GEOMETRY OF BIPERIODIC ALTERNATING LINKS

Transcription

1 GEOMETRY OF BIPERIODIC ALTERNATING LINKS ABHIJIT CHAMPANERKAR, ILYA KOFMAN, AND JESSICA S. PURCELL Abstract. A bipriodic altrnating link has an altrnating quotint link in th thicknd torus. In this papr, w focus on smi-rgular links, a class of bipriodic altrnating links whos hyprbolic structur can b immdiatly dtrmind from a corrsponding Euclidan tiling. Consquntly, w dtrmin th xact volums of smi-rgular links. W rlat thir commnsurability and arithmticity to th corrsponding tiling, and assuming a conjctur of Milnor, w show thr xist infinitly many pairwis incommnsurabl smi-rgular links with th sam invariant trac fild. W show that only two smi-rgular links hav totally godsic chckrboard surfacs; ths two links satisfy th Volum Dnsity Conjctur. Finally, w giv conditions implying that many additional bipriodic altrnating links ar hyprbolic and admit a positivly orintd, unimodular gomtric triangulation. W also provid sharp uppr and lowr volum bounds for ths links. 1. Introduction It is wll known, du to work of Mnasco in th 1980s, that thr xists a hyprbolic structur on th complmnt of any prim altrnating link in S 3 that is not a (2, q)-torus link [25]. Th altrnating diagram also provids a natural xplicit dcomposition of th link complmnt into two idal chckrboard polyhdra with facs idntifid [26, 6]. Aitchison and Rvs [7] studid altrnating links for which ths combinatorial polyhdra can b ralizd dirctly as idal hyprbolic polyhdra that can b glud togthr to obtain th complt hyprbolic structur on th link complmnt. Thy calld such links compltly ralizabl, and thy dscribd a larg but finit family of compltly ralizabl altrnating links, calld Archimdan links. Bcaus th gomtry of a compltly ralizabl link is xplicit, and matchs th combinatorics of th polyhdral dcomposition, it is asy to dtrmin various gomtric proprtis from its diagram, such as its xact hyprbolic volum. Howvr, only finitly many such links in S 3 ar known. In this papr w show that many infinit familis of altrnating links in T 2 I ar compltly ralizabl for an analogous notion of a polyhdral dcomposition. Morovr, th gomtric structur of ths links, calld smi-rgular links, can b immdiatly dtrmind from a corrsponding Euclidan tiling. Thus gomtric invariants, such as hyprbolic volum, arithmticity and commnsurability, can b computd dirctly from this tiling. Whn liftd to th univrsal covr of T 2 I, a link L in T 2 I bcoms a bipriodic link L in R 3. Convrsly, any bipriodic altrnating link L is invariant undr translations by a twodimnsional lattic Λ, such that L = L/Λ is an altrnating link in T 2 I. Thus, quivalntly, w prov that infinitly many bipriodic altrnating links ar compltly ralizabl, and w dtrmin thir gomtric proprtis. Not that th hyprbolic structurs of ths compltly ralizabl bipriodic altrnating links wr discovrd indpndntly by Adams, Caldron, and Mayr [3]. In addition, for a larg class of bipriodic links whos quotint admits a crtain kind of altrnating diagram on th torus, w find a hyprbolic structur on th complmnt with a possibly incomplt gomtric triangulation. That is, th complmnt dcomposs into 1

2 2 A. CHAMPANERKAR, I. KOFMAN, AND J. PURCELL ttrahdra that ar all positivly orintd with positiv volum. It is still unknown whthr or not vry link in S 3 admits a gomtric triangulation. Morovr, w prov that this triangulation is unimodular, and w find sharp lowr and uppr bounds for th hyprbolic volum. S Thorm 7.5. Th rsults on complt ralizability hav a numbr of intrsting consquncs: Exact volum and cusp shaps. W comput xact volums for smi-rgular links in trms of thir corrsponding Euclidan tiling. Thir cusp shaps can also b computd from this tiling. S Thorm 3.5 and Corollary Commnsurability and arithmticity. Rcall that two manifolds ar commnsurabl if thy admit a common finit shtd covr. Th trac fild of a hyprbolic manifold H 3 /Γ is th smallst fild containing th tracs of lmnts, and it is known to b a commnsurability invariant of link complmnts. Howvr, familis of links ar known to b pairwis incommnsurabl and yt hav th sam trac fild,.g. [15]. Using th gomtry of smi-rgular links, w show that this phnomnon also holds for such links. Infinitly many of thm hav trac fild Q(i, 3) but ar pairwis incommnsurabl, assuming a conjctur of Milnor on th Lobachvsky function. Convrsly, w also find infinitly many such links that ar commnsurabl to th figur-8 knot complmnt, and on commnsurabl to th Whithad link. S Thorm 4.1. Totally godsic chckrboard surfacs. It is an opn qustion whthr any altrnating knot admits two totally godsic chckrboard surfacs. Only a fw links in S 3 ar known to admit totally godsic chckrboard surfacs, including th Borroman rings [2]. Prtzl knots K(p, p, p) admit on totally godsic chckrboard surfac, but not two [5]. Our gomtric structurs allow us to show that two smi-rgular links hav totally godsic chckrboard surfacs, namly th infinit squar wav and th triaxial link. Howvr, w prov that no othr smi-rgular altrnating links hav this proprty. S Thorm 5.1. Volum Dnsity Conjctur. Th volum dnsity of a link in S 3 or T 2 I is dfind to b th ratio of its volum to crossing numbr. For a bipriodic altrnating link, th volum dnsity is that of its quotint link in T 2 I. In prvious work, th authors showd that if a squnc of knots or links in S 3 convrgs in an appropriat sns (s Dfinition 6.1) to th infinit squar wav, thn thir volum dnsitis approachs th volum dnsity of th infinit squar wav [13]. Th Volum Dnsity Conjctur (Conjctur 6.5) is that th sam rsult holds for any bipriodic altrnating link. In this papr, w prov th Volum Dnsity Conjctur for th triaxial link, which has consquncs for volum and dtrminant, as in [10, 12, 13]. S Thorm 6.7. Elswhr, bipriodic links hav bn calld tiling links [3], txtil links [9] and txtil structurs [28] Organization. In Sction 2, w dscrib th dcomposition of th link complmnt (T 2 I) L. In Sction 3, w prov that smi-rgular links ar compltly ralizabl, and thir gomtric structur and hyprbolic volum is dtrmind by th corrsponding Euclidan tiling. In Sction 4, w rlat th gomtry of th tiling to th commnsurability, arithmticity and invariant trac filds of th corrsponding smi-rgular links. Sctions 5 and 6 focus on two spcial links, namly th squar wav and th triaxial link. In Sction 5, w prov that ths ar th only two smi-rgular links that hav totally godsic chckrboard surfacs. In Sction 6, w prov th Volum Dnsity Conjctur for th triaxial link. Finally, Sction 7

