Canceling branch points and cusps on projections of knotted surfaces in -space

For a knotted surface in -space, its generic projection into -space has branch points as its singularities, and its successive projection into -space has fold points and cusps as its singularities. In this paper, we show that for non-orientable knotted surfaces, the numbers of branch points and cusps can be minimized by isotopy.

     

一类非凸的不可定向交叉数的序列

It is well known that finding the crossing number of a graph on nonplanar surfaces is very difficult.In this paper we study the crossing number of the circular graph C（10,4） on the projective plane and determine the nonorientable crossing number sequence of C（10,4）.On the basis of the result,we show that the nonorientable crossing number sequence of C（10,4） is not convex.     

关于循环图的曲面嵌入

该文集中探讨循环图的曲面嵌入性质.决定了所有循环图的最小亏格(其中包括可定向亏格与不可定向亏格)和最大亏格.对于固定的整数l(≥3)和充分大的 自然数n,只有一种方式将4 -正则循环图C(n,l)嵌入到环面上使得其每一个面都是4 -边形.特别地,循环图$C(2l+2,l)$在加入若干条新边后可以同时将环面与Klein瓶进行三角剖分.     

Line arrangements in

If is a hyperbolic manifold and is a simple closed geodesic, then lifts to a collection of lines in acted upon by . In this paper we show that such a collection of lines cannot contain a particular type of subset (called a bad triple) unless has orientation-reversing elements. This fact allows us to extend certain lower bounds on hyperbolic volume to the non-orientable case.

     

Symmetric extensions of dihedral quandles and triple points of non-orientable surfaces

Quandles with involutions that satisfy certain conditions, called good involutions, can be used to color non-orientable surface-knots. We use subgroups of signed permutation matrices to construct non-trivial good involutions on extensions of odd order dihedral quandles.For the smallest example of order 6 that is an extension of the three-element dihedral quandle R₃, various symmetric quandle homology groups are computed, and applications to the minimal triple point number of surface-knots are given.     

时空克莱因瓶上的热力学——从二维生物的奇妙旅行到共形量子态的路径积分

在克莱因瓶和莫比乌斯带上环游世界的二维生物会经历有趣的手征变换,这可归因于这些不可定向曲面的独特拓扑性质。作者最近在研究中发现,让共形场论中的量子态在这些曲面上做“时空旅行”(路径积分),也会得到新奇而普适的热力学性质。例如,克莱因瓶上的二维共形场论的自由能中包含一项普适的克莱因瓶熵。它仅依赖于共形场论的一些基本特征。利用克莱因瓶熵,可以精确地找到量子相变点,并刻画其普适类。更一般地,不可定向曲面上的普适热力学数据不仅对于研究凝聚态和统计物理中的临界现象有意义,对于其他领域,比如全息黑洞热力学等也有启发。