We discuss the properties of incompressible pairwise incompressible surfaces in a knot complement by using twist crossing number. Let K be a pretzel knot or rational knot that its twistindex is less than 6, and l...We discuss the properties of incompressible pairwise incompressible surfaces in a knot complement by using twist crossing number. Let K be a pretzel knot or rational knot that its twistindex is less than 6, and let F be an incompressible pairwise incompressible surface in S 3-K. Then F is a punctured sphere.展开更多
The central subject of studying in this paper is incompressible pairwise incompressible surfaces in link complements. Let L be a non-split prime link and let F be an incompressible pairwise incompressible surface in S...The central subject of studying in this paper is incompressible pairwise incompressible surfaces in link complements. Let L be a non-split prime link and let F be an incompressible pairwise incompressible surface in S3 - L. We discuss the properties that the surface F intersects with 2-spheres in S3 - L. The intersection forms a topological graph consisting of a collection of circles and saddle-shaped discs. We introduce topological graphs and their moves （R-move and S2-move）, and define the characteristic number of the topological graph for F∩S2±. The characteristic number is unchanged under the moves. In fact, the number is exactly the Euler Characteristic number of the surface when a graph satisfies some conditions. By these ways, we characterize the properties of incompressible pairwise incompressible surfaces in alternating （or almost alternating） link complements. We prove that the genus of the surface equals zero if the component number of F∩S2＋（or F∩S2-） is less than five and the graph is simple for alternating or almost alternating links. Furthermore, one can prove that the genus of the surface is zero if #（F） ≤8.展开更多
In this paper, we discuss mainly the properties of incompressible pairwise incomprcssiblc surfaccs in ahnost altcrnating link complcmcnts. Lct L bc a almost link and lct F be an incompressible pairwise incompressible ...In this paper, we discuss mainly the properties of incompressible pairwise incomprcssiblc surfaccs in ahnost altcrnating link complcmcnts. Lct L bc a almost link and lct F be an incompressible pairwise incompressible surface in S^3 - L. First, we give the properties that the surface F intersects with 2-spheres in S^3- L. The intersection consisting of a collection of circles and saddle-shaped discs is called a topological graph. One can compute the Euler Characteristic number of the surface by calculating the characteristic number of the graph. Next, we prove that if the graph is special simple, then the genus of the surface is zero.展开更多
文摘We discuss the properties of incompressible pairwise incompressible surfaces in a knot complement by using twist crossing number. Let K be a pretzel knot or rational knot that its twistindex is less than 6, and let F be an incompressible pairwise incompressible surface in S 3-K. Then F is a punctured sphere.
基金Supported by NSF of China (11071106)supported by Liaoning Educational Committee (2009A418)
文摘The central subject of studying in this paper is incompressible pairwise incompressible surfaces in link complements. Let L be a non-split prime link and let F be an incompressible pairwise incompressible surface in S3 - L. We discuss the properties that the surface F intersects with 2-spheres in S3 - L. The intersection forms a topological graph consisting of a collection of circles and saddle-shaped discs. We introduce topological graphs and their moves （R-move and S2-move）, and define the characteristic number of the topological graph for F∩S2±. The characteristic number is unchanged under the moves. In fact, the number is exactly the Euler Characteristic number of the surface when a graph satisfies some conditions. By these ways, we characterize the properties of incompressible pairwise incompressible surfaces in alternating （or almost alternating） link complements. We prove that the genus of the surface equals zero if the component number of F∩S2＋（or F∩S2-） is less than five and the graph is simple for alternating or almost alternating links. Furthermore, one can prove that the genus of the surface is zero if #（F） ≤8.
基金Supported by NSF of China(10171024)Supported by Liaoning Educational Committee(05L208)
文摘In this paper, we discuss mainly the properties of incompressible pairwise incomprcssiblc surfaccs in ahnost altcrnating link complcmcnts. Lct L bc a almost link and lct F be an incompressible pairwise incompressible surface in S^3 - L. First, we give the properties that the surface F intersects with 2-spheres in S^3- L. The intersection consisting of a collection of circles and saddle-shaped discs is called a topological graph. One can compute the Euler Characteristic number of the surface by calculating the characteristic number of the graph. Next, we prove that if the graph is special simple, then the genus of the surface is zero.
