NON-PERIPHERAL IDEAL DECOMPOSITIONS OF ALTERNATING KNOTS
NON-PERIPHERAL IDEAL DECOMPOSITIONS OF ALTERNATING KNOTSSTAVROS GAROUFALIDIS, IAIN MOFFATT, AND DYLAN P. THURSTONTM with one cusp is non-peripheralM . For such a triangulation, thegluing and completeness equations can be solved to recover the hyperbolic structure of M . A planarAbstract. An ideal triangulationif no edge ofTof a hyperbolic 3-manifoldis homotopic to a curve in the boundary torus ofprojection of a knot gives four ideal cell decompositions of its complement (minus 2 balls), two ofwhich are ideal triangulations that use 4 (resp., 5) ideal tetrahedra per crossing. Our main resultis that these ideal triangulations are non-peripheral for all planar, reduced, alternating projectionsof hyperbolic knots.Our proof uses the small cancellation properties of the Dehn presentationof alternating knot groups, and an explicit solution to their word and conjugacy problems.Inparticular, we describe a planar complex that encodes all geodesic words that represent elementsof the peripheral subgroup of an alternating knot group. This gives a polynomial time algorithmfor checking if an element in an alternating knot group is peripheral. Our motivation for this workcomes from the Volume Conjecture for knots.Contents1.Introduction11.1.Motivation: the Volume Conjecture21.2.Non-peripheral ideal triangulations of alternating knots21.3.Alternating knots and small cancellation theory3Acknowledgements32.4Small cancellation theory2.1.The (augmented) Dehn presentation of a knot group42.2.Square and grid presentations52.3.The word problem for square presentations62.4.The peripheral complex72.5.Some properties of the peripheral complex2.6.The peripheral word problem132.7.Proof of Theorem 1.3142.8.The peripheral complex and the Gauss code of an alternating knot15References9171. IntroductionDate : June 22, 2017.1991 Mathematics Subject Classi cation. Primary 57N10. Secondary 20F06, 57M25.Key words and phrases. ideal triangulations, knots, hyperbolic geometry, ideal tetrahedra,small cancellationtheory, Dehn presentation, alternating knots, Volume Conjecture.S.G. and D.T. were supported in part by National Science Foundation Grants DMS-15-07244 and DMS-14-06419respectively.1
2STAVROS GAROUFALIDIS, IAIN MOFFATT, AND DYLAN P. THURSTON1.1.Motivation: the Volume Conjecture.The motivation of our paper comes from the Kashaev'sVolume Conjecture for knots in 3-space, which states that for a hyperbolic knot K1Vol(K)log hKiN n n2πKashaev invariants of K ; see [Kas97, MM01].inS3we have:limwherehKiNis theThis gives a precise connectionbetween quantum topology and hyperbolic geometry. The Volume Conjecture has been veri ed foronly a handful hyperbolic knots: initially for the simplest hyperbolic41knot and now, due to the6work of Ohtsuki [Oht17], and Ohtsuki and Yokota [OY16], for all hyperbolic knots with at mostcrossings.The Volume Conjecture requires a common input for computing both the Kashaev invariant andthe hyperbolic volume. Such an input turns out to be a planar projection of a knotKwhich allowsone to express the Kashaev invariant as a multi-dimensional state sum whose summand is a ratioof quantum factorials (4 or 5, depending on the model used).On the other hand, a planar projection gives four ideal cell decompositions of its complement(minus 2 balls), two of which are ideal triangulations that use 4 (resp., 5) ideal tetrahedra percrossing. These ideal triangulations are well-known from the early days of hyperbolic geometry, andwere used by Weeks [Wee05] (in his computer programSnapPy [CDW]), by the third author [Thu99],Yokota [Yok02, Yok11], Sakuma-Yokota [SY] and others.An approach to the Volume Conjecture initiated by the third author in [Thu99], and also byYokota, Kashaev, Hikami, the rst author and others (see [Gar08, KY, Hik01, Yok02]), is to convertmulti-dimensional state-sum formulas for the Kashaev invariant to multi-dimensional state-integralformulas over suitable cycles, and then to apply a steepest descent method to study the asymptoticbehaviour of the Kashaev invariant.The summand (and hence, the integrand) depends on theplanar projection and the steepest descend method is applied to a leading term of the integrand,the so-called potential function.The critical points of the potential function have a geometricmeaning, namely they are solutions to thegluing equations.The latter are a special system ofpolynomial equations (studied by W. Thurston and Neumann-Zagier in [Thu77, NZ85]) that areassociated to the ideal triangulations of the knot complement discussed above. A suitable solutionto the gluing equations recovers the hyperbolic structure, and the value of the potential function isthe volume of the knot.The problem is that every planar projection leads to ideal triangulations, hence to gluing equations, and even if we know that the knot is hyperbolic, it is by no means obvious that those gluingequations have a suitable solution (or in fact, any solution) that recovers the complete hyperbolicstructure. It turns out that if a knot is hyperbolic, the lack of a suitable solution occurs only whenedges of the ideal triangulation are homotopic to peripheral curves in the boundary tori.1.2.Non-peripheral ideal triangulations of alternating knots. Ideal triangulationsof hy-perbolic 3-manifolds with cusps were introduced by W. Thurston in his study of Geometrizationof 3-manifolds; see [Thu77]. For thorough discussions, see [BP92, CDW, NZ85, Wee05]. An idealtriangulationTof a hyperbolic 3-manifoldMhomotopic to a curve in the boundary torus ofwith one cusp isM.non-peripheralif no edge ofTisFor such a triangulation, the gluing and com-pleteness equations of [NZ85] can be solved to recover the hyperbolic structure ofM.For a proof,see [Til12, Lem.2.2] and also the discussion in [DG12, Sec.3]. of a knot gives rise to four ideal cell decompositions of its complement ( ) and T ( )), the last two of which are ideal triangulations thatT2B ( ), TO ( ), T4T5TA planar projection(namely,use 4 (resp., 5) ideal tetrahedra per crossing.We will brie y recall these decompositions here,although their precise de nition is not needed for the statement and proof of Theorem 1.3 below. T2B ( )is a decomposition of the knot complement into one ball above and one ball below theplanar projection. These two balls have a cell-decomposition that matches the planar projection ofthe knot, and were originally studied by W. Thurston, and more recently by Lackenby [Lac04].
NON-PERIPHERAL IDEAL DECOMPOSITIONS OF ALTERNATING KNOTS TO ( )3is a decomposition of the knot complement minus two balls into ideal octahedra, one ateach crossing of .This was described by Weeks [Wee05], and also by the third author [Thu99],and by Yokota [Yok02, Yok11]. Each ideal octahedron can be subdivided into 4 ideal tetrahedra, or into 5 ideal tetrahedra. Thus, a subdivision of TO ( ) gives rise to two ideal triangulations of the knot complement minus two balls, denoted by T4T ( ) and T5T ( ).Theorem 1.1. If is a prime, reduced, alternating projection of a non-torus knot K , then the four ( ) and T ( ) are non-peripheral. Consequently, theideal cell decompositions T2B ( ), TO ( ), T4T5Tgluing equations have a solution that recovers the complete hyperbolic structure.1.3.Alternating knots and small cancellation theory.The above theorem follows from prov-ing that all edges of the above ideal triangulations are homotopically non-peripheral. Luckily, wecan describe those edges directly in terms of the planar projection of the knot as follows.De nition 1.2.R2 as the R23of R , andLetxy -planen crossings. Consider the projection planeK S 3 R3 { } obtained from by pulling be a knot diagram withconsider the knotthe overcrossing arcs above the plane and undercrossing arcs under the plane in the standard way.π1 (S 3 \ K)π1 (S 3 \ K).Fix a basepoint forkinds of loops in(1) AWirtinger arcin the unbounded region near one strand offollows the double of throughkK.crossings withWe distinguish four1 k 2nand thenreturns to the basepoint through either the upper or lower half-space.