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Hyperbolic volumes of the prime knots with at most ten crossings
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Numbers
$n$
$k$ 
$\mathrm{Vol}(S^3\setminus n_k)$
4
1:
2.029883212819307250042405108549040571883378615060599584034978213553194952516488044272940708456513390
comment: $4_1$, the figure-eight knot; the complement is m004 in the SnapPy census; alternating; $\mathrm{Vol}=2\,\mathrm{Cl}_2(\pi/3)$, the smallest volume of any hyperbolic knot complement
5
2:
2.828122088330783162763898809276634942770981317300649477043520327258802548322471630936947017929999107
comment: $5_2$, the three-twist knot; the complement is m015 in the SnapPy census; alternating; the volume agrees to the hundred digits listed with that of $12n_{242}$, the $(-2,3,7)$ pretzel knot
6
1:
3.163963228883143983991014715973154484812787671518115266582922441975565984963084839871725288561503347
comment: $6_1$, the stevedore knot; the complement is m032 in the SnapPy census; alternating
6
2:
4.400832516123046101441203865640928033983485418516465458521752513316600130972405316677122430240181317
comment: $6_2$, the Miller Institute knot; the complement is m289 in the SnapPy census; alternating; the volume agrees to the hundred digits listed with that of $12n_{121}$
6
3:
5.693021091281300765112483277481222926944301733006880037850870699995476072590906707654919542407040036
comment: $6_3$; the complement is s912 in the SnapPy census; alternating; the volume agrees to the hundred digits listed with that of $13n_{469}$
7
2:
3.331744231641114823914569108029712795546957909186004921221604455598741372842366515578862260348786284
comment: $7_2$; the complement is m053 in the SnapPy census; alternating
7
3:
4.592125697027062550196613994655473732354360326543270894426065765041363091864797025835868422067745209
comment: $7_3$; the complement is m340 in the SnapPy census; alternating
7
4:
5.137941201873417769841348339474845035649675092735712368597748060678580307623518886258443309869104970
comment: $7_4$, the endless knot; the complement is s648 in the SnapPy census; alternating
7
5:
6.443537380850573086143321716622058073873217368226322936963187980975322147558760438607110217714575595
comment: $7_5$; the complement is v3310 in the SnapPy census; alternating; the volume agrees to the hundred digits listed with that of $13n_{1153}$
7
6:
7.084925953510830364728651465842946535803269264618714410392357288452730211130714737471818401211189053
comment: $7_6$; the complement is t11291 in the SnapPy census; alternating
7
7:
7.643375172359955478221844448580771033642604991008608123921250359344872204900380176228775492582138714
comment: $7_7$; the complement is t12656 in the SnapPy census; alternating
8
1:
3.427205246274016219863539590053306030705839370723805347262267027006937570353148111703558168958342219
comment: $8_1$; the complement is m074 in the SnapPy census; alternating
8
2:
4.935242678280654357478067773841835986623623464823828986838376436713664618346604989388728944914177257
comment: $8_2$; the complement is s526 in the SnapPy census; alternating
8
3:
5.238684100798440155486473106366160923422533603912256567018583810872206426446036859555182233757580494
comment: $8_3$; the complement is s726 in the SnapPy census; alternating
8
4:
5.500486416347234559667725961124885523333947309365248022725842614407693324878554441656863189461001225
comment: $8_4$; the complement is s862 in the SnapPy census; alternating
8
5:
6.997189147792215478800577879145517688963840955647927083008781932873301739057752542646401798934554368
comment: $8_5$; the complement is t10932 in the SnapPy census; alternating
8
6:
7.475237429505242926190434594386674377390157910400196404167236590164642135147971799748567371426779187
comment: $8_6$; the complement is t12395 in the SnapPy census; alternating
8
7:
7.022196589095253263562473475305510109046101298463175818773042137984072003788309206031572984116399675
comment: $8_7$; the complement is t11034 in the SnapPy census; alternating
8
8:
7.801341224440062635745815173493825487317723290807983528499373926607454371684272866382676003979695728
comment: $8_8$; the complement is o9_37770 in the SnapPy census; alternating
8
9:
7.588180223641626717384987930579773944099813247303010420619739276712605630010356996411034983817094835
comment: $8_9$; the complement is t12587 in the SnapPy census; alternating
8
10:
8.651148558017081966171120266314370242184366775604459231026672122794048052391689077966097072258348937
comment: $8_{10}$; the complement is o9_43874 in the SnapPy census; alternating
8
11:
8.286316817806592588126824797751167162948744245411117139762993515683703595296757393453019207255602108
comment: $8_{11}$; the complement is o9_42258 in the SnapPy census; alternating
8
12:
8.935856927486689169342801339489742003701343394062741697940293108265540837345058667570108313934128204
