LEADER 03418nam 2200625Ia 450 001 9910438143803321 005 20200520144314.0 010 $a3-642-33302-8 024 7 $a10.1007/978-3-642-33302-6 035 $a(CKB)3400000000102777 035 $a(SSID)ssj0000831484 035 $a(PQKBManifestationID)11470845 035 $a(PQKBTitleCode)TC0000831484 035 $a(PQKBWorkID)10872882 035 $a(PQKB)10283002 035 $a(DE-He213)978-3-642-33302-6 035 $a(MiAaPQ)EBC3070786 035 $a(PPN)168324083 035 $a(EXLCZ)993400000000102777 100 $a20121223d2013 uy 0 101 0 $aeng 135 $aurnn|008mamaa 181 $ctxt 182 $cc 183 $acr 200 10$aGuts of surfaces and the colored Jones polynomial /$fDavid Futer, Efstratia Kalfagianni, Jessica Purcell 205 $a1st ed. 2013. 210 $aHeidelberg ;$aNew York $cSpringer$dc2013 215 $a1 online resource (X, 170 p. 62 illus., 45 illus. in color.) 225 1 $aLecture notes in mathematics,$x1617-9692 ;$v2069 300 $aBibliographic Level Mode of Issuance: Monograph 311 $a3-642-33301-X 320 $aIncludes bibliographical references (p. 163-166) and index. 327 $a1 Introduction -- 2 Decomposition into 3?balls -- 3 Ideal Polyhedra -- 4 I?bundles and essential product disks -- 5 Guts and fibers -- 6 Recognizing essential product disks -- 7 Diagrams without non-prime arcs -- 8 Montesinos links -- 9 Applications -- 10 Discussion and questions. 330 $aThis monograph derives direct and concrete relations between colored Jones polynomials and the topology of incompressible spanning surfaces in knot and link complements. Under mild diagrammatic hypotheses, we prove that the growth of the degree of the colored Jones polynomials is a boundary slope of an essential surface in the knot complement. We show that certain coefficients of the polynomial measure how far this surface is from being a fiber for the knot; in particular, the surface is a fiber if and only if a particular coefficient vanishes. We also relate hyperbolic volume to colored Jones polynomials. Our method is to generalize the checkerboard decompositions of alternating knots. Under mild diagrammatic hypotheses, we show that these surfaces are essential, and obtain an ideal polyhedral decomposition of their complement. We use normal surface theory to relate the pieces of the JSJ decomposition of the  complement to the combinatorics of certain surface spines (state graphs). Since state graphs have previously appeared in the study of Jones polynomials, our method bridges the gap between quantum and geometric knot invariants. 410 0$aLecture notes in mathematics (Springer-Verlag) ;$v2069. 606 $aKnot theory 606 $aThree-manifolds (Topology) 606 $aComplex manifolds 606 $aGeometry, Hyperbolic 615 0$aKnot theory. 615 0$aThree-manifolds (Topology) 615 0$aComplex manifolds. 615 0$aGeometry, Hyperbolic. 676 $a514.2242 700 $aFuter$b David$0479687 701 $aKalfagianni$b Efstratia$0521607 701 $aPurcell$b Jessica$0521608 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bMiAaPQ 906 $aBOOK 912 $a9910438143803321 996 $aGuts of surfaces and the colored Jones polynomial$9836978 997 $aUNINA