Normal Euler classes of knotted surfaces and triple points on projections

Abstract

Abstract. We present a new formula relating the normal Euler numbers of embedded surfaces in 4-space and the number of triple points on their projections into 3-space. This formula generalizes Banchoff's formula between normal Euler numbers and branch points on the projections.

On the number of singularities of a generic surface with boundary in a 3-manifold

“…(3) f|\partial M : \partial M -\partial N is a selftransverse immersion; i.e., it has only simple or transverse double points; (4) if x\in\partial M is a boundary point, then the planes Tf(T_{x}M) and T_{f(x)}\partial N are in general position in T_{f(x)}N . Note that the conditions ( 2)-( 4) imply the following: (5) if x_{1} and x_{2} are distinct boundary points of M with f(x_{1})=f(x_{2})= y , then the planes Tf(T_{x_{1}}M) , Tf(T_{x_{2}}M) and T_{y}(\partial N) are in general position in T_{y}N .…”

“…In [18], Sz\"ucs proves Theorem 1.1, using the theorem of Banchoff [1] concerning the number of triple points of an immersed surface in R^{3} and using a surgery technique. Here we give another proof using a recent result of Carter and Saito [4], [5] concerning the normal Euler number of an embedded surface in R^{4} .…”

Combinatorics of knotted surface diagrams

“…The sign of a triple point is defined as follows [8]. For the normal vectors v 1 , v 2 , v 3 of top, middle, bottom sheet, respectively, if the triple (v 1 , v 2 , v 3 ) matches the orientation of R 3 , then the sign is positive, and negative if otherwise.…”

“…Generalizations of Banchoff's formula appear in [12] and [19]. Relation between MRCN:57Q45 [4] and [8]. A further generalization is presented in this paper.…”