3 GEOMETRY OF BIPERIODIC ALTERNATING LINKS 3 is mor broad. W prov that if any link L in T 2 I admits a crtain kind of altrnating diagram on th torus, not just smi-rgular, thn (T 2 I) L is hyprbolic and admits a positivly orintd, gomtric triangulation. W thn find sharp lowr and uppr bounds for th hyprbolic volum of ths links. Acknowldgmnts. W thank Colin Adams for usful discussions, and not that part (1) of Thorm 3.5 was provd indpndntly in [3]. W thank th organizrs of th workshops Intractions btwn topological rcursion, modularity, quantum invariants and low-dimnsional topology at MATRIX and Low-dimnsional topology and numbr thory at MFO (Obrwolfach), whr part of this work was don. W thank Tom Run, whos figurs at [32] wr vry hlpful for this projct. Th first two authors acknowldg support by th Simons Foundation and PSC-CUNY. Th third author acknowldgs support by th Australian Rsarch Council. 2. Torihdra Lt I = ( 1, 1). Lt L b a link in T 2 I with an altrnating diagram on T 2 {0}, projctd to th 4 valnt graph G(L). First, w liminat a fw simpl cass. If G(L) is containd in a disk in T 2 {0}, thn th link complmnt will b rducibl, with an ssntial sphr nclosing a nighborhood of th disk. A similar argumnt shows that if a complmntary rgion of G(L) is a puncturd torus, th link complmnt is rducibl. Hnc, w may assum that all complmntary rgions in (T 2 {0}) G(L) ar disks or annuli. W will say th diagram is cllular if th complmntary rgions ar disks. W call th rgions th facs of L or of G(L). Similarly, if L is th bipriodic link with quotint link L, thn th facs of L rfr to th complmntary rgions of its diagram in R 2, which ar th rgions R 2 G(L). W will say th diagram of L on T 2 {0} is rducd if four distinct facs mt at vry crossing of G(L) in R 2. Not that a rducd cllular diagram has at last on crossing in T 2 {0}, and an altrnating, rducd, cllular diagram has at last two crossings. Throughout th papr, w will work with diagrams on T 2 {0} that ar altrnating, rducd and cllular. Dfinition 2.1. A torihdron is a con on th torus, i.. T 2 [0, 1]/(T 2 {1}), with a cllular graph G on T 2 {0}. An idal torihdron is a torihdron with th vrtics of G and th vrtx T 2 {1} rmovd. Hnc, an idal torihdron is homomorphic to T 2 [0, 1) with a finit st of points (idal vrtics) rmovd from T 2 {0}. Lt M = (T 2 I) L b th link complmnt. Th chckrboard polyhdral dcomposition of th complmnt of an altrnating link in S 3, as in [26, 6], can b gnralizd to that of an altrnating link in T 2 I. This was don for a singl xampl in [13], and much mor gnrally in [22]. Th following vrsion is appropriat for th links considrd hr: Thorm 2.2. Lt L b a link in T 2 ( 1, 1) with a rducd, cllular, altrnating diagram on T 2 {0}. Thn th link complmnt M = (T 2 I) L admits a dcomposition into two idal torihdra M 1 and M 2 with th following proprtis: (1) M 1 is homomorphic to T 2 [0, 1) and M 2 is homomorphic to T 2 ( 1, 0], ach with finitly many vrtics rmovd from T 2 {0}. Th points T 2 {±1} ar idal vrtics dnotd by ±. (2) T 2 {0} on M 1 and M 2 is labld with th diagram graph G(L), with vrtics rmovd. Thus, th graph has 4-valnt idal vrtics. (3) M is obtaind by gluing M 1 to M 2 along thir facs on T 2 {0}. Facs of G(L) ar chckrboard colord, and glud via a homomorphism that rotats th boundary of ach fac by on dg in a clockwis or anti-clockwis dirction, dpnding on whthr th fac is whit or shadd.

4 4 A. CHAMPANERKAR, I. KOFMAN, AND J. PURCELL dg Figur 1. Crossing dg at a 4 valnt vrtx, and corrsponding copis of on th uppr and lowr torihdra. Th arrows on th lowr diagram indicat th dirction of twisting for gluing facs. (4) Each dg class contains xactly four dgs, and ths dgs ar th crossing arcs of th link complmnt. At ach idal vrtx, a pair of opposit dgs ar idntifid, in both M 1 and M 2, with th opposit pair idntifid at th sam vrtx in M 1 and M 2. S Figur 1. W will rfr to M 1 and M 2 as th uppr and lowr torihdra of L. Whn thr is no ambiguity, th torihdron will b dnotd by P (L). Proof of Thorm 2.2. W procd xactly as in [26], but instad of two 3-balls, w start with a dcomposition of T 2 I into two thicknd tori T 2 [0, 1) and T 2 ( 1, 0]. Th torihdra ar formd by cutting along chckrboard surfacs; an arc of intrsction btwn such surfacs bcoms an idal dg of th dcomposition. Th form of th torihdra nar a crossing is illustratd in Figur 1. Sinc L has an altrnating, rducd and cllular diagram, th 1-sklton of ach torihdron is th sam as G(L), and its vrtics ar idal (rmovd). Th rmaindr of th construction for th cll dcompostion and th fac idntification in [26] is compltly local at vry crossing, and hnc gos through likwis in th toroidal cas. Ths dcompositions can b xtndd to bipriodic altrnating links. Thorm 2.3. Lt L b a bipriodic altrnating link whos quotint link L in T 2 I has a rducd, cllular, altrnating diagram on T 2 {0}. Thn R 3 L admits a dcomposition into two idntical infinit idal polyhdra. Each infinit polyhdron P (L) is homomorphic to R 2 [0, ), with a bipriodic planar graph idntical to G(L) in R 2 0 with vrtics rmovd. Th polyhdra ar glud via a homomorphism dtrmind by th chckrboard coloring of facs of G(L) as in Thorm 2.2. Proof. Lift th torihdral dcomposition of Thorm 2.2 to th covr R 3 L of (T 2 I) L Gomtric torihdra and stllation. W now wish to dtrmin conditions that guarant th torihdra admit a hyprbolic structur. In [22] it was shown that if such a link on T 2 I is wakly prim (s Dfinition 7.1) thn th link complmnt is hyprbolic. Howvr, th torihdra thmslsvs may not admit a hyprbolic structur. For xampl, if

5 GEOMETRY OF BIPERIODIC ALTERNATING LINKS 5 f f f f g f g f dgs = f f g dgs = f = g Figur 2. Top: Collapsing a bigon crats an dg with 3 valnt ndpoints. Bottom: Collapsing a squnc of bigons crats a chain of dgs with 3 valnt nd vrtics. G(L) or, quivalntly, G(L) has bigon facs, thr is no way th torihdra and polyhdra as constructd can admit a hyprbolic structur. Whil gomtric information can b obtaind vn if th torihdra ar not hyprbolic, w will obtain much strongr rsults whn thy ar. Thus w will modify th torihdra by collapsing bigons. Dfinition 2.4. Lt L b a bipriodic altrnating link whos quotint link L in T 2 I has a rducd, cllular, altrnating diagram on T 2 {0}. W dfin th graph T L on T 2 {0} by collapsing all bigons of th diagram graph G(L), as shown in Figur 2. If L has no bigons, thn T L = G(L). By lifting to th covr, w obtain a graph dnotd T L on R 2 {0} R 3 by collapsing all bigons of G(L). Lmma 2.5. Lt L b a link in T 2 I with a rducd, cllular, altrnating diagram on T 2 {0}. Lt M 1 and M 2 b th torihdra in Thorm 2.2. Collaps ach bigon in M 1 and M 2. Thn M = (T 2 I) L has a dcomposition into idal torihdra with th following proprtis: (1) T 2 {0} on M 1 and M 2 contains a graph T L as in Dfinition 2.4 with its vrtics rmovd. Thus, th valncs of idal vrtics ar 4, 3, and 2, dpnding on whthr th corrsponding vrtx of G(L) is adjacnt to 0, 1, or 2 bigons. (2) M is obtaind by gluing facs on M 2 to facs on M 1 by a homomorphism that rotats th boundary of ach fac by a singl rotation in on dg, clockwis or anti-clockwis dpnding on whthr th sam fac in G(L) is whit or shadd. (3) Each dg class corrsponds to a crossing arc as bfor, but crossing arcs associatd to crossings adjacnt across a bigon ar idntifid into a singl dg. Thus if a crossing is adjacnt to no bigons, th corrsponding dg has dgr four. If it is adjacnt to a twist rgion with n bigons, th corrsponding dg has dgr 4 + 2n. By lifting to th covr, a similar dcomposition holds for R 3 L, with th graph T L on R 2 {0} R 3. Proof. Th proof is by tracing th rsult of collapsing a bigon in torihdra M 1 and M 2, similar to th procss for collapsing bigons in th polyhdral dcomposition of an altrnating link in S 3. This is illustratd in Figur 2. W now turn th torihdral dcomposition into a triangulation.

