2017

Mediterr. J. Math.

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“…If q = 1, the prism manifold is denoted by M p (it has no exceptional fibers, just as L(p, 1)). Theorem 10 ( [12]). S 2,∞ (M p ) is a free R-module generated by ∅, x, x 2 ,.…”

Section: Results For Kauffman Bracket and Homflypt Skein Modules Usinmentioning

confidence: 99%

Link Diagrams in Seifert Manifolds and Applications to Skein Modules

Gabrovšek

Mroczkowski

2017

Springer Proceedings in Mathematics &Amp; Statistics

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In this survey paper we present results about link diagrams in Seifert manifolds using arrow diagrams, starting with link diagrams in F × S 1 and N×S 1 , where F is an orientable and N an unorientable surface. Reidemeister moves for such arrow diagrams make the study of link invariants possible. Transitions between arrow diagrams and alternative diagrams are presented. We recall results about the Kauffman bracket and HOMFLYPT skein modules of some Seifert manifolds using arrow diagrams, namely lens spaces, a product of a disk with two holes times S 1 , RP 3 #RP 3 , and prism manifolds. We also present new bases of the Kauffman bracket and HOMFLYPT skein modules of the solid torus and lens spaces.1 Arrow diagrams of links in products and twisted products of S 1 and a surface Let F be an orientable surface and N an unorientable surface. In this section we recall the construction of arrow diagrams for links in F × S 1 , introduced in [13], and N×S 1 , introducted in [11]. These diagrams are very similar to gleams introduced in [16]. Arrow diagrams of links in. By a general position argument we may assume that L intersects F 0 transversally in a finite number of points. In M the link L becomes L -a collection of circles and arcs with endpoints coming in pairs (x, 0) and (x, 1), x ∈ F . Let π be the vertical projection from M onto F . Then π(L ) is a collection of closed curves. Again, by a general position argument, we may assume that there are only transversal double points in π(L ) and the endpoints of arcs are projected onto points distinct from these double points. An arrow diagram D of the link L is π(L ) with some extra information: for double points P , the usual information of over-and undercrossing is encoded depending on the relative height of the two points π −1 (P ) in F × [0, 1]; for points Q that are projections of endpoints of arcs (x, 0) and (x, 1) in L , orient L in such a way that the height drops by 1 in L when the first coordinate crosses x, and put on Q an arrow indicating this orientation.Thus, an arrow diagram D is a collection of immersed curves in F , with under-and overcrossing information for double points and some arrows on these curves. For an example see the diagram on Figure 1.We call an arrow diagram regular, if none of the following forbidden positions appear on the diagram: i) cusps (Fig. 2a),ii) self-tangency points (Fig. 2b), iii) triple points (Fig. 2c), iv) two arrows coincide (Fig. 2d), v) arrows and crossings coincide (Fig. 2e).

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“…where h * is the induced homology map, see Figure 2. The Kirby diagram of L(p, q) is the unknot U bearing the framing −p/q, see Figure 3(a) (also see [12,21]). The unknot represents the meridian m 1 , on whose regular neighbourhood the surgery is performed.…”

Section: 2mentioning

confidence: 99%