(2) AWirtinger loopstarts at the basepoint, travels in either the upper (resp. lower) half-spaceto pass through a regionRof ,passes through a region adjacent toR,and then returnsthrough the upper (resp. lower) half-space to the basepoint. We forbid the short loop aroundthe strand near the basepoint, which is manifestly a meridian.(3) ADehn arcstarts at the basepoint, travels in the upper (resp. lower) half-space through a and then returns to the basepoint through the lower (resp. upper) half-spaceregion ofwithout passing through the projection plane.(4) Ashort arcfollows the double of from the basepoint until some crossing, where it jumpsto the other strand in the crossing and then follows the double back to the basepoint.There four types of arc are illustrated in Figure 1.These arcs are denoted by the lettersA, B , CandDin [SY].Theorem 1.3. If is a prime, reduced, alternating projection of a non-torus knot K , then allWirtinger arcs, Wirtinger loops, Dehn arcs and short arcs are non-peripheral.Theorem 1.1 immediately follows from Theorem 1.3, since all of the arcs that appear in any ofthe decompositions in Theorem 1.1 are of one of the four types in Theorem 1.3.The proof of Theorem 1.3 uses the small cancellation property of the Dehn presentation ofhyperbolic alternating knots. Curiously, our proof uses an explicit solution to the conjugacy problemof the Dehn presentation of a prime reduced alternating planar projection .See Remark 2.17below. More generally, we emphasise that the approach we take in this paper is an algorithmic one.Moreover our algorithms run in polynomial-time (in the length of the word).Acknowledgements.A rst draft of this paper was written in 2002 and was completed in 2007,but unfortunately remained unpublished.During a conference in Waseda University in 2016 inhonour of the 20th anniversary of the Volume Conjecture, an alternative proof of the results ofour paper (using cubical complexes) was announced by Sakuma-Yokota [SY], and with the samemotivation as ours. We thank Sakuma-Yokota for their encouragement to publish our results, andthe organisers of the Waseda conference (especially Jun Murakami) for their hospitality.
4STAVROS GAROUFALIDIS, IAIN MOFFATT, AND DYLAN P. THURSTONWirtinger arcWirtinger loopDehn arcShort arcFigure 1. Four types of loop in a knot complement.2. Small cancellation theory2.1.The (augmented) Dehn presentation of a knot group.We begin with a discussion of theaugmented Dehn presentation of a knot diagram. As it turns out, the augmented Dehn presentation(de ned below) is a small cancellation group and this structure provides a quick and implementablesolution to its word problem. Background on small cancellation groups and combinatorial grouptheory can be found in [LS77].Throughout this paper we implicitlywe write a set of relatorsR,symmetrizeall group presentations. This means that whenwe actually mean the set of all relators which can be obtained fromRby inversion and cyclic permutation.Letn 1 be ancrossing planar diagram of a linkL.Of theof these regions are bounded. Assign a unique labelregion and the label0n 2 regions of the diagram , exactly1, 2, . . . , n 1 to each of these boundedto the unbounded region. We identify each region with its label.We obtain a group presentation from the labelled diagramfor each regioni 0, 1, 2, . . . , n 1 of .Take one relatorRi Xin crossings of whichas follows. Take one generatorfor each of theis read from the diagram thusadbcXa Xb 1 Xc Xd 1If we choose a base point above the projection plane, and we choose a pointregioni.Then the generatorXipiin the interior of eachcan be described geometrically by a loop in the knot complementwhich passes from the base point, downwards through the regionpithen back up to the base point