6 6 A. CHAMPANERKAR, I. KOFMAN, AND J. PURCELL vrtical dg horizontal dg Fac of T L Pyramids Bipyramid formd by gluing two pyramids Stllating dg Figur 3. Horizontal, vrtical and stllating dgs of th stllatd bipyramid triangulation. Lmma 2.6. Lt L b a link in T 2 I with a rducd, cllular, altrnating diagram on T 2 {0}. Thn th link complmnt M = (T 2 I) L admits a dcomposition into idal ttrahdra with th following proprtis: (1) Edgs ar labld as horizontal, vrtical, and stllating. Each horizontal dg corrsponds to an dg of T L, and is idntifid to othr horizontal dgs. Each vrtical dg runs from an idal vrtx of M 1 or M 2 to ±. Each stllating dg runs through th cntr of a non-triangular fac in th complmnt of T L, from to +. (2) For n 3, ach n gon rgion in th complmnt of T L on T 2 {0} is dividd into n idal ttrahdra. Dnot th collction of idal ttrahdra of Lmma 2.6 by T. W will call T th stllatd bipyramid triangulation. Proof. Con ach fac of T L on T 2 {0} to ±, rspctivly, obtaining a bipyramid. Edgs of th pyramids that ar incidnt to ± bcom vrtical dgs. Edgs of th pyramids that ar dgs of T L bcom horizontal dgs. S Figur 3. Now, for ach fac of T L, glu th uppr pyramid on that fac to th lowr pyramid according to th chckrboard coloring, as dscribd in Lmma 2.5. This provids a dcomposition of (T 2 I) L into bipyramids on th facs of T L. Obtain an idal triangulation T of (T 2 I) L by stllating th bipyramids into ttrahdra by adding stllating dgs, as shown in Figur 3. By taking th covr of (T 2 I) L, Lmma 2.6 givs a dcomposition of R 3 L into idal ttrahdra. This dcomposition was usd in [13] to triangulat th infinit squar wav. It was xtndd to altrnating links in S 3 in [4] and to links in thicknd highr gnus surfacs in [3]. Our work hr for links in T 2 I was don indpndntly. 3. Smi-rgular altrnating links In this sction, w study a class bipriodic altrnating links whos gomtric structur and hyprbolic volum can b immdiatly dtrmind from a corrsponding Euclidan tiling. For th tilings considrd in this sction, th vrtics can only b 3 valnt or 4 valnt; i.. th links L or L can hav at most on bigon pr twist rgion.

7 GEOMETRY OF BIPERIODIC ALTERNATING LINKS 7 Figur 4. A tiling with isolatd 3 valnt vrtics. Figur modifid from [32]. Dfinition 3.1. A bipriodic altrnating link L is calld smi-rgular if T L is isomorphic, as a plan graph, to a bipriodic dg-to-dg Euclidan tiling with convx rgular polygons. W also say th quotint link L is smi-rgular, and rfr to th quotint tiling of T 2 as T L. Not that a smi-rgular link is rducd and cllular. If th link has no bigons, thn T L is isomorphic to G(L). Sinc thr is at most on bigon in vry twist rgion of L, for vry bigon, both ndpoints of th corrsponding dg in T L ar 3 valnt. Thus, if V 4 (T L ) dnots th subst of 4 valnt vrtics, thn th graph T L V 4 (T L ) admits a prfct matching, dfind as follows. Dfinition 3.2. A prfct matching on a graph is a pairing of adjacnt vrtics that includs vry vrtx xactly onc. Figur 4 shows an xampl of a tiling with isolatd 3 valnt vrtics. Hnc, this tiling cannot b associatd with a smi-rgular bipriodic altrnating link. W now discuss local shaps for Euclidan tilings with 3 and 4 valnt vrtics. Lt T b a bipriodic dg-to-dg Euclidan tiling with convx rgular polygons. Grünbaum and Shphard [20] classifid such tilings according to th pattrn of polygons at ach vrtx. Thy dscribd 21 vrtx typs which can occur in T. If an a gon, b gon, c gon, tc. appar in cyclic ordr around th vrtx, thn its typ is dnotd by a.b.c.... For xampl, th squar tiling has vrtx typ Lmma 3.3. Lt T b any bipriodic dg-to-dg Euclidan tiling with convx rgular polygons, with vrtics of valnc 3 and 4, such that T V 4 (T ) admits a prfct matching. (i) If T has only 4 valnt vrtics, thn th only polygons that can occur in T ar triangls, squars and hxagons, and vry vrtx of T has on of fiv vrtx typs: , , , , or (ii) If T has only 4 valnt vrtics, th numbr of triangls is twic th numbr of hxagons in th fundamntal domain. (iii) If T has 3 valnt vrtics, thn th only polygons that can occur in T ar triangls, squars, hxagons, octagons, and dodcagons. Th allowabl vrtx typs for part (i) ar shown in Figur 5. Proof. Part (i): Among th 21 vrtx typs classifid in [20], thr ar only svn 4 valnt vrtx typs. Ths ar th fiv typs discussd abov, plus two mor typs that includ a dodcagon: and But allowing only ths svn typs of 4 valnt vrtics maks it impossibl to xtnd a tiling from a vrtx of th dodcagon to a tiling of th whol

