Alexander numbering of knotted surface diagrams

A generic projection of a knotted oriented surface in 4-space divides 3-space into regions. The number of times (counted with sign) that a path from infinity to a given region intersects the projected surface is called the Alexander numbering of the region. The Alexander numbering is extended to branch and triple points of the projections. A formula that relates these indices is presented.

[1]  S. Kamada.  2-Dimensional Braids and Chart Descriptions , 1993 .

[2]  Grant Cairns,et al.  A new formula for winding number , 1993 .

[3]  J W Alexander,et al.  A Lemma on Systems of Knotted Curves. , 1923, Proceedings of the National Academy of Sciences of the United States of America.

[4]  Alexander’s and Markov’s theorems in dimension four , 1994, math/9407217.

[5]  T. Banchoff.  Double tangency theorems for pairs of submanifolds , 1981 .

[6]  M. Saito,et al.  Canceling branch points on projections of surfaces in 4-space , 1992 .

[7]  Thomas Banchoff,et al.  Triple points and surgery of immersed surfaces , 1974 .

[8]  W. Marar,et al.  The Euler characteristic of a generic wave front in a 3-manifold , 1992 .

[9]  J. C. Cantrell,et al.  Topology of Manifolds , 1970 .

[10]  W. Massey.  Proof of a conjecture of Whitney. , 1969 .

[11]  H. Whitney.  On regular closed curves in the plane , 1937 .

[12]  J. Carter,et al.  Triple points of immersed surfaces in three dimensional manifolds , 1989 .

[13]  S. Kamada.  Nonorientable surfaces in 4-space , 1989 .

[14]  Oleg Viro,et al.  Some integral calculus based on Euler characteristic , 1988 .

[15]  V. Goryunov.  Local invariants of mappings of surfaces into three-space , 1997 .

[16]  S. Kamada.  SURFACES IN R4 OF BRAID INDEX THREE ARE RIBBON , 1992 .

[17]  L. Rudolph.  Braided surfaces and seifert ribbons for closed braids , 1983 .

[18]  S. Kamada.  A characterization of groups of closed orientable surfaces in 4-space , 1994 .

[19]  H. Whitney.  On the Topology of Differentiable Manifolds , 1992 .

[20]  M. Saito,et al.  BRAIDS AND MOVIES , 1996 .

[21]  M. Polyak,et al.  Whitney formula in higher dimensions , 1996 .

[22]  M. Saito,et al.  Normal Euler classes of knotted surfaces and triple points on projections , 1997 .

[23]  M. Saito,et al.  Knotted Surfaces and Their Diagrams , 1997 .

[24]  Gui-Song Li.  On self-intersections of immersed surfaces , 1998 .

[25]  On Knots , 1990, Acta Applicandae Mathematicae.