NON-PERIPHERAL IDEAL DECOMPOSITIONS OF ALTERNATING KNOTSthrough the pointp05which lies in the unbounded region. Dehn showed thatdefD h X0 , X1 , . . . , Xn 1 R1 , R2 , . . . Rn , X0 iis a presentation for the knot groupread from the diagram .π1 (S 3 \ L).Dehn presentationWe call this theofπ1 (S 3 \ L)In what follows, we use a minor modi cation of the Dehn presentationwhich has better small cancellation properties.Theaugmented Dehn presentation, A , of is the group presentationdefA h X0 , X1 , . . . , Xn 1 R1 , R2 , . . . Rn i .The augmented Dehn presentation arises as a Dehn presentation of a link. Given a labelled linkdiagram ,construct a new labelled link diagram Oby adding a zero-crossing componentO,augmented link diagram. The augmented Dehn presentation of is a presentation for the for the augmented link group π1 (S 3 \ (K O)), i.e., .which boundsThis is called theA π1 (S 3 \ K) Z . D O (1)We will solve the word problem inD by solving it inA .For completeness, let us say afew words about why it is su cient to solve the word problem inA .This is a consequence ofsome standard facts about group presentations that can be found in, for example, [LS77].LetPG h g1 , . . . , gk r1 , . . . rj i and PH h h1 , . . . , hl s1 , . . . sm i be presentations for groups G andH respectively. Then the standard presentation, which we denote by PG PH , for the free productG H isPG PH h g1 , . . . , gk , h1 , . . . , hl r1 , . . . rj , s1 , . . . sm i .A standard consequence of theandPG PHif and only ifas above, ifwnormal form for free products (again see [LS77]) is that with PG , PHg1 , . . . , gk and their inverses, then w G 1problem in D π1 (S 3 \ K) can be solved by theis a word in the generatorsw G H 1. Thus, by (1), theA π1 (S 3 \ K) Z.wordword problem inAn an explicit isomorphism of the augmented Dehn presentation with a standard presentationfor the free productπ1 (S 3 \ K) Zis given byφ : A D hY i(2)where φ : Xi 7 YXi Y 1i 0otherwise.φ corresponds to isotoping the component O of the augmentedsubdiagram so that it bounds a disc in the projection plane.Geometrically,theiflink inS3away fromRemark2.1. Let ι : D D hY i denote the natural inclusion. Given a projection l of a loop π1 (S 3 \ K) in the diagram , we can read o a representative φ 1 (ι(w)) as follows: follow theloop l from its basepoint in the direction of its orientation. When l passes downwards through aregion i of assign a generator Xi ; and whenever l passes upwards through a region i of assign 1a generator Xi . The word thus obtained clearly represents the loop . Thus, w 6 π1 (S 3 \K) 1 if and 1 (ι(w)) 6 only if φA 1.2.2.Square and grid presentations.alternating knot diagram hasThe augmented Dehn presentation of a prime, reduced,small cancellationproperties, as was rst observed by Weinbaum in[Wei71].LetG hX Ri be a symmetrized group presentation. We call a non-emptyR if there exist distinct words s, t R such that s ru and t rv .wordrapiecewithrespect toDe nition 2.2.(a) A symmetrized presentationthe following two small cancellation conditions:hX Ri is called a square presentation if it satis es
6STAVROS GAROUFALIDIS, IAIN MOFFATT, AND DYLAN P. THURSTONCondition C 00 (4).All relators have length four and no de ning relator is a product of fewer thanfour pieces.Condition T (4) .Letr1 , r2andr3be any three de ning relators such that no two of the words areinverses to each other, then one of(b) A symmetrized presentationand in additionXr1 r2 , r2 r3hX Riorr3 r1is called ais freely reduced without cancellation.grid presentation if it is a square presentationis colored by two colors (black or white) and every relator alternates in the twocolors and in taking inverse.Remark.2.3There does not appear to be a standard terminology of the above de nition. In [Wei71],Weinbaum calls square presentationsthe termC 00 (4) T (4) presentations.In [Joh97, Joh00], Johnsgard usesparity to denote the black/white coloring of a grid presentation.In [Wis06, Defn.3.1] and[Wis07, Defn.2.2], Wise uses the terms squared presentations and VH presentations for our squarepresentations and grid presentations.We may depict a relatorrof a grid presentation by a Euclidean square as follows:cab 1 cd 1 dbaIt is easy to see that in a grid presentation the following holds: Relator squares have oriented edges, labelled fromin each relator square. We call a two letter subword of a relator aX.pair.There are two sinks and two sourcesTheC 00 (4)condition says that a pairuniquely determines a relator up to cyclic permutation and