comment: $8_{12}$; alternating
8
13:
8.531232201460415724515566872038674069840850540315561291533303578030894910094808238890951396601383627
comment: $8_{13}$; the complement is o9_43592 in the SnapPy census; alternating
8
14:
9.217800316021928668826315392572382719713716730305618878551082624220790606432264347897962632103602439
comment: $8_{14}$; alternating
8
15:
9.930648293796182981002132155583115280481932549940267397025098490557586318306082669148367939793254813
comment: $8_{15}$; alternating
8
16:
10.57902191689927032580868455008867289850488527959708081750186108063491054630380415328384963860879257
comment: $8_{16}$; alternating
8
17:
10.98590760628482030368843975918306940395979412778451352137193797193860955529037961343290352379839065
comment: $8_{17}$; alternating
8
18:
12.35090620915820017473630443842615201419925670411999628061247880954755424482189928349282608428662587
comment: $8_{18}$, the Carrick mat; alternating; the volume agrees to the hundred digits listed with that of $12n_{276}$
8
20:
4.124903251807676155505899848869378162929477646664116556055337236197309623828308031039561706007263026
comment: $8_{20}$; the complement is m222 in the SnapPy census; non-alternating; the volume agrees to the hundred digits listed with that of $11n_{38}$
8
21:
6.783713519835126515708813942086034048831469456592638291597331406498241570022911079977087577633297417
comment: $8_{21}$; the complement is v3505 in the SnapPy census; non-alternating
9
2:
3.486660146295043589800612891536700319842118666423966061501787018213007923890021024805084591950091120
comment: $9_2$; the complement is m094 in the SnapPy census; alternating
9
3:
4.994856404125713723884371557220584682996719984574057832572370502395623456599176748911621294414524803
comment: $9_3$; the complement is s558 in the SnapPy census; alternating
9
4:
5.556518816346559286667047653914881572932237070135089084742897149435118315706721378369908694594948423
comment: $9_4$; the complement is s870 in the SnapPy census; alternating
9
5:
5.698441750940057506475340307000851186065219396546904900550887978475028767717083634707426649708661207
comment: $9_5$; the complement is v2284 in the SnapPy census; alternating
9
6:
7.203600761623164963763030817599115460286508916778425118893622800926303703797962838214808660569651006
comment: $9_6$; the complement is t11675 in the SnapPy census; alternating
9
7:
8.014861457829953897612056440674773172391235023594149057179364674298901971062601944619520587123193151
comment: $9_7$; the complement is o9_40064 in the SnapPy census; alternating
9
8:
8.192347962434355610506211273193284921428262794663113006216460288467221308414227153973987735561531123
comment: $9_8$; the complement is o9_41611 in the SnapPy census; alternating
9
9:
8.016815565678015928433319760944587413158151099220670178389567151012873778308280600818263229960511989
comment: $9_9$; the complement is o9_40076 in the SnapPy census; alternating
9
10:
8.773457282055681093380645643263778793967523222762466426057483664385918737540551331819826900002999083
comment: $9_{10}$; the complement is o9_44057 in the SnapPy census; alternating
9
11:
8.288589042858491027206866435559555446363042030373466030494426274883863465670654026104626612843608982
comment: $9_{11}$; the complement is o9_42277 in the SnapPy census; alternating
9
12:
8.836642343885841864901929245036422308889245726246846192725402089687740427175329252103771209929410767
comment: $9_{12}$; alternating
9
13:
9.135094037937276623780862329774183905035919673370930314502697581786978774579574713768655715710277121
comment: $9_{13}$; alternating
9
14:
8.954989262002549026653522886700382765848482459353780760409923520775619365908058295251513449160463685
comment: $9_{14}$; alternating
9
15:
9.885498660334439236200342895963562730220069444550909147401790032396141553971003211778528850165134057
comment: $9_{15}$; alternating
9
16:
9.883006960018826840837164201107837275263335478902569864113023201531377212841241009328770602345200697
comment: $9_{16}$; alternating
9
17:
9.474580453499224038101853829811005947373958203365565697318768651178284975488172246591384321612036992
comment: $9_{17}$; alternating
9
18:
10.05772963558100035154051203733712785468284252289186371407335730734099811595698167894331365523463303
comment: $9_{18}$; alternating
9
19:
10.03254744783669672127003669315336888518484414702447992164942678440040867135818359436537881132881390
comment: $9_{19}$; alternating
9
20:
9.644304073800876129193872430793992400779848595043427152708951459592885407619010126663116484887757914
comment: $9_{20}$; alternating
9
21:
10.18326553568108705480629469633108238996808694473326878840634286672992925727238002174115151432893399
comment: $9_{21}$; alternating
9
22:
10.62072702123947284583849629980525124567915069613648517772783372362272522041313111329669434395570336
comment: $9_{22}$; alternating
9
23:
10.61134829405250931492112757555730619048621840619318659745539789121655665991891485218424739505336283