8 8 A. CHAMPANERKAR, I. KOFMAN, AND J. PURCELL Figur 5. Allowabl vrtx typs for tilings with only 4 valnt vrtics. plan. A straightforward cas-by-cas analysis shows that if a dodcagon is prsnt, thn all th vrtics of th dodcagon cannot b of typ or , thus ruling out a dodcagon among th vrtx typs for a 4 valnt tiling. Part (ii): If t, s, h ar th numbrs of triangls, squars and hxagons, rspctivly, in th fundamntal domain, thn w hav 4v = 2 = 6h + 4s + 3t and f = h + s + t. Now, sinc th fundamntal domain givs a tiling of th torus, v + f = 0 which implis that t = 2h. Part (iii): Again w considr th 21 vrtx typs classifid in [20]. Tn of ths giv a 3-valnt vrtx, and thos of typ 4.8.8, 6.6.6, , and satisfy th conditions of th lmma. W show that th othrs cannot appar. For ach of th othr typs, all but contain an n gon that appars in no othr vrtx typs, namly n = 15, 18, 20, 24, and 42. In ach of ths cass, sinc w allow only 3 and 4-valnt vrtics, ach of th vrtics mting th n gon must b of th sam vrtx typ, namly th uniqu 3-valnt vrtx mting that n gon. But now w stp through ach cas and chck that no tiling of th plan xists with ths vrtics on th n gon, and only 3 and 4-valnt vrtx typs away from th n gon. Finally, onc w hav ruld out th othr typs, th sam argumnt applis for th cas : all vrtics must b of typ , and w cannot complt th tiling. Thorm 3.4. Evry bipriodic dg-to-dg Euclidan tiling T with convx rgular polygons, with vrtics of valnc 3 or 4, such that T V 4 (T ) admits a prfct matching, is th tiling T L for a smi-rgular bipriodic link L. Proof. Givn a tiling, not that if all vrtics of T ar 4 valnt, thn T = G(L) = T L for a smi-rgular link L with no bigons simply by rplacing vrtics with crossings in an altrnating mannr. Othrwis, if th tiling T has 3 valnt vrtics, thn th prfct matching condition implis that th 3 valnt vrtics of T can b covrd by a subst of mutually disjoint dgs. W doubl ach of ths dgs to obtain th projction G(L) of a smi-rgular link L with bigons. Thn th graph T L of Lmma 2.5 is xactly T. Lt v tt and v oct b th hyprbolic volums of th rgular idal ttrahdron and th rgular idal octahdron, rspctivly. Also lt v and v dnot th volum of a hyprbolic idal bipyramid ovr a rgular octagon and rgular dodcagon, rspctivly, with 16 and 24 facs. Th main rsult in this sction is th following thorm. Part (1) was provd indpndntly in [3]. Thorm 3.5. Lt L b any smi-rgular bipriodic link, with altrnating quotint link L in T 2 I, and T L as in Dfinition 2.4. Thn (1) (T 2 I) L has a complt hyprbolic structur coming from a dcomposition into rgular idal bipyramids on th facs of T L.

9 GEOMETRY OF BIPERIODIC ALTERNATING LINKS 9 (2) If T L is 4 valnt, thn th volum of (T 2 I) L, dnotd vol(l), satisfis vol(l) = 10H v tt + S v oct, whr th fundamntal domain of L contains H hxagons and S squars. (3) If T L also has 3-valnt vrtics, thn vol(l) = (6H + 2T ) v tt + S v oct + Ω v 16 + D v 24, whr H, T, S, Ω and D dnot th numbr of hxagons, triangls, squars, octagons and dodcagons, rspctivly, in th fundamntal domain of L. Th proof of th thorm uss angl structurs. Dfinition 3.6. Givn an idal triangulation { i } of a 3 manifold, an angl structur is a choic of thr angls (x i, y i, z i ) (0, π) 3 for ach ttrahdron i, assignd to thr dgs of i mting in a vrtx, such that (1) x i + y i + z i = π; (2) opposit dgs in ach i ar assignd th sam angl; (3) for vry dg in th triangulation, th angl sum about th dg is 2π. Associatd to any angl structur is a volum, obtaind by summing th volums of hyprbolic idal ttrahdra ralizing th spcifid angls. This dfins a volum functional ovr th spac of angl structurs for a givn idal triangulation of a manifold. Th volum functional is convx, and thus has a maximum. If th maximum of th volum functional occurs in th intrior of th spac of angl structurs, thn it follows from work of Casson and Rivin that th angl structurs at th maximum giv th uniqu complt, finit volum hyprbolic structur on th manifold [31] (s also [19]). W will prov Thorm 3.5 by finding th uniqu maximum in th intrior of th spac of angl structurs. W now slightly modify th stllatd bipyramid triangulation T from Lmma 2.6 by prforming a 3-2 mov on vry stllatd bipyramid ovr a trianglular fac. As a rsult, vry triangl of T L corrsponds to two idal ttrahdra, on abov and on blow th triangular fac. Lt T dnot this modifid triangulation. Lmma 3.7. Lt L b a smi-rgular link with no bigons. Thn th triangulation T of (T 2 I) L admits an angl structur with th following proprtis: (1) Th ttrahdra coming from stllatd hxagons ar all rgular idal ttrahdra. (2) Th ttrahdra coming from ach stllatd octahdron glu to form a rgular idal octahdron. (3) Th ttrahdra abov and blow vry triangular fac of T L ar rgular idal ttrahdra. Proof. Sinc L has no bigons, G(L) is isomorphic to T L. Rcall that all ttrahdra of T com from stllating bipyramids ovr all facs of T L, including th triangular facs. W first assign angls to vrtical dgs. Each vrtical dg of th triangulation runs from a vrtx of G(L) to ±. Th vrtx is also a vrtx of T L, a Euclidan tiling. Th four facs adjacnt to this vrtx hav four intrior angls coming from th Euclidan tiling. Thr ar ight ttrahdra coming from th stllation adjacnt to this vrtical dg: two ttrahdra pr fac adjacnt to that vrtx in T L. For ach ttrahdron, assign to its vrtical dg half of th Euclidan intrior angl from T L of th associatd fac. Bcaus T L is a Euclidan tiling, it immdiatly follows that th angl sum around vry vrtical dg is 2π. In addition, w obtain furthr information about adjacnt facs. A rgular Euclidan n gon has intrior angl (n 2)π/n. If th facs adjacnt to a vrtx of T L ar an n 1 gon,

10 10 A. CHAMPANERKAR, I. KOFMAN, AND J. PURCELL n 4 gon θ 4 n 3 gon θ 3 θ 1 n 1 gon θ 2 n 2 gon α 4 α 3 α 1 α 2 (a) (b) Figur 6. (a) Vrtx angls θ i at a 4 valnt vrtx of th tiling T L. (b) Dihdral angls α i = 2π/n i at stllating dgs of T. an n 2 gon, an n 3 gon, and an n 4 gon, with intrior angls θ 1, θ 2, θ 3, and θ 4, rspctivly, thn th sum of th intrior angls at that vrtx satisfis (3.8) 4 θ i = i=1 4 i=1 (n i 2)π n i = 2π = 1 n n n n 4 = 1. S Figur 6 (a). Nxt w assign angls to th stllating dgs. Ths run from to + through th cntr of an n gon fac, and ar adjacnt to n ttrahdra. As th facs of T L ar rgular Euclidan polygons, w assign ach dg mting th stllating dg an angl 2π/n, so th angl sum around stllating dgs is 2π. Bcaus opposit dihdral angls on ach ttrahdron ar qual, this also assigns angls to horizontal dgs: ach has angl 2π/n. Now considr th angl sum ovr all angls assignd to dgs idntifid to a fixd horizontal dg. Thr ar xactly four ttrahdra incidnt to, shown in Figur 6 (b), with on ttrahdron in ach of th four facs adjacnt to a vrtx of T L. Th dihdral angl on th horizontal dg coming from th i-th ttrahdron is α i = 2π/n i. Thus, th angl sum at is 2π + 2π + 2π + 2π = 2π, n 1 n 2 n 3 n 4 whr th quality follows by quation (3.8). Thrfor, th angls assignd to th horizontal, vrtical and stllating dgs mak th angl sum 2π on vry dg of T, which givs an angl structur. Finally, w us this angl structur to gt an angl structur on T. Stllatd bipyramids ovr a triangular fac consist of thr ttrahdra, ach with angl 2π/3 on thir horizontal and stllating dgs, and angl π/6 on vrtical dgs. Rplacing ths thr ttrahdra via a 3-2 mov, th stllating dg is rmovd. At vrtical dgs, two facs glu to on, so th angl bcoms π/3. At horizontal dgs, th angl is halvd, bcoming π/3. This rsults in two idal ttrahdra with all angls π/3, which ar rgular idal ttrahdra. Lmma 3.9. Lt L b a smi-rgular link with at most on bigon pr twist rgion. Thn th triangulation T of (T 2 I) L admits an angl structur with th proprtis of Lmma 3.7, and additionally (4) Ttrahdra coming from octagons glu to form a bipyramid ovr a rgular idal octagon, and thos coming from dodcagons form a bipyramid ovr a rgular idal dodcagon.

