Motive of the representation varietes of torus knots for low rank affine groups

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We compute the motive of the variety of representations of the torus knot of type (m, n) into the affine groups AGL1(C) and AGL2(C). For this, we stratify the varieties and show that the motives lie in the subring generated by the Lefschetz motive q = [C].
[1] M. Culler, P.B. Shalen, Varieties of group representations and splitting of 3-manifolds, Ann. of Math. (2) 117, 109–146, (1983). [2] P. Deligne, Th´eorie de Hodge II, Publ. Math. I.H.E.S. 40, 5–58, (1971). [3] P. Deligne, Th´eorie de Hodge III, Publ. Math. I.H.E.S. 44, 5–77, (1974). [4] ´ A. Gonz´alez-Prieto, Pseudo-quotients of algebraic actions and their application to character varieties, [5] A. Gonz´alez-Prieto, M. Logares, V. Mu˜noz, Representation variety for the rank one affine group. In eds. I.N. Parasidis, E. Providas and Th.M. Rassias, Mathematical Analysis in Interdisciplinary Research, Springer, to appear. [6] A. Gonz´alez-Prieto, V. Mu˜noz, Motive of the SL4-character variety of torus knots, [7] T. Hausel, M. Thaddeus, Mirror symmetry, Langlands duality, and the Hitchin system, Invent. Math. 153, 197–229, (2003). [8] N.J. Hitchin, The self-duality equations on a Riemann surface, Proc. London Math. Soc. (3) 55, 59–126, (1987). [9] S. Lawton, V. Mu˜noz, E-polynomial of the SL(3,C)-character variety of free groups, Pacific J. Math. 282, 173–202, (2016). 10] M. Logares, V. Mu˜noz, P.E. Newstead, Hodge polynomials of SL(2,C)-character varieties for curves of small genus, Rev. Mat. Complut. 26, 635–703, (2013). [11] A. Lubotzky, A.R. Magid, Varieties of representations of finitely generated groups, Mem. Amer. Math. Soc. 58 (1985). [12] J Mart´ın-Morales, A M Oller-Marc´en, Combinatorial aspects of the character variety of a family of one-relator groups, Topology Appl. 156, 2376–2389, (2009). [13] J. Mart´ınez, V. Mu˜noz, E-polynomials of the SL(2,C)-character varieties of surface groups, Internat. Math. Research Notices 2016, 926–961, (2016). [14] J. Mart´ınez, V. Mu˜noz, E-polynomial of the SL(2,C)-character variety of a complex curve of genus 3, Osaka J. Math. 53, 645–681, (2016). [15] V. Mu˜noz, The SL(2,C)-character varieties of torus knots, Rev. Mat. Complut. 22, 489–497, (2009). [16] V. Mu˜noz, J. Porti, Geometry of the SL(3,C)-character variety of torus knots, Algebraic and Geometric Topology 16, 397–426, (2016). [17] M. Nagata, Invariants of a group in an affine ring, J. Math. Kyoto Univ. 3, 369–377, (1963/1964). [18] P. E. Newstead, Introduction to moduli problems and orbit spaces, Tata Institute of Fundamental Research Lectures on Mathematics and Physics 51, (TIFR, 1978). [19] D. Rolfsen, Knots and links, Mathematics Lecture Series 7, (Publish or Perish, 1990).