comment: $9_{23}$; alternating
9
24:
10.83372910894525973596949080435891568017906965633399460952514840103051602200505919055825226836806637
comment: $9_{24}$; alternating
9
25:
11.39030514777365466287618180417180083186753564033560738055005893603197459355712160867716687803264469
comment: $9_{25}$; alternating
9
26:
10.59584051499937158843300795984586106143088580150982932153256502883379591871292072811861586619728027
comment: $9_{26}$; alternating
9
27:
10.99998095828711975926842263077699358923018898248028381466697627887296281682777758950878414389084260
comment: $9_{27}$; alternating
9
28:
11.56317701626512347086099735656757511959927094897728436311943864091477505975249035284192502927107223
comment: $9_{28}$; alternating
9
29:
12.20585616511471979816417040355329404969126040141113228084116259767250534355685842120582886566872438
comment: $9_{29}$; alternating
9
30:
11.95452696823203372437771965436219011941310722962672952396921064688343341668386078267926902810439167
comment: $9_{30}$; alternating
9
31:
11.68631220787810518200412709691748289072132338277264350096776684495860292776499267048078122702294917
comment: $9_{31}$; alternating
9
32:
13.09989984589252689342245135549631402433260588216766879110371326495918232862154256225604509175147217
comment: $9_{32}$; alternating
9
33:
13.28045563625478793997374382357100210091620058737852360164305869396071962116647451098425311183372730
comment: $9_{33}$; alternating
9
34:
14.34458138778896259381763305535457637398038905143423288899698453230439216544636891058042764624126340
comment: $9_{34}$; alternating
9
35:
7.940579247781310788020939116417835696058867861587489743881140177077561150911538202044332258516659057
comment: $9_{35}$; the complement is o9_39339 in the SnapPy census; alternating
9
36:
9.884578653127015299080035856783255500340247711905342396313504867565293589268443121110551675955251923
comment: $9_{36}$; alternating
9
37:
10.98944959257654075898366956140664375829291627013551046609596876346389691846167893966531855050549583
comment: $9_{37}$; alternating
9
38:
12.93285870225282765054430309681514524827064715134238024824947833663060995314661727589972931765817680
comment: $9_{38}$; alternating
9
39:
12.81031000331111537379569007733684751696168114546506361123501878506953377125203397165932307587291121
comment: $9_{39}$; alternating
9
40:
15.01834285786511784337632386362403541959937861865467743544649587570391502899367871358111179544817081
comment: $9_{40}$; alternating
9
41:
12.09893602599078738356455696387624160295557377848341191207713147082541960996893625200183011481533545
comment: $9_{41}$; alternating
9
42:
4.056860224236820144181924184017659447160664377093809560226560015443973033695891463348932385667550031
comment: $9_{42}$; the complement is m199 in the SnapPy census; non-alternating; the volume agrees to the hundred digits listed with that of $10_{132}$
9
43:
5.904085858508103343399291567743601164111449093592832770722914563550225063785513115220930351775731906
comment: $9_{43}$; the complement is v2623 in the SnapPy census; non-alternating; the volume agrees to the hundred digits listed with that of $12n_{243}$
9
44:
7.406767572367503230450694635253254663268340770408296705786041956403800617046027155017112835470358424
comment: $9_{44}$; the complement is t12271 in the SnapPy census; non-alternating
9
45:
8.602031166401500513187133737531962565549767481110625641947737945500080131861777027549736056592294886
comment: $9_{45}$; the complement is o9_43771 in the SnapPy census; non-alternating
9
46:
4.751701965517899516974944380705063021022097529240014292878498556746809005463622163435573924694403983
comment: $9_{46}$; the complement is m372 in the SnapPy census; non-alternating
9
47:
10.04995786127436987664325808516065451432923952691131355592452250381132061663657587830246616276612479
comment: $9_{47}$; non-alternating
9
48:
9.531879835800977929061482874843656213289902660898744630842645988987809354349090398790009950139637899
comment: $9_{48}$; non-alternating
9
49:
9.427073627769277209212996030922116475903271057668831590145067757529341827741572103123156726433330358
comment: $9_{49}$; non-alternating
10
1:
3.526195990735375717449975565714271633063532767933819127806321364157317348886160031294411339813906516
comment: $10_1$; the complement is s016 in the SnapPy census; alternating
10
2:
5.114841460302400721438106628468470523914735245629326811523719108779649554034504844499930511307337653
comment: $10_2$; the complement is v1217 in the SnapPy census; alternating
10
3:
5.732104786782051402516945130177559742109586626308535181487068856083600881297637716706907424182538628
comment: $10_3$; the complement is v2362 in the SnapPy census; alternating
10
4:
5.817129692825428970294282856281853700890621758403352248199628845418509238317142010395439176464646613
comment: $10_4$; the complement is v2488 in the SnapPy census; alternating
