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# Communications in Analysis and Geometry

## Volume 13 (2005)

### Number 3

### Square numbers, spanning trees and invariants of achiral knots

Pages: 591 – 631

DOI: http://dx.doi.org/10.4310/CAG.2005.v13.n3.a5

#### Author

#### Abstract

We give constructions to realize an odd number, which can be represented as sum of two squares, as determinant of an achiral knot, thus proving that these are exactly the numbers occurring as such determinants. Later, we study which numbers occur as determinants of prime alternating achiral knots, and obtain a complete result for perfect squares.

Using the checkerboard coloring, then an application is given to the number of spanning trees in planar self-dual graphs. Another application is some enumeration results on achiral rational knots.

A new formula for the volume of a hyperbolic and spherical tetrahedron is obtained from the quantum 6*j*-symbol. This formula is of symmetric form with respect to the symmetry of the tetrahedron.