Topic
Crosscap number
About: Crosscap number is a research topic. Over the lifetime, 66 publications have been published within this topic receiving 921 citations.
Papers published on a yearly basis
Papers
TL;DR: In this paper, the isotopy classification of 2-bridges knots in S3-Kp/q is studied, where the 2-bridge knots are assumed to be drawn on a square "pillowcase".
Abstract: To each rational number p/q, with q odd, there is associated the 2-bridge knot Kp/q shown in Fig. 1. QI bl Fig. 1. The 2-bridge knot Kp/q In (a), the central grid consists of lines of slope +p/q, which one can imagine as being drawn on a square \"pillowcase\". In (b) this \"pillowcase\" is punctured and flattened out onto a plane, making the two \"bridges\" more evident. The knot drawn is K3/5, which happens to be the figure eight knot. (We assume q odd in order to get a knot rather than a two-component link.) The double cover of S 3 branched along Kp/q is the lens space Lq,p. With this observation, attributed in [16] to Seifert, the isotopy classification of 2-bridge knots follows easily from the classification [14] of oriented lens spaces: K~/q =gp,/q, if and only if q'=q and p,_p+_l (modq). Basic references for 2-bridge knots are [2, 16, 17]. We shall derive in this paper the isotopy classification of the incompressible surfaces, orientable or not, in S3-Kp/q. As an application, we obtain some information about the manifold resulting from Dehn surgery on Kp/q: Excluding the cases when Kp/q is a torus knot (Dehn surgery on torus knots was completely analyzed in [10]), every Dehn surgery on K m yields an irreducible manifold, and all but finitely many Dehn surgeries yield non-Haken (i.e., not sufficiently large) non-Seifert-fibered manifolds. The case of the figure eight
409 citations
TL;DR: In particular, for each g > 3, there are only finitely many vertex-transitive graphs of genus g which can be drawn on S but not on any surface of smaller genus (respectively crosscap number) as discussed by the authors.
Abstract: We describe all regular tiings of the torus and the Klein bottle. We apply this to describe, for each orientable (respectively nonorientable) surface S, all (but finitely many) vertex-transitive graphs which can be drawn on S but not on any surface of smaller genus (respectively crosscap number). In particular, we prove the conjecture of Babai that, for each g > 3, there are only finitely many vertex-transitive graphs of genus g. In fact, they all have order 2, there are only finitely many groups that act on the surface of genus g . We also derive a nonorientable version of Hurwitz' theorem.
133 citations
TL;DR: In this article, the authors investigate the crosscap number of the non-orientable compact surface which a commutative ring R can be embedded in and illustrate all finite commutive rings R (up to isomorphism) such that Γ ( R ) is projective.
39 citations
TL;DR: In this paper, the authors present three methods for computing crosscap number that offer varying trade-offs between precision and speed: (i) an algorithm based on Hilbert basis enumeration and (ii) an exact integer programming, both of which either compute the solution precisely or reduce it to two possible values, and (iii) a fast but limited precision integer programming algorithm that bounds the solution from above.
Abstract: The crosscap number of a knot is an invariant describing the non-orientable surface of smallest genus that the knot bounds. Unlike knot genus (its orientable counterpart), crosscap numbers are difficult to compute and no general algorithm is known. We present three methods for computing crosscap number that offer varying trade-offs between precision and speed: (i) an algorithm based on Hilbert basis enumeration and (ii) an algorithm based on exact integer programming, both of which either compute the solution precisely or reduce it to two possible values, and (iii) a fast but limited precision integer programming algorithm that bounds the solution from above.
The first two algorithms advance the theoretical state of the art, but remain intractable for practical use. The third algorithm is fast and effective, which we show in a practical setting by making significant improvements to the current knowledge of crosscap numbers in knot tables. Our integer programming framework is general, with the potential for further applications in computational geometry and topology.
26 citations
TL;DR: In this article, the cross-cap number of alternating links in terms of their Jones polynomial was derived for infinite families of alternating knots with up to twelve crossings and generalizations of their results for classes of non-alternating links.
20 citations
Related Topics (5)
Performance Metrics
| Year | Papers |
|---|---|
| 2021 | 3 |
| 2020 | 11 |
| 2019 | 4 |
| 2018 | 6 |
| 2017 | 5 |
| 2016 | 2 |