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Experimental evidence for the Volume Conjecture for the simplest hyperbolic non-2-bridge knot. (English) Zbl 1092.57005
The Volume Conjecture for hyperbolic knots consists of two parts: (a) it states that the limit of a sequence of complex numbers (involving the $$n$$-th colored Jones polynomial evaluated at the primitive complex $$n$$-th root of unity) exists and (b) it identifies the limit essentially with the hyperbolic volume of the knot. The Volume Conjecture is known only for the simplest hyperbolic knot $$4_1$$. The authors develop an efficient formula for the colored Jones function of the $$k4_3$$ knot, which is the simplest hyperbolic non $$2$$-bridge knot. Then using that formula they present numerical evidence (via a Fortran program) for the Volume Conjecture for the $$k4_3$$ knot. For this knot they also provide among other computations a presentation of its fundamental group and peripheral system, the Alexander and $$A$$-polynomial, and the rank of the Heegaard-Floer Homology.

##### MSC:
 57M25 Knots and links in the $$3$$-sphere (MSC2010) 57N10 Topology of general $$3$$-manifolds (MSC2010)
##### Keywords:
Hyperbolc Volume Conjecture; Jones Polynomial
##### Software:
qZeil; Snap; SnapPea
Full Text:
##### References:
 [1] D H Bailey, A Fortran-90 based multiprecision system, ACM Trans. Math. Software 21 (1995) 379 · Zbl 0883.68017 · doi:10.1145/212066.212075 · www.acm.org [2] D W Boyd, Mahler’s measure and invariants of hyperbolic manifolds, A K Peters (2002) 127 · Zbl 1030.11055 [3] P J Callahan, J C Dean, J R Weeks, The simplest hyperbolic knots, J. Knot Theory Ramifications 8 (1999) 279 · Zbl 0933.57010 · doi:10.1142/S0218216599000195 [4] D Cooper, M Culler, H Gillet, D D Long, P B Shalen, Plane curves associated to character varieties of 3-manifolds, Invent. Math. 118 (1994) 47 · Zbl 0842.57013 · doi:10.1007/BF01231526 · eudml:144229 [5] O Costin, S Garoufalidis, in preparation [6] J C Dean, Small Seifert-fibered Dehn surgery on hyperbolic knots, Algebr. Geom. Topol. 3 (2003) 435 · Zbl 1021.57002 · doi:10.2140/agt.2003.3.435 · emis:journals/UW/agt/AGTVol3/agt-3-14.abs.html · eudml:122904 · arxiv:math/0306232 [7] S Garoufalidis, T T Q Lê, The colored Jones function is $$q$$-holonomic, Geom. Topol. 9 (2005) 1253 · Zbl 1078.57012 · doi:10.2140/gt.2005.9.1253 · emis:journals/UW/gt/GTVol9/paper29.abs.html · eudml:125449 · arxiv:math/0309214 [8] S Garoufalidis, On the characteristic and deformation varieties of a knot, Geom. Topol. Monogr. 7, Geom. Topol. Publ., Coventry (2004) 291 · Zbl 1080.57014 · emis:journals/UW/gt/GTMon7/paper12.abs.html [9] S Garoufalidis, J S Geronimo, Asymptotics of $$q$$-difference equations, Contemp. Math. 416, Amer. Math. Soc. (2006) 83 · Zbl 1144.57005 [10] W Gehrke, Fortran 90 language guide, Springer, New York (1995) · Zbl 0830.68016 [11] H Goda, H Matsuda, T Morifuji, Knot Floer homology of $$(1,1)$$-knots, Geom. Dedicata 112 (2005) 197 · Zbl 1081.57011 · doi:10.1007/s10711-004-5403-2 [12] S Gukov, Three-dimensional quantum gravity, Chern-Simons theory, and the $$A$$-polynomial, Comm. Math. Phys. 255 (2005) 577 · Zbl 1115.57009 · doi:10.1007/s00220-005-1312-y · arxiv:hep-th/0306165 [13] R M Kashaev, The hyperbolic volume of knots from the quantum dilogarithm, Lett. Math. Phys. 39 (1997) 269 · Zbl 0876.57007 · doi:10.1023/A:1007364912784 [14] L H Kauffman, S L Lins, Temperley-Lieb recoupling theory and invariants of 3-manifolds, Annals of Mathematics Studies 134, Princeton University Press (1994) · Zbl 0821.57003 [15] V F R Jones, Hecke algebra representations of braid groups and link polynomials, Ann. of Math. $$(2)$$ 126 (1987) 335 · Zbl 0631.57005 · doi:10.2307/1971403 [16] T T Q Lê, The colored Jones polynomial and the $$A$$-polynomial of knots, Adv. Math. 207 (2006) 782 · Zbl 1114.57014 · doi:10.1016/j.aim.2006.01.006 [17] G Masbaum, P Vogel, 3-valent graphs and the Kauffman bracket, Pacific J. Math. 164 (1994) 361 · Zbl 0838.57007 · doi:10.2140/pjm.1994.164.361 [18] K Morimoto, M Sakuma, Y Yokota, Identifying tunnel number one knots, J. Math. Soc. Japan 48 (1996) 667 · Zbl 0869.57008 · doi:10.2969/jmsj/04840667 [19] H Murakami, J Murakami, The colored Jones polynomials and the simplicial volume of a knot, Acta Math. 186 (2001) 85 · Zbl 0983.57009 · doi:10.1007/BF02392716 [20] H Murakami, The asymptotic behavior of the colored Jones function of a knot and its volume · Zbl 0983.57009 · arxiv:math.GT/0004036 [21] P Ozsváth, Z Szabó, Heegaard Floer homology and alternating knots, Geom. Topol. 7 (2003) 225 · Zbl 1130.57303 · doi:10.2140/gt.2003.7.225 · emis:journals/UW/gt/GTVol7/paper6.abs.html · eudml:128404 · arxiv:math/0209149 [22] J Weeks, SnapPea · www.geometrygames.org [23] D Coulson, O A Goodman, C D Hodgson, W D Neumann, Snap · www.ms.unimelb.edu.au [24] W P Thurston, The geometry and topology of 3-manifolds, lecture notes, MSRI (1979) · www.msri.org [25] V G Turaev, The Yang-Baxter equation and invariants of links, Invent. Math. 92 (1988) 527 · Zbl 0648.57003 · doi:10.1007/BF01393746 · eudml:143581 [26] H S Wilf, D Zeilberger, An algorithmic proof theory for hypergeometric (ordinary and “$$q$$”) multisum/integral identities, Invent. Math. 108 (1992) 575 · Zbl 0739.05007 · doi:10.1007/BF02100618 · eudml:144006
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