Symmetries

As shown in figure 10.0.1, there are various kinds of symmetries on knots. In the first half of this chapter, we study some relationships between symmetries and the polynomial invariants. As an application, we explain the proof of [Kawauchi 1979] on the non-invertibility of 8₁₇ (see figure 10.0.2). In the latter half of this chapter, we study the symmetry group of a knot, which essentially controls the symmetries of a knot. We explain a (still unpublished) theory of F. Bonahon and L. Siebenmann (cf. [Bonahon-Siebenmann *]) for a canonical decomposition of a knot, which gives us good insight into the knot and enables us to determine the symmetry groups of algebraic knots including 817 and the Kinoshita-Terasaka knot K _KT (see figure 3.8.1a).