inversion. T (4) says that if ab and b 1 c are pairs then ac is not. If a, b and c are letters such that ab and b 1 c are both pairs (with b 6 c), then the word acis called a sister-set. By the T (4) condition, no pair is a sister-set. The edges of a relator square have an additional coloring: they are vertical or horizontal.Moreover, going around a relator square we alternate between black and white. We can invoke a convention that the black and white colorings correspond to horizontal andvertical line placement in our drawings of relator squares. A rotation or re ection of a relator square corresponds to the cyclic permutation or inversionof a relator.We can now state Weinbaum's theorem.Theorem 2.4.The augmented Dehn presentation of a prime, reduced, alternating knotdiagram is a grid presentation.[Wei71]In [LS77] Lyndon and Schupp show that square and grid presentations have have solvable wordand conjugacy problems. Since the appearance of that work, polynomial time algorithms have beengiven for the word (see [Joh97, Sec.7])) and conjugacy problems ([Joh97]) of these groups. We usethese more e cient algorithms here.2.3.The word problem for square presentations.word problem of square presentations.astandard 2-complex KIn this section we recall the solution to theTo any group presentationin the usual way:Kloop representingrK (1) ,we can associateconsists of one 0-cell, one labelled 1-cell for eachgenerator and one 2-cell for each relator, where the 2-cellattached to the 1-skeleton,G hX RiDrr R is Dr with arepresenting the relatorby a continuous map which identi es the boundaryin the 1-skeleton. We impose a piece-wise Euclidean structure on the standard2-complex and set all 1-cells to be of unit length.
NON-PERIPHERAL IDEAL DECOMPOSITIONS OF ALTERNATING KNOTSwA word ,7G if and only if there is a simply connected planar 2-complexφ : (D, D) (K, K (1) ) such that the 0-cells are mapped to 0-cells, open i-cells are(1) . Such aopen i-cells, for i 1, 2 and D is mapped to the loop representing w in Krepresents the identity inand a mapmapped to2-complex, labelled in the natural way, is called aDehn diagram.Throughout this text we use two concepts of labels of edge-paths of the standard 2-complex,peripheral complex (introduced below) or Dehn diagram. The label of an edge-path is the sequenceof letters determined by the edge-path, where travelling along an edge labelleda.letteracontributes theThis is distinct from the word labelling an edge-path, which is the word in the groupa against the orientation contributesa.determined by the path, where travelling along an edge labelledthe lettera 1 ,and travelling with the orientation, the letterA word in a group presentation is said to beall representatives of the same word, i.e.,wgeodesic if it contains the least number of letters overis geodesic if w min{ w0 w G w0 }.A geodesicword represents the identity if and only if it is the empty word. A word in a group presentation isgeodesic if and only if it labels a geodesic edge-path in the standard two complex of the presentation.A key result of small cancellation theory is the followingGeodesic Characterisation Theorem; see[Joh97, Sec.3] and also [Kap97, Lem.3.2].Theorem 2.5. A word in a square presentation is geodesic if and only if it is freely reduced andcontains no subword x1 . . . xn which is part of ax2chainx3:xn 1x1The word x1 . . . xn is called aRemark.2.6xnchain word.Observe that the Geodesic Characterisation Theore
is that these ideal triangulations are non-peripheral for all planar, reduced, alternating projections of hyperbolic knots. Our proof uses the small cancellation properties of the Dehn presentation of alternating knot groups, and an explicit solution to their word and conjugacy problems. In
Polarimetric Decompositions Pauli, the Krogager and the Cameron decompositions. As mentioned above, these decompositions of the scattering matrix can be only employed to characterize coherent scatterers. Therefore, we will finish this section presenting an algorithm to detect such a
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