On Bott polynomials. (English) Zbl 0847.57004

Wang, Zhenghan

J. Knot Theory Ramifications 3, No. 4, 537-546 (1994).

Summary: R. Bott [Port. Math. 11, 35-40 (1952; Zbl 0047.42003)] defined two combinatorial invariants for finite cell complexes in the 1950’s. As described in [R. Bott, in “Topological methods in modern mathematics,” Proc. of a Symposium in honor of John Milnor’s sixtieth birthday, 125-135 (1993; Zbl 0817.57003)], they fit beautifully into a state model framework. Recently, it has been observed that for finite graphs, the coefficients of this polynomial are the same set of invariants as defined by H. Whitney in [Am. J. Math. 55, 231-235 (1933; Zbl 0006.37003)]. In section 1, we define the Bott-Whitney polynomial and prove some of its basic properties. In section 2, we show that for a planar connected graph, the Bott-Whitney polynomial is essentially the chromatic polynomial of its dual graph. In section 3, an interpretation of the coefficients of the Bott-Whitney polynomial is given following Whitney [loc. cit.]. In section 4, we study more general Bott polynomials.

MSC:

57M15	Relations of low-dimensional topology with graph theory
05C99	Graph theory

Keywords:

graphs; Bott polynomial; chromatic polynomial; state model

Citations:

Zbl 0006.37003; Zbl 0047.42003; Zbl 0817.57003