NONTRIVIAL PROJECTIONS OF THE TRIVIAL KNOT

T. Homma, the author and M. Takahashi proved in [HOT] that there is a good algorithm for recognizing the standard 3-sphere S3 in 3-manifolds with Heegaard splittings of genus two, and later T. Homma and the author proved in [HO] that any nontrivial 3-bridge knot diagrams of the trivial knot T always have waves but generally speaking there are many knot diagrams of T without waves (see also [Mo]). In this paper, we define the concept of n-waves and 0-waves mean waves. Then it is shown that there exist knot diagrams of T with no n-waves, where n is an arbitrary non-negative integer. Furthermore we consider a method to distinguish by the computer whether knot diagrams with a certain range of crossings give the trivial knot. Of course, the method does permit us to distinguish whether 3-bridge knot diagrams to be trivial and so at this present a plenty of knot diagrams including the one given by Figure 2 (but not Figure 3 and Figure 4) are distinguished to be trivial by the computer. The author would like to thank S. Suzuki for helpful discussions and S. Yamada, T. Yamada and Y. Ozaki for programming aids.