曲面結び目のカンドルコサイクル不変量の多重化公式

Abstract

A surface knot is the image of a smooth embedding of a connected closed surface in 4-dimensional Euclidean space. A framing of a surface knot is a section of the unit normal bundle of the surface knot. The n-parallel of a framed surface knot is the union of n parallel copies of the surface knot obtained by pushing it along the framing. In this paper, we show that the quandle cocycle invariant using a finite quandle X of the n-parallel of a framed surface knot can be presented by the rack cocycle invariants using finitely many racks which are determined by X (independently of n) of the framed surface knot (Theorem 4.5). In particular, when X is the dihedral quandle R_p of order of odd prime p, we show that the R_p coloring number of the n-parallel of a framed surface knot F can be presented in terms of the R_p coloring number of F and the divisibility of the framing of F in the first cohomology group of F (Theorem 4.6 (1)), where we naturally identify the set of framings of F and the first cohomology group of F (Proposition 3.7). Further, for the Mochizuki 3-cocycle of R_p, we show that the quandle cocycle invariant of the n-parallel of a framed surface knot F for odd n can be presented in terms of the quandle cocycle invariant of F and the divisibility of the framing of F (Theorem 4.6 (2)). Further, when X is the tetrahedral quandle Q_4, we show that the Q_4 coloring number of the n-parallel of a framed surface knot F can be presented in terms of the coloring numbers of F for Q_4 and a certain rack determined by Q_4 and the divisibility of the framing of F (Theorem 4.15). In order to show the above results, we develop a method to describe framed surface knots in terms of their diagrams. We present a framed surface knot by a surface knot diagram with arcs written on it, and show that two framed surface knots are isotopic if and only if their diagrams with arcs are related by certain moves (Theorem 3.6), which is a generalization of a theorem of Roseman for usual surface knot diagrams. Further, we introduce the rack cocycle invariants of framed surface knots, and show its isotopy invariance by using Theorem 3.6 (Theorem 3.11).