Linear growth conjecture for character varieties of Montesinos knots

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Let m=m(a1/b1,a2/b2,…,at/bt)\mathfrak{m}=\mathfrak{m}(a_1/b_1,a_2/b_2,\ldots,a_t/b_t) be a Montesinos knot with tt tangles. Consider its PSL⁡2(C)\operatorname{PSL}_2(\mathbb C)- and SL⁡2(C)\operatorname{SL}_2(\mathbb C)-character varieties. Linear growth conjecture. The dimensions of the PSL⁡2(C)\operatorname{PSL}_2(\mathbb C)- and SL⁡2(C)\operatorname{SL}_2(\mathbb C)-character varieties of m\mathfrak{m} grow linearly with the number of tangles tt. The conjecture is motivated by the linear growth of the dimension of the SL⁡2(R)\operatorname{SL}_2(\mathbb R) character variety of the associated orbifold as the number of tangles increases; the source does not state a resolution.

References

Primary source

Thomas W. Mattman, “The Culler-Shalen seminorms of the (-2,3,n) pretzel knot”, arXiv:math/9911085 (2001).

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