11 GEOMETRY OF BIPERIODIC ALTERNATING LINKS 11 n 4 gon θ 4 θ 3 θ 1 n 3 gon dg n 1 gon θ 3 θ 1 θ 2 n 2 gon α 4 α 3 α 3 α 1 α1 α 2 (a) (b) Figur 7. (a) Vrtx angls θ i at a pair of 3 valnt vrtics of th tiling T L. (b) Dihdral angls α i = 2π/n i 1 i 4, at stllating dgs of T. Proof. As in Lmma 3.7, w stllat bipyramids, and assign angls to vrtical dgs of ttrahdra using th vrtx angls of th Euclidan tiling T L. To stllating dgs, hnc also to horizontal dgs, assign angls 2π/n. Again, w nd to chck that th angl sum around ach dg class is 2π. For vrtical and stllating dgs, and for horizontal dgs with no 3-valnt ndpoints, th proof of Lmma 3.7 applis as bfor to giv angl sum 2π as dsird. Considr a horizontal dg with 3-valnt ndpoints. Rcall th ffct of collapsing bigons, as in Lmma 2.5: th horizontal dg will b idntifid to th crossing arcs of a bigon fac, as shown in Figur 2 (top). Thr ar four facs surrounding th two 3-valnt vrtics of : an n 1 gon, an n 2 gon, an n 3 gon, and an n 4 gon, as illustratd in Figur 7 (a). Th vrtx angl of ach rgular Euclidan n i gon is θ i = (n i 2)π n i for 1 i 4. Hnc, th following quations xprss th angl sum around th two ndpoints of dg : (n 1 2)π + (n 2 2)π + (n 3 2)π = 2π n 1 n 2 n 3 = = 1 n 1 n 2 n 3 (n 1 2)π + (n 3 2)π + (n 4 2)π = 2π n 1 n 3 n 4 = = 1 n 1 n 3 n 4 Adding th quations on th right, w obtain: (3.10) = 1 n 1 n 2 n 3 n 4 Now, th horizontal dg associatd to th crossing arc of a bigon is idntifid to six ttrahdra, as shown in Figur 7 (b). Th angl sum at dg is 4π + 2π + 4π + 2π = 2π, n 1 n 2 n 3 n 4 whr th quality follows by Equation (3.10). Thus, this angl assignmnt givs th dsird angl structur. To complt th proof, prform a 3-2 mov on th stllatd bipyramids ovr triangls, as in th proof of Lmma 3.7. As bfor, w obtain two rgular idal ttrahdra for ach triangular fac of T L. Lmma Lt L b a smi-rgular link. For th triangulation T of (T 2 I) L, lt A(T ) b th st of angl structurs on T. Thn th volum functional vol: A(T ) R is maximizd at th angl structur of Lmma 3.7 or Lmma 3.9, dpnding on whthr or not T L has 3-valnt vrtics.

12 12 A. CHAMPANERKAR, I. KOFMAN, AND J. PURCELL Proof. If L has no bigons, thn by Lmma 3.3 (i), th only rgular n gons that can aris in T L hav n = 3, 4, or 6. If L has bigons, thn 3 valnt vrtics appar in pairs, and octagons and dodcagons can also aris. Th ttrahdra coming from rgular triangls and hxagons hav th angls of a rgular idal ttrahdron. Sinc th volum of a ttrahdron is maximizd by th rgular idal ttrahdron, th givn angl structur maximizs th volum of ths ttrahdra. For n = 4, 8, or 12, w claim that th ttrahdra obtaind by stllating th rgular 4, 8, and 12 bipyramids also maximiz volum. Th rgular 4 bipyramid is th rgular idal octahdron, which has volum v oct. If w combin th ttrahdra obtaind by stllating th squar bipyramids, w obtain a dcompostion of (T 2 I) L into rgular ttrahdra and octahdra. In [14, Lmma 3.3], w provd that for any angl structur on an idal octahdron P, th volum of that angl structur satisfis vol(p ) v oct. Th sam argumnt applis as wll for n = 8 and n = 12: by work of Rivin [31], an angl structur will hav volum boundd by th uniqu complt hyprbolic structur on th n bipyramid with that angl assignmnt. But th maximal volum of a complt idal n bipyramid is obtaind uniquly by th rgular n bipyramid (s [1, Thorm 2.1]). Thrfor, whthr or not L has bigons, th angl structur on T maximizs volum. Proof of Thorm 3.5. In Lmmas 3.7 and 3.9, w found an angl structur on an idal triangulation of th complmnt of any smi-rgular bipriodic link, and by Lmma 3.11, that angl structur maximizs volum. Work of Casson and Rivin implis that th gluing of hyprbolic idal ttrahdra that raliz this angl structur givs th complt finit volum structur on (T 2 I) L. This complts th proof of part (1). To prov parts (2) and (3), it rmains to apply Lmma 3.3, dscribing which Euclidan tilings can occur as T L. For part (2), by Lmma 3.3 (ii), thr ar twic th numbr of triangls as hxagons. A bipyramid on a rgular hxagon dcomposs into six rgular idal ttrahdra, which contribut 6v tt to th volum. Sinc vry triangl contributs two rgular idal ttrahdra, it follows that vry hxagon contributs 10v tt to th volum. Thus, if T L has H hxagons and S squars pr fundamntal domain, thn th volum dnsity of L is 10H v tt + S v oct pr fundamntal domain. Similarly for part (3), for ach rgular n gon fac of T L, th contribution to th volum is that of a rgular n bipyramid. This provs th rsult. Corollary Lt L b any smi-rgular bipriodic link, with altrnating quotint link L in T 2 I. Thn th cusps of (T 2 I) L satisfy th following. (1) Cusps corrsponding to T 2 {±1} hav fundamntal domain idntical to a fundamntal domain of th corrsponding Euclidan tiling T L. (2) Cusps corrsponding to componnts of L ar tild by rgular triangls and squars in th cas L has no bigons, and othrwis by rgular triangls and squars, and triangls obtaind by stllating a rgular octagon and a rgular dodcagon. Proof. Th scond itm follows from th fact that (T 2 I) L is obtaind by gluing rgular ttrahdra and octahdra, and th ttrahdra obtaind by stllating bipyramids ovr rgular octagons and dodcagons in th cas of bigons. Th first follows from th fact that th pattrn of such polygons mting th cusps corrsponding to T 2 {±1} coms from th Euclidan tiling.

