It is known that if there isn't a closed essential surface in $S^3 \setminus K$, the dimension of $SL(2,\mathbb C)$ character variety is $1$. (In fact, it holds for a general 3-manifold, not only for a knot complement. )
Is there any known counter-example of a knot for the converse, i.e., a knot $K$ where a closed essential surface exists in $S^3\setminus K$ and $\dim_{\mathbb C}( \chi_{SL(2,\mathbb C)}(S^3\setminus K))=1$?
In Incompressible surfaces in tunnel number one knot complements by Mario Eudave-Muñoz, Figure 7 gives an example of a knot with a closed essential surface. The author guesses it is the least volume example. The volume computed is 6.2597017011.
Using SnapPy, one sees that knot 343 on the census also has this volume, and so the knot in Figure 7 appears to be K8_143 (I thank Mario Eudave-Muñoz for pointing this out to me).
From SnapPy this knot has fundamental group $$\langle a,b\ | \ aaaabAAAABaaaBAAAAbaaaabbbb\rangle$$ where $A=a^{-1}$ and $B=b^{-1}.$
In my paper, with JP Burelle and Caleb Ashley, Rank 1 character varieties of finitely presented groups an effective algorithm is described to compute the $\mathrm{SL}(2, \mathbb{C})$-character variety of any finitely presented group $\Gamma$. This algorithm has been integrated into SnapPy.
Using SnapPy, I computed relations that cut-out the $\mathrm{SL}(2, \mathbb{C})$-character variety of K8_143. Here is the code:
M=CensusKnots[343]
G=M.fundamental_group()
G.character_variety_vars_and_polys()
I then used Macaulay2 to compute the dimension. Here is the code (the second input comes from the output from SnapPy):
R=QQ[Ta,Tb,Tab]
I=ideal((-Tb^2 + 1)*Ta^20 + (3*Tab*Tb^3 - 4*Tab*Tb)*Ta^19 + ((-3*Tab^2 - 3)*Tb^4 + (3*Tab^2 + 24)*Tb^2 + (2*Tab^2 - 20))*Ta^18 + ((Tab^3 + 6*Tab)*Tb^5 + (2*Tab^3 - 60*Tab)*Tb^3 + (-6*Tab^3 + 68*Tab)*Tb)*Ta^17 + ((-3*Tab^2 - 3)*Tb^6 + (-2*Tab^4 + 42*Tab^2 + 54)*Tb^4 + (3*Tab^4 - 30*Tab^2 - 236)*Tb^2 + (Tab^4 - 32*Tab^2 + 170))*Ta^16 + (3*Tab*Tb^7 + (-6*Tab^3 - 84*Tab)*Tb^5 + (Tab^5 - 40*Tab^3 + 480*Tab)*Tb^3 + (-2*Tab^5 + 80*Tab^3 - 484*Tab)*Tb)*Ta^15 + (-Tb^8 + (27*Tab^2 + 40)*Tb^6 + (19*Tab^4 - 222*Tab^2 - 396)*Tb^4 + (-26*Tab^4 + 90*Tab^2 + 1252)*Tb^2 + (-12*Tab^4 + 212*Tab^2 - 800))*Ta^14 + (-28*Tab*Tb^7 + (2*Tab^3 + 468*Tab)*Tb^5 + (-10*Tab^5 + 272*Tab^3 - 2020*Tab)*Tb^3 + (20*Tab^5 - 436*Tab^3 + 1872*Tab)*Tb)*Ta^13 + (9*Tb^8 + (-94*Tab^2 - 212)*Tb^6 + (-71*Tab^4 + 560*Tab^2 + 1538)*Tb^4 + (84*Tab^4 + 37*Tab^2 - 3939)*Tb^2 + (58*Tab^4 - 753*Tab^2 + 2275))*Ta^12 + (104*Tab*Tb^7 + (64*Tab^3 - 1344*Tab)*Tb^5 + (40*Tab^5 - 898*Tab^3 + 4886*Tab)*Tb^3 + (-80*Tab^5 + 1251*Tab^3 - 4266*Tab)*Tb)*Ta^11 + (-32*Tb^8 + (152*Tab^2 + 576)*Tb^6 + (129*Tab^4 - 641*Tab^2 - 3430)*Tb^4 + (-113*Tab^4 - 778*Tab^2 + 7565)*Tb^2 + (-145*Tab^4 + 1550*Tab^2 - 4004))*Ta^10 + (-192*Tab*Tb^7 + (-194*Tab^3 + 2118*Tab)*Tb^5 + (-81*Tab^5 + 1592*Tab^3 - 6884*Tab)*Tb^3 + (162*Tab^5 - 2018*Tab^3 + 5792*Tab)*Tb)*Ta^9 + (56*Tb^8 + (-95*Tab^2 - 851)*Tb^6 + (-109*Tab^4 + 94*Tab^2 + 4425)*Tb^4 + (19*Tab^4 + 1784*Tab^2 - 8752)*Tb^2 + (199*Tab^4 - 1861*Tab^2 + 4290))*Ta^8 + (176*Tab*Tb^7 + (235*Tab^3 - 1762*Tab)*Tb^5 + (86*Tab^5 - 1502*Tab^3 + 5376*Tab)*Tb^3 + (-172*Tab^5 + 1797*Tab^3 - 4494*Tab)*Tb)*Ta^7 + (-48*Tb^8 + (-21*Tab^2 + 655)*Tb^6 + (23*Tab^4 + 446*Tab^2 - 3124)*Tb^4 + (99*Tab^4 - 1793*Tab^2 + 5754)*Tb^2 + (-145*Tab^4 + 1238*Tab^2 - 2640))*Ta^6 + (-64*Tab*Tb^7 + (-116*Tab^3 + 640*Tab)*Tb^5 + (-44*Tab^5 + 664*Tab^3 - 2001*Tab)*Tb^3 + (88*Tab^5 - 796*Tab^3 + 1786*Tab)*Tb)*Ta^5 + (16*Tb^8 + (40*Tab^2 - 216)*Tb^6 + (16*Tab^4 - 319*Tab^2 + 1028)*Tb^4 + (-80*Tab^4 + 786*Tab^2 - 1881)*Tb^2 + (48*Tab^4 - 396*Tab^2 + 825))*Ta^4 + ((12*Tab^3 - 41*Tab)*Tb^5 + (8*Tab^5 - 89*Tab^3 + 233*Tab)*Tb^3 + (-16*Tab^5 + 134*Tab^3 - 294*Tab)*Tb)*Ta^3 + ((-4*Tab^2 + 12)*Tb^6 + (-4*Tab^4 + 38*Tab^2 - 97)*Tb^4 + (12*Tab^4 - 96*Tab^2 + 222)*Tb^2 + (-4*Tab^4 + 40*Tab^2 - 100))*Ta^2 + (2*Tab*Tb^5 + (2*Tab^3 - 14*Tab)*Tb^3 + (-4*Tab^3 + 20*Tab)*Tb - 1)*Ta + (Tb^4 - 4*Tb^2 + 2),
(-Tb^3 + 2*Tb)*Ta^19 + (3*Tab*Tb^4 - 7*Tab*Tb^2 + Tab)*Ta^18 + ((-3*Tab^2 - 3)*Tb^5 + (6*Tab^2 + 26)*Tb^3 + (2*Tab^2 - 39)*Tb)*Ta^17 + ((Tab^3 + 6*Tab)*Tb^6 + (Tab^3 - 63*Tab)*Tb^4 + (-9*Tab^3 + 115*Tab)*Tb^2 + (2*Tab^3 - 17*Tab))*Ta^16 + ((-3*Tab^2 - 3)*Tb^7 + (-2*Tab^4 + 42*Tab^2 + 54)*Tb^5 + (5*Tab^4 - 63*Tab^2 - 262)*Tb^3 + (-36*Tab^2 + 320)*Tb)*Ta^15 + (3*Tab*Tb^8 + (-5*Tab^3 - 81*Tab)*Tb^6 + (Tab^5 - 33*Tab^3 + 501*Tab)*Tb^4 + (-3*Tab^5 + 117*Tab^3 - 778*Tab)*Tb^2 + (Tab^5 - 26*Tab^3 + 120*Tab))*Ta^14 + (-Tb^9 + (24*Tab^2 + 38)*Tb^7 + (17*Tab^4 - 210*Tab^2 - 384)*Tb^5 + (-40*Tab^4 + 228*Tab^2 + 1372)*Tb^3 + (-5*Tab^4 + 256*Tab^2 - 1435)*Tb)*Ta^13 + (-25*Tab*Tb^8 + (-2*Tab^3 + 421*Tab)*Tb^6 + (-9*Tab^5 + 238*Tab^3 - 2016*Tab)*Tb^4 + (27*Tab^5 - 602*Tab^3 + 2803*Tab)*Tb^2 + (-9*Tab^5 + 136*Tab^3 - 455*Tab))*Ta^12 + (8*Tb^9 + (-73*Tab^2 - 186)*Tb^7 + (-56*Tab^4 + 480*Tab^2 + 1406)*Tb^5 + (120*Tab^4 - 252*Tab^2 - 4129)*Tb^3 + (40*Tab^4 - 938*Tab^2 + 3824)*Tb)*Ta^11 + (82*Tab*Tb^8 + (59*Tab^3 - 1098*Tab)*Tb^6 + (32*Tab^5 - 755*Tab^3 + 4505*Tab)*Tb^4 + (-96*Tab^5 + 1582*Tab^3 - 5795*Tab)*Tb^2 + (32*Tab^5 - 367*Tab^3 + 999*Tab))*Ta^10 + (-25*Tb^9 + (98*Tab^2 + 456)*Tb^7 + (86*Tab^4 - 453*Tab^2 - 2877)*Tb^5 + (-152*Tab^4 - 420*Tab^2 + 7356)*Tb^3 + (-126*Tab^4 + 1925*Tab^2 - 6169)*Tb)*Ta^9 + (-131*Tab*Tb^8 + (-145*Tab^3 + 1521*Tab)*Tb^6 + (-56*Tab^5 + 1215*Tab^3 - 5625*Tab)*Tb^4 + (168*Tab^5 - 2244*Tab^3 + 6852*Tab)*Tb^2 + (-56*Tab^5 + 539*Tab^3 - 1269*Tab))*Ta^8 + (38*Tb^9 + (-37*Tab^2 - 595)*Tb^7 + (-52*Tab^4 - 66*Tab^2 + 3312)*Tb^5 + (32*Tab^4 + 1428*Tab^2 - 7633)*Tb^3 + (196*Tab^4 - 2232*Tab^2 + 5880)*Tb)*Ta^7 + (99*Tab*Tb^8 + (143*Tab^3 - 1045*Tab)*Tb^6 + (48*Tab^5 - 965*Tab^3 + 3612*Tab)*Tb^4 + (-144*Tab^5 + 1624*Tab^3 - 4269*Tab)*Tb^2 + (48*Tab^5 - 409*Tab^3 + 868*Tab))*Ta^6 + (-28*Tb^9 + (-36*Tab^2 + 396)*Tb^7 + (-8*Tab^4 + 420*Tab^2 - 2022)*Tb^5 + (96*Tab^4 - 1440*Tab^2 + 4316)*Tb^3 + (-152*Tab^4 + 1380*Tab^2 - 3082)*Tb)*Ta^5 + (-24*Tab*Tb^8 + (-48*Tab^3 + 250*Tab)*Tb^6 + (-16*Tab^5 + 288*Tab^3 - 876*Tab)*Tb^4 + (48*Tab^5 - 464*Tab^3 + 1082*Tab)*Tb^2 + (-16*Tab^5 + 128*Tab^3 - 260*Tab))*Ta^4 + (8*Tb^9 + (28*Tab^2 - 110)*Tb^7 + (16*Tab^4 - 218*Tab^2 + 548)*Tb^5 + (-64*Tab^4 + 536*Tab^2 - 1131)*Tb^3 + (48*Tab^4 - 374*Tab^2 + 758)*Tb)*Ta^3 + (-4*Tab*Tb^8 + (-4*Tab^3 + 24*Tab)*Tb^6 + (12*Tab^3 - 31*Tab)*Tb^4 - 19*Tab*Tb^2 + (-4*Tab^3 + 17*Tab))*Ta^2 + (4*Tb^7 + (4*Tab^2 - 33)*Tb^5 + (-16*Tab^2 + 82)*Tb^3 + (12*Tab^2 - 55)*Tb)*Ta + (-Tab*Tb^4 + 3*Tab*Tb^2 - Tb - Tab),