10
5:
7.373935134274567903172552946198823272684579010974206473983182899140132823539406375447246043951840642
comment: $10_5$; the complement is o9_32208 in the SnapPy census; alternating
10
6:
8.390937606724453661780200585682171476961038450782098923692611040667806986133248692651760682131082490
comment: $10_6$; the complement is o9_42963 in the SnapPy census; alternating
10
7:
9.115906395629990533260304661908353753969464772481909234526713634725519506353922995724187363970014779
comment: $10_7$; alternating
10
8:
6.083234837064976885744176250038135961094007519171639990711173662174830970951796980747201026761790000
comment: $10_8$; the complement is v2858 in the SnapPy census; alternating
10
9:
8.294099675229288513531627364159880187917501921983815098541205346359035889088065644455795128143828594
comment: $10_9$; the complement is o9_42320 in the SnapPy census; alternating
10
10:
9.180573644138947661124025266005455954328527976646533987953887234120935537795271790495806742574382684
comment: $10_{10}$; alternating
10
11:
9.370442441399364576419231928680427901405766836521190701659850260307239320927635641775632068723167684
comment: $10_{11}$; alternating
10
12:
9.817495190695845949672230938133235775828873777758044460750913858907071126234831285058671097996328162
comment: $10_{12}$; alternating
10
13:
10.57848019046471844182992445892523393853859031129301211138339272825505108459351303791275667186171436
comment: $10_{13}$; alternating
10
14:
10.93768941441755376423776512842977875718912382737187031247767247118967551209890337511623795431801795
comment: $10_{14}$; alternating
10
15:
8.973449326001361128847540260090232402252543670038245976756524605642970918739740616664341552321975863
comment: $10_{15}$; alternating
10
16:
9.546642305028445763040152864481971626975375493188920196013638172604390874471209186005939950133261682
comment: $10_{16}$; alternating
10
17:
8.536755599193234612901770434618599262454960647390983453984975635885774194350205740950434384809506513
comment: $10_{17}$; alternating
10
18:
10.63984427125763276011148711454740386216694906060603964648982902473225289889833347031802971383195621
comment: $10_{18}$; alternating
10
19:
9.844771302667735179370159569287583040684566703052749667123703745754547546367244901614388005008463770
comment: $10_{19}$; alternating
10
20:
8.317378718415006765366428468866786808707484575778927192880563309246697554503012036545395026299519101
comment: $10_{20}$; the complement is o9_42467 in the SnapPy census; alternating
10
21:
9.675141144155305623994598519701651667930802331996283934503043436352545908626524935479426096312426484
comment: $10_{21}$; alternating
10
22:
9.981866517896332269796645772745178114703604123836556561962498139864421561010864390610554512286373488
comment: $10_{22}$; alternating
10
23:
11.39322463488506089190149706283395463581940355541070680271339742446563526911754466591996337919731814
comment: $10_{23}$; alternating
10
24:
10.97745473682175564981979319609961751408365499726041102004549229819658582428180425271203291693407654
comment: $10_{24}$; alternating
10
25:
11.87577960365207953409865472134765255019630366476709824202514252274447780250403553949404632908873523
comment: $10_{25}$; alternating
10
26:
11.35201755798783659188545022630704516064811592373264598021537138923672403502618630050297093392204880
comment: $10_{26}$; alternating
10
27:
12.38413196612856967046055167778900639303602924913391035647845619702438420285747854152740965412113870
comment: $10_{27}$; alternating
10
28:
10.26467494650644454577027339966245559312739710416979518647780266065116360153738553307728972635242994
comment: $10_{28}$; alternating
10
29:
11.60290524666798164739678172082450230200807073057229793629135643083505727838365165320680453147697066
comment: $10_{29}$; alternating
10
30:
11.82875756950365124376800613506841234659401518477658254689923584789850853712297287893048019110748789
comment: $10_{30}$; alternating
10
31:
11.04426403104338553336980865678698014492716237119899955092580555418382528836291272453467250365955839
comment: $10_{31}$; alternating
10
32:
12.09093687193191809716378288672043161513907487668289150749755531327424895499351473329505482184626463
comment: $10_{32}$; alternating
10
33:
11.53567393228233262155266978946846054673257152692913709591527891683307534117082426616074933003186128
comment: $10_{33}$; alternating
10
34:
8.422266667868570266183123407796570454094923931408609450049690392133058883972192853593943309278838933
comment: $10_{34}$; alternating
10
35:
10.39449678560532192742499330622210286128130524674647418308318026705884740250119515336041254226608712
comment: $10_{35}$; alternating
10
36:
10.47618554030524176132672512698724512276324046979202752257976706246826102559606545146112980456131865
comment: $10_{36}$; alternating
10
37:
10.96581053242464396508202520198396473440253942927989432648562382280370443447312624299946471038702710
