Closed incompressible surfaces in the complements of positive knots

We show that any closed incompressible surface in the complement of a positive knot is algebraically non-split from the knot, positive knots cannot bound non-free incompressible Seifert surfaces and that the splittability and the primeness of positive knots and links can be seen from their positive diagrams. Received: June 28, 2000     

Unknotting Tunnels and Seifert Surfaces

Let K be a knot with an unknotting tunnel and suppose thatK is not a 2-bridge knot. There is an invariant = p/q Q/2Z,with p odd, defined for the pair (K, ). The invariant has interesting geometric properties. It is oftenstraightforward to calculate; for example, for K a torus knotand an annulus-spanning arc, (K, ) = 1. Although is definedabstractly, it is naturally revealed when K is put in thinposition. If 1 then there is a minimal-genus Seifert surfaceF for K such that the tunnel can be slid and isotoped to lieon F. One consequence is that if (K, ) 1 then K > 1. Thisconfirms a conjecture of Goda and Teragaito for pairs (K, )with (K, ) 1. 2000 Mathematics Subject Classification 57M25,57M27.     

The Canonical Genus of a Classical and Virtual Knot

A diagram D of a knot defines the corresponding Gauss Diagram G _D . However, not all Gauss diagrams correspond to the ordinary knot diagrams. From a Gauss diagram G we construct closed surfaces F _G and S _G in two different ways, and we show that if the Gauss diagram corresponds to an ordinary knot diagram D, then their genus is the genus of the canonical Seifert surface associated to D. Using these constructions we introduce the virtual canonical genus invariant of a virtual knot and find estimates on the number of alternating knots of given genus and given crossing number.     

几乎交错环链补中的不可压缩曲面

In this paper,we discuss mainly the properties of incompressible pairwies incom- prcssiblc surfaccs in almost altcrnating link complcmcnts. Lct L bca almost link and lct F be an incompressible palrwise 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.     

A Class of Planar Surfaces in Surface×I

Ｗｅｒｅｓｔｒｉｃｔｏｕｒｃｏｎｓｉｄｅｒａｔｉｏｎｔｏｏｒｉｅｎｔａｂｌｅａｎｄｃｏｍｐａｃｔｃａｔｅｇｏｒｙ .Ｆｏｒｄｅｆｉｎｉｔｉｏｎｓａｎｄｔｅｒｍｉｎｏｌｏｇｙ ,ｓｅｅｆｏｒｅｘａｍｐｌｅ ,[1 ]ａｎｄ [2 ].　ＬｅｔＳｂｅａｃｌｏｓｅｄｏｒｉｅｎｔａｂｌｅｓｕｒｆａｃｅｗｉｔｈｐｏｓｉｔｉｖｅｇｅｎｕｓ .Ｉｎｔｈｉｓｐａｐｅｒ ,ＭａｌｗａｙｓｄｅｎｏｔｅｓＳ×Ｉ ,ａｎｄＳ0 =Ｓ× 0 ,Ｓ1=Ｓ× 1 .ＡｎａｎｎｕｌｕｓＡｉｎＭｗｉｔｈｉｔｓｔｗｏｂｏｕｎｄａｒｉｅｓｌｙｉｎｇｉｎＳ0ａｎｄＳ1ｒ…     

Braiding a link with a fixed closed braid

We show that every oriented link diagram with a closed braid diagram as a sublink diagram can be deformed into a closed braid diagram by a deformation keeping the sublink diagram and, under a mild condition, the number of Seifert circles fixed. As an application, we give an upper bound for the braid index of the link obtained by reversing the orientation of its sublink by using only the information of an original link.     

Equivalent Curves in Surfaces

We consider various notions of equivalence for homotopy classes of curves on hyperbolic surfaces based on topological, algebraic, and geometric structures, and find the relationships between these equivalences.     

Reducing a Representation of Homotopy 3-spheres

In the present note we show that a representation of homotopy 3-spheres can be somewhat simplified under some circumstances.     

纽结补中的不可压缩分段不可压缩曲面

本文利用扭转交叉数（twist-erossing number)讨论扭结补中的不可压缩分段不可压缩曲面的性质,设K是一个排叉结（pretzel knot)或者是一个扭转交叉数少于6的有理纽结,如果F是S^3-K中的不可压缩分段不可压缩曲面,那么F是一个穿孔球面。     

Paralinearization of the Dirichlet-Neumann operator and applications to progressive gravity waves

This note presents a paradifferential approach to the analysis of the water waves equations.     

Generalized Neuwirth Groups and Seifert Fibered Manifolds

The topological properties of the generalized Neuwirth groups, _n^k are discussed. For examp, we demonstrate that the group, _n^k is the fundamental group of the Seifert fibered space _n^k. Moreover, discuss some other invariants and algebraic properties of the above groups.This work was supported by Polish grant (BW-5100–5–0259–9) and the Russian Foundation for Basic Research (grant number 98–01–00699).2000 Mathematics Subject Classification: 20F34, 57M05, 57M60     

The Witt classes of torsion linking forms of (4n−1)-manifolds with pseudo-free circle actions

In this paper we compute the Witt class of torsion linking forms of (4n−1)-dimensional closed oriented manifolds admitting orientation preserving pseudo-free circle actions.     

第二型面积分与流量问题

从物理和数学的角度讨论了一些在第二型面积分的教学中遇到的问题,通过讨论可消除学生在学习这部分内容时有关不可压缩流体的误解,并完善一些教科书的相关表述.