Character varieties of even classical pretzel knots

Abstract

For each even classical pretzel knot P (2k 1 + 1, 2k 2 + 1, 2k 3 ), we determine the character variety of irreducible SL(2, C)-representations, and clarify the steps of computing its A-polynomial.

The ${\rm SL}(2,\mathbb{C})$-character variety of the Borromean link

“…Concrete computations are still rare in the literature. The results for knots whose groups are generated by at least three generators mainly belong to the first author [2][3][4]. Till now there have been no results for links with at least three components.…”

The ${\rm SL}(2,\mathbb{C})$-character variety of a Montesinos knot

“…Existing results include: Torus knots [10], double twist knots [9], double twist links [12], (−2, 2m + 1, 2n)-pretzel links and twisted Whitehead links [16]. For classical pretzel knots, the author [3,4] found their character varieties. the irreducible character variety of each classical pretzel knot can be embedded in C 5 , consisting of a finite set of points, some conics, and a high-genus algebraic curve; the defining equations can be explicitly written down.…”