comment: $10_{37}$; alternating
10
38:
11.34931354723682674024948344095535343472225349194193589253705964790594044101484751800510031048179778
comment: $10_{38}$; alternating
10
39:
11.58951872736846304264766544342348567427758336753062291743947607985285668523665100993802804036565827
comment: $10_{39}$; alternating
10
40:
12.88874033027648608014252820973658982129232356855146846058843081483972005356644798032207200971560582
comment: $10_{40}$; alternating
10
41:
12.37661549863504100524891793812042884382143575927193176801541940545256019828470845732570649446910034
comment: $10_{41}$; alternating
10
42:
13.23984836679564762114472509892880523016677426485132735021515207736781938549614970079946685427483779
comment: $10_{42}$; alternating
10
43:
12.60259611426207676475592569493654139847721466128695489341552164383665225712803531327072465797028474
comment: $10_{43}$; alternating
10
44:
12.96899420503249995576009051591499926351847746946927979808125434551622733969295960781511488828781048
comment: $10_{44}$; alternating
10
45:
13.71607598475272999898568237483179998632492681722606199885411760592197437464552866951190141413500352
comment: $10_{45}$; alternating
10
46:
7.716999809324509682365664832843835279178643978819574987944506600578980337439621525209719550956323785
comment: $10_{46}$; the complement is o9_36697 in the SnapPy census; alternating
10
47:
9.385194156727692223146595182360097446935385914803642402659313693054286273195531694628681608912626903
comment: $10_{47}$; alternating
10
48:
10.37890144901783195882934554147416836323267733808984350731708659751747592119332837499438103578734441
comment: $10_{48}$; alternating
10
49:
11.45319336621992796211370392421014383567064463679628078154600029312602736285908646076332407098959920
comment: $10_{49}$; alternating
10
50:
11.19888979208256562184849426747661009492838519966422213004726119017601806163722752682152976441641455
comment: $10_{50}$; alternating
10
51:
12.63137951994410936349701983145630455871615096418859023274090270094911514554703244116591943526739868
comment: $10_{51}$; alternating
10
52:
11.53754818249897204651925884942957411247208857133952535820505950128279740874795403402075686137181622
comment: $10_{52}$; alternating
10
53:
12.88684742018346427562821266494009817659207660520002542954933202017121506971514125716035392026471958
comment: $10_{53}$; alternating
10
54:
10.59131113363908975260847652507768125248629602983193108872398025115196012443520894251295736916416614
comment: $10_{54}$; alternating
10
55:
12.18554062703973553039459766179439407744541622264998938782152343396061455823662954225624501314196321
comment: $10_{55}$; alternating
10
56:
12.39880215580871573157268482591476600018303655252601339473892502778970727535698133546571319237266298
comment: $10_{56}$; alternating
10
57:
13.58855850222604919932297481863861088103012098721379083578883165295935990819086356662729607327444483
comment: $10_{57}$; alternating
10
58:
12.72133069853759993150286807513988373513471086021039282996379221828992957373077978881076223515305181
comment: $10_{58}$; alternating
10
59:
13.38994414143272117632219458004732152490774986301358444356232500587514263613608944447271162627904472
comment: $10_{59}$; alternating
10
60:
13.98004153570804666967087284852772441562663473661040761858072066295319719202751218996936601470485758
comment: $10_{60}$; alternating
10
61:
8.458580267591309133489405511575629668355192794412479110348510998493172431108359007802879012545148598
comment: $10_{61}$; alternating
10
62:
10.14146903630351166670886471530944135234089539983793380100657496449193635949427747175681149226922938
comment: $10_{62}$; alternating
10
63:
11.51169129183895222385314362306082520242282809250262718788177942501761951879353084669206500352133960
comment: $10_{63}$; alternating
10
64:
10.86809271305099405350864792778532982086276930337628618692278089909027189275016887973240748125433850
comment: $10_{64}$; alternating
10
65:
12.07646143448946181682531155369893428366909576956406874131784160726848132106382779217412400459800623
comment: $10_{65}$; alternating
10
66:
13.02926887083449583949450709610272134475387763069937086098848628726102761307006513917686081967881012
comment: $10_{66}$; alternating
10
67:
12.42163050542115179808146167971465233239580089199159055342451933295366608029259312687880648946703910
comment: $10_{67}$; alternating
10
68:
11.63703522930269749746392357236850323856894936452558322132165836955421482341428610438822305883320376
comment: $10_{68}$; alternating
10
69:
14.12650517067014817645749416311195448904347969019611466774798088254754801870257263578538118419125004
comment: $10_{69}$; alternating
10
70:
12.51088773042080247352261566661876842167176596562612277242765329632489893615398929735194995155922644
comment: $10_{70}$; alternating
10
71:
13.38522874603995483277585991759952666580596689898038092781226241323574253061224275691195161213901061