13 GEOMETRY OF BIPERIODIC ALTERNATING LINKS 13 Figur 8. Figur 4 from [20] showing thr of th infinitly many smi-rgular links with triangls and hxagons Figur 9. Lft: Tiling for on of th infinitly many smi-rgular links commnsurabl with th figur-8 knot. Right: Tiling for L 2, on of a pairwis incommnsurabl family of smi-rgular links L i with th sam invariant trac fild. Figurs modifid from [32]. 4. Commnsurability and arithmticity of smi-rgular links Th proof of Thorm 3.5 nabls us to comput th invariant trac filds of smi-rgular links without bigons. In Thorm 4.1 blow, w rlat th gomtry of th tiling to th commnsurability, arithmticity and invariant trac filds of th corrsponding links. Milnor [27] conjcturd that, xcpt for crtain wll-known rlations, valus of th Lobachvsky function at rational multipls of π ar rationally indpndnt. In particular, its valus at π/3 and π/4 ar conjcturd to b rationally indpndnt. Assuming this, w show that thr xist infinitly many smi-rgular links that ar pairwis incommnsurabl, but hav th sam invariant trac fild. Thorm 4.1. For a smi-rgular link L with no bigons, with altrnating quotint link L, lt M = (T 2 I) L and lt k(m) dnot its invariant trac fild. (1) If th fundamntal domain of T L contains only squars, thn k(m) = Q(i), and M is commnsurabl to th Whithad link complmnt. Hnc, M is arithmtic. In this cas, L is th uniqu smi-rgular link calld th squar wav W blow. (2) If th fundamntal domain of T L contains only triangls and hxagons, thn k(m) = Q(i 3), and M is commnsurabl to th figur-8 knot complmnt. Hnc, M is arithmtic. In this cas, L is on of infinitly many smi-rgular links. Exampls ar shown in Figurs 8 and 9 (lft). (3) If th fundamntal domain of T L contains at last on hxagon and on squar, thn k(m) = Q(i, 3). Hnc M is not arithmtic. Assuming v tt and v oct ar rationally indpndnt, thr ar infinitly many commnsurability classs of smi-rgular links with this invariant trac fild (.g. s Figur 9 (right)).

14 14 A. CHAMPANERKAR, I. KOFMAN, AND J. PURCELL Figur 10. Lft: Four paralllograms which compris strips of th tils and Right: On of th paralllograms fitting on th til Proof. W first comput th invariant trac filds in all th thr cass. Anothr way of looking at th proof of Thorm 3.5 (1) is that th dg gluing quations for T hav solutions with ttrahdral paramtrs iπ/3 and/or iπ/2. Sinc th invariant trac fild for cuspd manifolds is gnratd by th ttrahdral paramtrs [29], th invariant trac filds ar as givn abov. W now addrss th commnsurability of ths links. In cas (1), a dirct computation in Snap [17] vrifis that on such M is arithmtic, namly th quotint of th squar wav. Any link with fundamntal domain consisting only of squars must b commnsurabl to this on. Sinc M is cuspd, has finit volum, is arithmtic, and has th sam invariant trac fild as th Whithad link complmnt, M is commnsurabl to th Whithad link complmnt (s.g. [23, Thorm 8.2.3]). For cas (2), sinc T L contains only triangls and hxagons, it uss tils only of th typ or by Lmma 3.3. As obsrvd in Sction 1 of [20], by stacking horizontal strips mad up of only on typ of til, w can obtain infinitly many bipriodic tilings; xampls ar shown in Figurs 8 and 9 (lft). Morovr, in any bipriodic tiling with just triangls and hxagons, such a horizontal strip always xists, and translation along th strip is on of th dirctions of th bipriodic action. Each strip of th tils or consists of strips of four typs of paralllograms, ach of which consists of half of th hxagon and a triangl, as shown in Figur 10. Ths four paralllograms ar rlatd by rflctions and π rotations. Sinc w ar using only tils of th typ or , w can construct th bipriodic tiling using just ths four paralllograms. Bcaus thy ar all rlatd by rflctions and π rotations, this implis all such tilings ar commnsurabl. A dirct computation in Snap [17] vrifis that on of ths links is arithmtic: namly th complmnt of th quotint of th triaxial link M (s Figur 11). Sinc M is cuspd, has finit volum, is arithmtic, and has th sam invariant trac fild as th figur-8 knot complmnt, M is commnsurabl to th figur-8 knot complmnt [23]. Hnc th commnsurability claim follows for all bipriodic tilings containing only triangls and hxagons. For cas (3), lt L 1 and L 2 b any two smi-rgular links with no bigons, and at last on hxagon and on squar pr fundamntal domain. By Thorm 3.5 (2), vol(l 1 ) = p 1 v tt + q 1 v oct and vol(l 2 ) = p 2 v tt + q 2 v oct for crtain positiv intgrs p 1, q 1, p 2, q 2. If L 1 and L 2 ar commnsurabl, thn thr xist intgrs A and B such that A(p 1 v tt + q 1 v oct ) = B(p 2 v tt + q 2 v oct ) = (A p 1 B p 2 )v tt = (B q 2 A q 1 )v oct.

15 GEOMETRY OF BIPERIODIC ALTERNATING LINKS 15 Figur 11. Lft to right: Triaxial link L with a fundamntal domain (blu squar), trihxagonal tiling T L, and a figur from Gauss 1794 notbook [30]. Figur 12. K (lft and cntr) and minimally twistd 5 chain link (right). Assuming v tt and v oct ar rationally indpndnt, this quation implis that A B = p 2 p 1 and A B = q 2 q 1 = p 1 q 2 q 1 p 2 = 0. By Thorm 3.4, thr xist infinitly many smi-rgular links L i with no bigons, such that for any pair L i1, L i2, p i1 q i2 q i1 p i2 0. Hnc ths links L i ar pairwis incommnsurabl. Figur 9 (right) shows th tiling for L 2, which is part of a family L j with on hxagon and 4j squars pr fundamntal domain. All th L j s hav th sam invariant trac fild. 5. Th triaxial link Th triaxial link L is shown in Figur 11 with its projction, th trihxagonal tiling. It has long bn usd for waving, and appars to hav bn considrd mathmatically by Gauss, who drw it in his 1794 notbook; s [30]. Lt L b th altrnating quotint of th triaxial link in T 2 I. This can b dscribd as a link K in S 3 by drawing a fundamntal domain on a Hgaard torus in S 3, thn adding th Hopf link givn by th cors of th two Hgaard tori, as in Figur 12 (lft). Aftr isotopy, th diagram of K appars as in Figur 12 (cntr). This link complmnt is isomtric to th complmnt of th minimally twistd 5-chain link, shown in Figur 12 (right), although th links ar not isotopic: K is th link L12n2232, and th minimally twistd 5-chain link is L10n113 in th Host-Thistlthwait cnsus of links up to 14 crossings [21]. Among its many intrsting proprtis, S 3 K is conjcturd to b th 5 cuspd manifold with th smallst hyprbolic volum, and most of th hyprbolic manifolds in th cuspd cnsus can b obtaind as its Dhn fillings [24].