(-Tb^2 + 1)*Ta^19 + (3*Tab*Tb^3 - 4*Tab*Tb)*Ta^18 + ((-3*Tab^2 - 3)*Tb^4 + (3*Tab^2 + 23)*Tb^2 + (2*Tab^2 - 19))*Ta^17 + ((Tab^3 + 6*Tab)*Tb^5 + (2*Tab^3 - 57*Tab)*Tb^3 + (-6*Tab^3 + 64*Tab)*Tb)*Ta^16 + ((-3*Tab^2 - 3)*Tb^6 + (-2*Tab^4 + 39*Tab^2 + 51)*Tb^4 + (3*Tab^4 - 27*Tab^2 - 214)*Tb^2 + (Tab^4 - 30*Tab^2 + 152))*Ta^15 + (3*Tab*Tb^7 + (-5*Tab^3 - 78*Tab)*Tb^5 + (Tab^5 - 38*Tab^3 + 426*Tab)*Tb^3 + (-2*Tab^5 + 74*Tab^3 - 424*Tab)*Tb)*Ta^14 + (-Tb^8 + (24*Tab^2 + 37)*Tb^6 + (17*Tab^4 - 186*Tab^2 - 348)*Tb^4 + (-23*Tab^4 + 66*Tab^2 + 1059)*Tb^2 + (-11*Tab^4 + 184*Tab^2 - 665))*Ta^13 + (-25*Tab*Tb^7 + (-2*Tab^3 + 396*Tab)*Tb^5 + (-9*Tab^5 + 236*Tab^3 - 1645*Tab)*Tb^3 + (18*Tab^5 - 368*Tab^3 + 1504*Tab)*Tb)*Ta^12 + (8*Tb^8 + (-73*Tab^2 - 178)*Tb^6 + (-56*Tab^4 + 407*Tab^2 + 1236)*Tb^4 + (64*Tab^4 + 82*Tab^2 - 3055)*Tb^2 + (48*Tab^4 - 595*Tab^2 + 1729))*Ta^11 + (82*Tab*Tb^7 + (59*Tab^3 - 1016*Tab)*Tb^5 + (32*Tab^5 - 696*Tab^3 + 3571*Tab)*Tb^3 + (-64*Tab^5 + 945*Tab^3 - 3076*Tab)*Tb)*Ta^10 + (-25*Tb^8 + (98*Tab^2 + 431)*Tb^6 + (86*Tab^4 - 355*Tab^2 - 2471)*Tb^4 + (-66*Tab^4 - 677*Tab^2 + 5266)*Tb^2 + (-106*Tab^4 + 1089*Tab^2 - 2717))*Ta^9 + (-131*Tab*Tb^7 + (-145*Tab^3 + 1390*Tab)*Tb^5 + (-56*Tab^5 + 1070*Tab^3 - 4366*Tab)*Tb^3 + (112*Tab^5 - 1319*Tab^3 + 3614*Tab)*Tb)*Ta^8 + (38*Tb^8 + (-37*Tab^2 - 557)*Tb^6 + (-52*Tab^4 - 103*Tab^2 + 2793)*Tb^4 + (-20*Tab^4 + 1288*Tab^2 - 5321)*Tb^2 + (124*Tab^4 - 1121*Tab^2 + 2508))*Ta^7 + (99*Tab*Tb^7 + (143*Tab^3 - 946*Tab)*Tb^5 + (48*Tab^5 - 822*Tab^3 + 2765*Tab)*Tb^3 + (-96*Tab^5 + 945*Tab^3 - 2252*Tab)*Tb)*Ta^6 + (-28*Tb^8 + (-36*Tab^2 + 368)*Tb^6 + (-8*Tab^4 + 384*Tab^2 - 1682)*Tb^4 + (88*Tab^4 - 1092*Tab^2 + 2946)*Tb^2 + (-72*Tab^4 + 600*Tab^2 - 1254))*Ta^5 + (-24*Tab*Tb^7 + (-48*Tab^3 + 226*Tab)*Tb^5 + (-16*Tab^5 + 240*Tab^3 - 674*Tab)*Tb^3 + (32*Tab^5 - 272*Tab^3 + 586*Tab)*Tb)*Ta^4 + (8*Tb^8 + (28*Tab^2 - 102)*Tb^6 + (16*Tab^4 - 190*Tab^2 + 454)*Tb^4 + (-48*Tab^4 + 374*Tab^2 - 763)*Tb^2 + (16*Tab^4 - 134*Tab^2 + 285))*Ta^3 + (-4*Tab*Tb^7 + (-4*Tab^3 + 20*Tab)*Tb^5 + (8*Tab^3 - 15*Tab)*Tb^3 + (4*Tab^3 - 22*Tab)*Tb)*Ta^2 + (4*Tb^6 + (4*Tab^2 - 29)*Tb^4 + (-12*Tab^2 + 57)*Tb^2 + (4*Tab^2 - 19))*Ta + (-Tab*Tb^3 + 2*Tab*Tb - 2))
dim I
The resulting output is 1.