comment: $10_{71}$; alternating
10
72:
12.92959468704969729477308342628222064213901490790491545953365372757967876808954655082909917030375762
comment: $10_{72}$; alternating
10
73:
13.70688084237152535342494971083973980375323628535092557577256994689087030440523449483281888983248294
comment: $10_{73}$; alternating
10
74:
12.00603699837807391017414032743607990145138074734991433975909236280774573408809233415932183320322487
comment: $10_{74}$; alternating
10
75:
13.43074878428476458343544387297374453120090586305007864641671300353180200764208965364793182883008423
comment: $10_{75}$; alternating
10
76:
11.51286041446981421779837120304107290241259244958298059146574886271249685468481136557822052977093237
comment: $10_{76}$; alternating
10
77:
12.07471168144469044317601452157251905626358224806962540495958369654386659221393726302397727304056041
comment: $10_{77}$; alternating
10
78:
12.50209919698768930012694129893289326366196660058550574655959652328909135979433195423247072502502328
comment: $10_{78}$; alternating
10
79:
12.54029521848435258689887168048800590622097033918390249747013832673160710629983391969667725680936412
comment: $10_{79}$; alternating
10
80:
13.39404447151597507955631681213695255413245207678008489595755028008523583252027098309121700258669008
comment: $10_{80}$; alternating
10
81:
14.49266703158334180334648489407645305650316009559966232315146084968620014633596367470985751532689072
comment: $10_{81}$; alternating
10
82:
12.43147953132618675708293591187474135266216490869341286925267575167406855761433816464088157180081531
comment: $10_{82}$; alternating
10
83:
14.25805184918628542019505246991001576829006367997060252720719145575644793609835447438663248505403402
comment: $10_{83}$; alternating
10
84:
14.70989878155416425279146718246562564963516474385980832659590783287080083183943989629491409432694675
comment: $10_{84}$; alternating
10
85:
11.79777365992886533210550426859675408934440771888881079080021169755933187865891750409256582454139585
comment: $10_{85}$; alternating
10
86:
14.34125613950009527952144516672850183255918764852895856680753607903929550229470367174397121072242449
comment: $10_{86}$; alternating
10
87:
14.27364459807036895639671236836032423160535841053752614526891038803824732743935826662513981609734562
comment: $10_{87}$; alternating
10
88:
15.64664917178012861066908972711267525417980995539921986021946503131495572039423850995509625899941377
comment: $10_{88}$; alternating
10
89:
15.56605892312014018318709163767915538869624976426809373953925354324242643687976432597754179793370582
comment: $10_{89}$; alternating
10
90:
13.86614986327110161639848256197626939449805077684306593766601089355758889327885569599629597706705748
comment: $10_{90}$; alternating
10
91:
13.48702242029779274241420107425691520505802284301009895063231398842532962318786401534601545016172975
comment: $10_{91}$; alternating
10
92:
14.85535017159680095618212263975762284454504296534617408308557038487177682051785009808850190432756559
comment: $10_{92}$; alternating
10
93:
13.01646716490768640302480016708943843966623796598435819321763813202958638791336732935721308468816375
comment: $10_{93}$; alternating
10
94:
13.31156652496759115978602869730853626196249860995666023748518497522483667504951082314583829698668871
comment: $10_{94}$; alternating
10
95:
15.04785288458121338820911577373562417618740532274819410215272061540360600362988753823341616933451702
comment: $10_{95}$; alternating
10
96:
15.17785142772461242709291458454449386962531002497245561699858341194635121327834333758964682843144469
comment: $10_{96}$; alternating
10
97:
14.85274670837589520225453929522495394290458825443116048370114876847066367895089629492566246650067089
comment: $10_{97}$; alternating
10
98:
14.41291902355025838069407310980140024568670306330125381203014376660745138211832288644889926084549973
comment: $10_{98}$; alternating
10
99:
14.33434451904500796204785704187655087543724142216561812013367938095249609861362789080219237462086189
comment: $10_{99}$; alternating
10
100:
12.81088256646548713234477184005564640398786612314103260108771617998617791698190038913163547913327236
comment: $10_{100}$; alternating
10
101:
14.68750683762170960379597471599080967766818036546615760711779053917648728158285493371973740905640171
comment: $10_{101}$; alternating
10
102:
13.72734329713671502858905919266794768240233767760420471594287247820618863110658587457766545110911753
comment: $10_{102}$; alternating
10
103:
13.87478956993313535168911716079752022366515804884281950694455797951763925603637973697335060866370162
comment: $10_{103}$; alternating
10
104:
14.10712668755234567488717604537326454979301863647494501344343027298827917035392616721145632610956460
comment: $10_{104}$; alternating
10
105:
15.18169550461262693853445689896848366246439874247258592805851849543303186681487175198636338068936510
comment: $10_{105}$; alternating
10
106:
13.93301899431438670723516591827112315499018101006738477446819288170787208207258347650085785849406097
comment: $10_{106}$; alternating
10
107:
15.35285269701981287287885679130917231594446129533726695862595537326527140837294058021008304003726099
comment: $10_{107}$; alternating
10
108:
12.90462395290652058636164799482918864313421358599429798038337556586815797330443957394467106791808955
comment: $10_{108}$; alternating
10
109:
14.90020860302684151470291481581187049462304575593247206423294955839538359320727867713685427278463089
comment: $10_{109}$; alternating
10
110:
14.77746266507286337655972967468847547544531627846390232857911720478640905731253409953633966014284347
comment: $10_{110}$; alternating
10
111:
14.26502301349320587293946374030880326052804784229843559398089333743207360726872867087157341601960712
comment: $10_{111}$; alternating
10
112:
14.75588407595319018461934792392277779537965925468808238821617541352197037656007730251634613980792194
comment: $10_{112}$; alternating
10
113:
16.47347366146883833103885468924130549182548731669497936817789036398861590455896853234173603133993002
comment: $10_{113}$; alternating
10
114:
15.30490434344304123785519990502457976084957855732656471622835025928586439468412691187873254942983493
comment: $10_{114}$; alternating
10
115:
16.63803805643663711666685492987873960382526695220968051552783174131494268548883100652086996975997633
comment: $10_{115}$; alternating
10
116:
15.42386650731443553442346176000165213669926990367571629252215509268650797559488142699445052195228919
comment: $10_{116}$; alternating
10
117:
16.12544352397696331055801464873775163705670274870449627560662487670770288418694634563488294775035677
comment: $10_{117}$; alternating
10
118:
15.54521490567671657059785209980407457786438037818364745303604974250153904615094852786878184071986974
comment: $10_{118}$; alternating
10
119:
15.93869412628264817254097668693066187669046145517525361677226633457919089295408530595229681500308892
comment: $10_{119}$; alternating
10
120:
16.27136828215934116894311729521209712766641639834145341251779945026712488361912827673430309967832128
comment: $10_{120}$; alternating
10
121:
16.97487703219090261210248038900671313856993918564967765218033495184758594917800285046361394613810057
comment: $10_{121}$; alternating
10
122:
16.41082315861027821612783140976962166715762016593829820535743466227676935519483436206479641502414415
comment: $10_{122}$; alternating
10
123:
17.08570948298286127690097484048365482503835960943062901443617629029186273886386122904543945202181708
comment: $10_{123}$; alternating; the largest volume among the prime knots with at most ten crossings
10
125:
4.611961374497318894786808121309016075928663352688345621776343428258385686465130171268454575234632437
comment: $10_{125}$; the complement is s385 in the SnapPy census; non-alternating
10
126:
6.904256123806480923772397319865608429707616673437677909112473837709516573189096729537443491768330013
comment: $10_{126}$; the complement is t10499 in the SnapPy census; non-alternating; the volume agrees to the hundred digits listed with that of $13n_{912}$
10
127:
8.896816759631431060775666479919038402471197664377774554688570321337641893333012131729346497722506541
comment: $10_{127}$; non-alternating
10
128:
5.860539301740001833791749551598084603064070395282505607309256309087478867309791803411145454852533832
comment: $10_{128}$; the complement is v2553 in the SnapPy census; non-alternating; the volume agrees to the hundred digits listed with that of $11n_{57}$
10
129:
8.901516637278768975131436950806497346511158989036552838267342655186497708689499202595874631121352813
comment: $10_{129}$; non-alternating
10
130:
6.778198890046892416334421271835782395339037998616188570223820004850967418594181293187397456311616898
comment: $10_{130}$; the complement is t09901 in the SnapPy census; non-alternating; the volume agrees to the hundred digits listed with that of $13n_{1021}$
10
131:
9.465021657265467676707601598019298982471501473711893148974042053245632134025051211886978747838732766
comment: $10_{131}$; non-alternating
10
132:
4.056860224236820144181924184017659447160664377093809560226560015443973033695891463348932385667550031
comment: $10_{132}$; the complement is m201 in the SnapPy census; non-alternating; the volume agrees to the hundred digits listed with that of $9_{42}$
10
133:
7.798300232166505743143839685065691706877998939243686967821225190138633486741443878952033230022446015
comment: $10_{133}$; the complement is o9_37732 in the SnapPy census; non-alternating
10
134:
8.392922508336978318098338674728254250247643431028423631609083422530267723102182253748918261299658992
comment: $10_{134}$; the complement is o9_42974 in the SnapPy census; non-alternating; the volume agrees to the hundred digits listed with that of $13n_{1164}$
10
135:
10.68717496833962782761300037281679227633215340753983305196204192636006577541987634780734229455602651