16 16 A. CHAMPANERKAR, I. KOFMAN, AND J. PURCELL For a smi-rgular bipriodic altrnating link, Thorm 3.5 provids a dcomposition of th link complmnt into idal hyprbolic torihdra. W say a torihdron is right-angld if it admits a hyprbolic structur in which all dihdral angls on dgs qual π/2. In this sction w prov that only two smi-rgular bipriodic altrnating links admit a dcomposition into right-angld torihdra: th squar wav W studid in [13], and th triaxial link L. Thorm 5.1. Th squar wav W and th triaxial link L ar th only smi-rgular links such that right-angld torihdra giv th complt hyprbolic structur. Thus, th links W and L hav totally godsic chckrboard surfacs. Thorm 5.1 implis that no smi-rgular bipriodic altrnating links hav right-angld torihdra bsids th squar wav and th triaxial link. It is still unknown whthr thr ar bipriodic altrnating links that ar not smi-rgular with a right-angld torihdral dcomposition. Qustion 5.2. Bsids th squar wav W and th triaxial link L, do thr xist any othr right-angld bipriodic altrnating links? Proof of Thorm 5.1. First, w claim that if L has bigons thn th torihdra cannot b right-angld. For if thr is a bigon, Lmma 2.5 implis that thr is an dg of dgr six. If th torihdra wr right-angld, th angl sum on this dg would b (π/2) 6 = 3π, which is impossibl. Thus L has no bigons, and all vrtics of T L ar 4-valnt. By Thorm 3.5, thr is a dcomposition of any smi-rgular altrnating link into rgular idal bipyramids on th facs of T L. In this cas, by Lmma 3.3, th only facs that occur ar triangls, squars, and hxagons. W obtain th torihdral dcomposition from th bipyramid dcomposition of Thorm 3.5 by first splitting ach bipyramid into two pyramids, and thn gluing vrtical facs. Splitting into pyramids cuts in half th angl at th corrsponding horizontal dg. Rcall that a horizontal dg of a rgular idal bipyramid ovr an n gon has angl 2π/n (s th proof of Lmma 3.7). Thus, splitting along a squar givs angl π/4, splitting along a hxagon givs angl π/6. W alrady hav two pyramids ovr triangular facs, which ar rgular idal ttrahdra with angl π/3. Now, whn w glu along vrtical facs, th angl coming from on sid of th shard fac is addd to th angl coming from th othr to giv th nw angl on th horizontal dg. Thr ar six cass: (1) A hxagon is adjacnt to a hxagon. Thn th angl on th adjacnt horizontal dg is π/6 + π/6 = π/3. (2) A hxagon is adjacnt to a triangl. Th angl is π/6 + π/3 = π/2. (3) A hxagon is adjacnt to a squar. Th angl is π/6 + π/4 = 5π/12. (4) A triangl is adjacnt to a triangl. Th angl is π/3 + π/3 = 2π/3. (5) A triangl is adjacnt to a squar. Th angl is π/3 + π/4 = 7π/12. (6) A squar is adjacnt to a squar. Th angl is π/4 + π/4 = π/2. Figur 13 shows th angls on horizontal dgs of th torihdra for all fiv vrtx typs from Lmma 3.3. Not that th only cass that yild right angls ar th dgs whr two adjacnt squars mt and th dgs whr hxagons ar adjacnt to triangls. In ordr for th ntir torihdron to b right-angld, ths ar th only fac adjacncis possibl. Thn th only possibl vrtx typs from Lmma 3.3 ar and Ths smi-rgular links ar xactly th triaxial link and th squar wav, rspctivly.

17 GEOMETRY OF BIPERIODIC ALTERNATING LINKS 17 7π/12 π/2 π/2 π/2 π/2 π/3 π/2 2π/3 π/2 5π/12 π/2 7π/12 5π/12 π/2 Figur 13. Angls on horizontal dgs for th fiv vrtx typs. 6. Proof of th Volum Dnsity Conjctur for th triaxial link In [10, 12, 13], w considrd bipriodic altrnating links as limits of squncs of finit hyprbolic links. W focusd on th asymptotic bhavior of two basic invariants, on gomtric and on diagrammatic, for a hyprbolic link K: Th volum dnsity of K is dfind as vol(k)/c(k), and th dtrminant dnsity of K is dfind as 2π log dt(k)/c(k), whr c(k) dnots crossing numbr. Th volum dnsity is known to b boundd by th volum of th rgular idal octahdron, v oct , and th sam uppr bound is conjcturd for th dtrminant dnsity. In [10], w dfind th following notion of convrgnc of links, and provd that for any squnc of altrnating links K n that convrg to a bipriodic altrnating link L in this sns, th dtrminant dnsitis of K n convrg to a typ of dtrminant dnsity of L. Dfinition 6.1 ([10, 13]). W will say that a squnc of altrnating links K n Følnr F convrgs almost vrywhr to th bipriodic altrnating link L, dnotd by K n L, if th rspctiv projction graphs {G(K n )} and G(L) satisfy th following: Thr ar subgraphs G n G(K n ) such that (i) G n G n+1, and G n = G(L), (ii) lim G n / G n = 0, whr dnots numbr of vrtics, and G n G(L) consists n of th vrtics of G n that shar an dg in G(L) with a vrtx not in G n, (iii) G n G(L) (nλ), whr nλ rprsnts n 2 copis of th Λ-fundamntal domain for th lattic Λ such that L = L/Λ, (iv) lim G n /c(k n ) = 1. n Dfinition 6.2 (Dfinition 4.4 [13]). A diagram has no cycl of tangls if whnvr a disk mbddd in th torus mts th diagram transvrsly in xactly four dgs, thn th disk contains a singl twist rgion; i.. a squnc of bigons or xactly on crossing. As abov, lt W b th infinit squar wav. Using th right-angld hyprbolic structur of W in an ssntial way, in [13] w provd: Thorm 6.3. [13] Lt K n b any altrnating hyprbolic link diagrams with no cycls of F tangls such that K n W. Thn for Kn, th volum and dtrminant dnsitis satisfy: vol(k n ) 2π log dt(k n ) lim = lim = v oct. n c(k n ) n c(k n )

18 18 A. CHAMPANERKAR, I. KOFMAN, AND J. PURCELL Th volum dnsity of a bipriodic altrnating link is dfind as vol((t 2 I) L)/c(L), whr c(l) is th crossing numbr of th rducd altrnating projction of L on th torus, F which is minimal. Hnc, as K n W, th volum dnsitis of Kn convrg to th volum dnsity of W, which is v oct ; s [13]. For dtrminant dnsity, thr is a toroidal invariant of W that appars as th limit of th dtrminant dnsity, namly th Mahlr masur of th two-variabl charactristic polynomial of th toroidal dimr modl on an associatd bipriodic graph, which masurs th ntropy of th dimr modl. In [10], this diagrammatic rsult for W was xtndd to any bipriodic altrnating link L: Thorm 6.4. [10] Lt L b any bipriodic altrnating link, with altrnating quotint link L. Lt p(z, w) b th charactristic polynomial of th associatd toroidal dimr modl. Thn F log dt(k n ) K n L = lim = n c(k n ) m(p(z, w)). c(l) Th following conjcturs ar motivatd by Thorms 6.3 and 6.4: Conjctur 6.5 (Volum Dnsity Conjctur). Lt L b any bipriodic altrnating link, with F altrnating quotint link L. Lt K n b altrnating hyprbolic links such that K n L. Thn vol(k n ) lim = vol((t 2 I) L) n c(k n ) c(l) Conjctur 6.6 (Toroidal Vol-Dt Conjctur). Lt L b any bipriodic altrnating link, with altrnating quotint link L. Lt K n b altrnating hyprbolic links such that K n F L. Thn vol((t 2 I) L) 2π m(p(z, w)). Th Toroidal Vol-Dt Conjctur is studid in mor dtail in forthcoming work [11]. Thorm 6.3 provs both conjcturs with quality for th squar wav W. In Thorm 6.7, w stablish both conjcturs with quality for th triaxial link as wll. By Thorm 5.1, ths ar th only smi-rgular links such that right-angld torihdra giv th complt hyprbolic structur. Thorm 6.7. Lt L b th triaxial link. Lt K n b any altrnating hyprbolic link diagrams F with no cycls of tangls such that K n L. Thn vol(k n ) 2π log dt(k n ) lim = lim = vol((t 2 I) L) n c(k n ) n c(k n ) c(l) Proof. By Thorm 3.5, By [10, Exampl 4.2], vol((t 2 I) L) c(l) = 10 v tt. 3 2π log dt(k n ) 2π m(p(z, w)) lim = n c(k n ) c(l) = 10 v tt. 3 It rmains to prov Conjctur 6.5 for th triaxial link L. Namly, F vol(k n ) K n L = lim n c(k n ) = vol((t 2 I) L). c(l) = 10 v tt. 3 W adapt th proof of [13, Thorm 1.4], which provs Conjctur 6.5 for W.

