Non-standard components of the character variety for a family of Montesinos knots (Q2848493)

A Montesinos link consists of \(n\) rational tangles \(\beta_i/\alpha_i\). It is a knot if and only if all \(\alpha_i\) are odd or exactly one of them is even. The latter type is called a Montesinos knot of Kinoshita-Terasaka type.NEWLINENEWLINEThe main result shows that for any integers \(n>3\) and \(1\leq d\leq n-3\), there exist Montesinos knots of Kinoshita-Terasaka type with \(n\) tangles, whose \(\mathrm{SL}_2(\mathbb{C})\)-character varieties contain arbitrarily many irreducible components of dimension \(d\). In addition, the trace of the meridian is not constant on these components. A point is the fact that such a Montesinos knot yields a connected sum of two-bridge knots by changing some crossings in the rational tangle with even denominator, and so representations of the factors yield those of the original Montesinos knot.