LEADER 04286nam 22005655 450 001 9910300121903321 005 20200705073352.0 010 $a981-13-1150-1 010 $a978-981-13-1150-5 024 7 $a10.1007/978-981-13-1150-5 035 $a(CKB)4100000005679292 035 $a(DE-He213)978-981-13-1150-5 035 $a(MiAaPQ)EBC5494014 035 $a(PPN)229913172 035 $a(EXLCZ)994100000005679292 100 $a20180815d2018 u| 0 101 0 $aeng 135 $aurnn|008mamaa 181 $ctxt$2rdacontent 182 $cc$2rdamedia 183 $acr$2rdacarrier 200 10$aVolume Conjecture for Knots$b[electronic resource] /$fby Hitoshi Murakami, Yoshiyuki Yokota 205 $a1st ed. 2018. 210 1$aSingapore :$cSpringer Singapore :$cImprint: Springer,$d2018. 215 $a1 online resource (IX, 120 p. 98 illus., 18 illus. in color.) 225 1 $aSpringerBriefs in Mathematical Physics,$x2197-1757 ;$v30 311 $a981-13-1149-8 320 $aIncludes bibliographical references and index. 327 $a1. Preliminaries (knots and links, braids, hyperbolic geometry) -- 2. R-matrix, the Kashaev invariant and the colored Jones polynomimal -- 3. Volume conjecture -- 4. Triangulation of a knot complement and hyperbolicity equation -- 5. Idea of the ?proof? -- 6. Representations of a knot group into SL(2;C) and their Chern-Simons invariant -- 7. Generalization of the volume conjecture. 330 $aThe volume conjecture states that a certain limit of the colored Jones polynomial of a knot in the three-dimensional sphere would give the volume of the knot complement. Here the colored Jones polynomial is a generalization of the celebrated Jones polynomial and is defined by using a so-called R-matrix that is associated with the N-dimensional representation of the Lie algebra sl(2;C). The volume conjecture was first stated by R. Kashaev in terms of his own invariant defined by using the quantum dilogarithm. Later H. Murakami and J. Murakami proved that Kashaev?s invariant is nothing but the N-dimensional colored Jones polynomial evaluated at the Nth root of unity. Then the volume conjecture turns out to be a conjecture that relates an algebraic object, the colored Jones polynomial, with a geometric object, the volume. In this book we start with the definition of the colored Jones polynomial by using braid presentations of knots. Then we state the volume conjecture and give a very elementary proof of the conjecture for the figure-eight knot following T. Ekholm. We then give a rough idea of the ?proof?, that is, we show why we think the conjecture is true at least in the case of hyperbolic knots by showing how the summation formula for the colored Jones polynomial ?looks like? the hyperbolicity equations of the knot complement. We also describe a generalization of the volume conjecture that corresponds to a deformation of the complete hyperbolic structure of a knot complement. This generalization would relate the colored Jones polynomial of a knot to the volume and the Chern?Simons invariant of a certain representation of the fundamental group of the knot complement to the Lie group SL(2;C). We finish by mentioning further generalizations of the volume conjecture. 410 0$aSpringerBriefs in Mathematical Physics,$x2197-1757 ;$v30 606 $aMathematical physics 606 $aTopology 606 $aHyperbolic geometry 606 $aMathematical Physics$3https://scigraph.springernature.com/ontologies/product-market-codes/M35000 606 $aTopology$3https://scigraph.springernature.com/ontologies/product-market-codes/M28000 606 $aHyperbolic Geometry$3https://scigraph.springernature.com/ontologies/product-market-codes/M21030 615 0$aMathematical physics. 615 0$aTopology. 615 0$aHyperbolic geometry. 615 14$aMathematical Physics. 615 24$aTopology. 615 24$aHyperbolic Geometry. 676 $a530.15 700 $aMurakami$b Hitoshi$4aut$4http://id.loc.gov/vocabulary/relators/aut$0768009 702 $aYokota$b Yoshiyuki$4aut$4http://id.loc.gov/vocabulary/relators/aut 906 $aBOOK 912 $a9910300121903321 996 $aVolume Conjecture for Knots$91963846 997 $aUNINA