19 GEOMETRY OF BIPERIODIC ALTERNATING LINKS 19 Figur 14. Circl pattrn for P (L) for th triaxial link L. By Thorm 5.1, th chckrboard surfacs of L ar totally godsic, with th facs of th torihdra lifting to totally godsic hyprplans in th univrsal covr H 3 of R 3 L, mting at right angls. This forcs ths hyprplans to mt H 3 = Ĉ in th circl pattrn shown in Figur 14. Now procd as in [13]. If K n S 3 is a squnc of altrnating hyprbolic links such that F K n L, thn by [13, Thorm 4.13] th volums of Kn ar boundd blow by twic th volum of th polyhdron obtaind from th chckrboard polyhdron of K n by assigning right angls to all th dgs. Dnot this by P n. Now w rpat th proof of [13, Lmma 5.3], rplacing th disk pattrn coming from W with that of Figur 14 coming from L. In that proof, ach instanc of v oct (th volum dnsity of W) must b rplacd by (10v tt /3), th volum dnsity of L. Finally, stp through th proof of Thorm 1.4 in [13, sction 5]. Again, first chck that th hypothss ar satisfid for th lmma analogous to [13, Lmma 5.3] (with v oct rplacd by 10v tt /3). Not for condition (2), th argumnt is th sam but th constant 4 is rplacd by 6 (this dos not affct th limit). For (1), B(x, l) = 6l. Othrwis, th argumnt is idntical. Thn this lmma and [13, Thorm 4.13] imply vol(k n ) lim n c(k n ) = 10v tt Hyprbolicity for altrnating links in T 2 I In this sction, w gnraliz th rsults on hyprbolicity and triangulations to widr classs of links on T 2 I. W prov that if a link L in T 2 I admits a crtain kind of altrnating diagram on th torus, thn (T 2 I) L is hyprbolic. Morovr, for a (possibly incomplt) hyprbolic structur on th complmnt, w find a triangulation that satisfis Thurston s gluing quations. W show that this triangulation is positivly orintd and unimodular. This lads to lowr and uppr bounds on th hyprbolic volum of (T 2 I) L, dnotd by vol(l). To dscrib th links, w nd a fw dfinitions. First, th dfinition of connctd sums of knots in S 3 can b xtndd to knots in T 2 I. W ar concrnd with knots with rducd diagrams that ar not connctd sums, as in th following dfinition. Dfinition 7.1. A diagram is wakly prim if whnvr a disk mbddd in th diagram surfac mts th diagram transvrsly in xactly two dgs, thn th disk contains a simpl dg of th diagram and no crossings. In addition, w will also nd Dfinition 6.2 and th following:

20 20 A. CHAMPANERKAR, I. KOFMAN, AND J. PURCELL Dfinition 7.2. A godsic idal triangulation satisfying th dg gluing quations is calld unimodular if vry ttrahdron has an dg paramtr z that satisfis z = 1. Dfinition 7.3. Lt B n dnot th hyprbolic rgular idal bipyramid whos link polygons at th two coning vrtics ar rgular n gons. For a fac f of G(L), lt f dnot th dgr of th fac. Lt L b a link in T 2 I with an altrnating diagram on T 2 {0}, and lt T L b th toroidal graph dfind in Lmma 2.5. Dfin th bipyramid volum of L as vol (L) = vol(b f ). f {facs of T L } Not that for all smi-rgular links, Thorm 3.5 implis vol(l) = vol (L). Dfinition 7.4. Lt L b a link in T 2 I with an altrnating diagram on T 2 {0}, and lt P (L) b th torihdron as in Thorm 2.2. Assum P (L) admits an idal right-angld hyprbolic structur. Dfin th right-angld volum of L as vol (L) = 2vol(P (L)). Not that for th squar wav and triaxial link, Thorm 5.1 implis vol(l) = vol (L). Suppos K is a link in S 3 with a prim, altrnating, twist-rducd diagram with no cycl of tangls, and with bigons rmovd. In [13, Thorm 4.13], w provd that th two chckrboard polyhdra coming from K, whn givn an idal hyprbolic structur with all right angls, hav volum providing a lowr bound for vol(s 3 K). In Thorm 7.5 blow, w xtnd [13, Thorm 4.13] to any link in T 2 I satisfying similar conditions. Th mthods involvd in proving Thorm 7.5 ar diffrnt from thos usd in [13], which rlid on volum bounds via guts of 3 manifolds cut along ssntial surfacs. Hr, th proof of Thorm 7.5 involvs circl packings, gomtric structurs on triangulations, and th convxity of volum. Thorm 7.5. Lt L b a link in T 2 I with a wakly prim altrnating diagram on T 2 {0} with no bigons. If L has no cycl of tangls, thn (1) (T 2 I) L is hyprbolic. (2) (T 2 I) L admits a (possibly incomplt) hyprbolic structur obtaind from an idal, positivly orintd, unimodular triangulation. (3) Undr th structur of itm (2), th torihdron P (L) is right-angld. (4) vol (L) vol(l) vol (L). Not that by Thorms 3.5 and 5.1, both inqualitis in (4) bcom qualitis for th squar wav and th triaxial link: all thr giv th volum of th complt structur. Thus both uppr and lowr bounds of Thorm 7.5, (4) ar sharp. In gnral, th bounds in (4) ar volums of incomplt idal hyprbolic structurs on (T 2 I) L. Rmark 7.6. Thorm 7.5 should b compard to th rsults of [22]. That papr also implis that th links in Thorm 7.5 ar hyprbolic, and givs a lowr bound on volum in trms of th numbr of twist rgions of th diagram. Sinc th diagram hr has no bigons, this also amounts to a lowr volum bound in trms of th crossing numbr. Howvr, th rsults of Thorm 7.5 ar strongr in this cas bcaus thy giv an xplicit hyprbolic structur, although that structur is likly incomplt. Morovr, th volum bound of Thorm 7.5 is known to b sharp for th squar wav and triaxial link. Th proof of Thorm 7.5 will procd in th ordr (3), (2), (1), (4). Th proof rlis on a fundamntal rsult about th xistnc of crtain circl pattrns on th torus, du to Bobnko and Springborn [8]. Th following spcial cas of [8, Thorm 4] applis.