comment: $10_{135}$; non-alternating
10
136:
7.746274546233127565391446018747172204555087335955424175164526464979871483959839814365679846839722815
comment: $10_{136}$; the complement is o9_37080 in the SnapPy census; non-alternating
10
137:
9.250556262699874205862355877943722662165709529296099327706645792898930759518288995826826501653543279
comment: $10_{137}$; non-alternating
10
138:
10.46724624267089805362062144469683392044429303243233433835755514932205658781679448651577417014690211
comment: $10_{138}$; non-alternating
10
139:
4.851170757332737567058327052115312478845283027769987822105853729464035695794418026804214648831525592
comment: $10_{139}$; the complement is m389 in the SnapPy census; non-alternating
10
140:
5.212566822320545235321430058742189690886047784272239160446067909250942073034416263237054959669310104
comment: $10_{140}$; the complement is s704 in the SnapPy census; non-alternating
10
141:
7.936474229212679348100470069899929988949996532401176141202234276902618921485010830892822638553236282
comment: $10_{141}$; the complement is o9_39277 in the SnapPy census; non-alternating
10
142:
6.770816780739823289358143482493052137704993017289757673075293957535588044158930755220814774600195386
comment: $10_{142}$; the complement is t09859 in the SnapPy census; non-alternating
10
143:
9.070899270019317958966915814457138955564117640669401311420617606704396241773057531518054168561552564
comment: $10_{143}$; non-alternating
10
144:
10.79659498413383620848339266634719611254160513657264676066536381460570242317764771340021098697324460
comment: $10_{144}$; non-alternating
10
145:
5.044899162918904312925592386244805975361583861173506086197210464163859503877026709507482938504143706
comment: $10_{145}$; the complement is s580 in the SnapPy census; non-alternating
10
146:
10.56101675491469860610732216477543698740004421043683280641882486821205559926500776230859090669322352
comment: $10_{146}$; non-alternating
10
147:
9.417590916016745131918297063516263957959951873190792781979205105619309984853466543673769967363543715
comment: $10_{147}$; non-alternating
10
148:
10.26024343631367197964958083706444292718275020758246590549853298123895937331929255589532988766304834
comment: $10_{148}$; non-alternating
10
149:
11.44272677886196902410857437352990543638133075971343758321873132497299894921169737261705533898449660
comment: $10_{149}$; non-alternating
10
150:
10.08136285656081831039970820723434656333996771477206878438447266934888975809813465202624711121924581
comment: $10_{150}$; non-alternating
10
151:
11.84304475168114478290612523313936322020069031063226869199372225145367106646076172873286755190223128
comment: $10_{151}$; non-alternating
10
152:
8.536065347205608603144181920549325994964991396914008018730723688281131342697980603601150568551725114
comment: $10_{152}$; the complement is o9_43609 in the SnapPy census; non-alternating
10
153:
7.374343889306711754881819019588113347930382123822712140332006664185143806087425035170653238964118708
comment: $10_{153}$; the complement is t12200 in the SnapPy census; non-alternating
10
154:
9.249887443861750234495968288244023722650320885457918638204253065053466235504931433681123040688619563
comment: $10_{154}$; non-alternating
10
155:
9.250541613012098776688797885254060095679416759181827490495550912731820047801992282273540023416857781
comment: $10_{155}$; non-alternating
10
156:
11.16339064217632597062741800961293955989822872405803310297135840936343266053708044964783977960291490
comment: $10_{156}$; non-alternating
10
157:
12.66533328499850268932823081503352502718442075748092512964080297108119370864130627340689746941486200
comment: $10_{157}$; non-alternating
10
158:
12.27123635844697224298380649872997553749502668583198144055424593372851572308339796769591419887709452
comment: $10_{158}$; non-alternating
10
159:
11.74064103791391607160567892758650776171307956362481348405530315828554014136063230593703773227401009
comment: $10_{159}$; non-alternating
10
160:
9.203916606090773025161662698774173265075109604459124536993748429979893333809038341236597919848671792
comment: $10_{160}$; non-alternating
10
161:
5.638772948853053747877990920924033500382748146209878546172131025908646044290062414270934020809676370
comment: $10_{161}$, the Perko pair, listed twice by Rolfsen as $10_{161}$ and $10_{162}$; the complement is v2166 in the SnapPy census; non-alternating
10
162:
10.69336054671985524299401866899790896658146809067207952737321537842804036441499466211032243948530505
comment: $10_{162}$; non-alternating
10
163:
13.29000306859835797891170741635590606384401259763168538604931425240195220979653737355960707888099948
comment: $10_{163}$; non-alternating
10
164:
12.50668793089236135653192281825397164733488552195477130144269339128587638806577292043294797656380203
comment: $10_{164}$; non-alternating
10
165:
11.60308464976907662459170011576233319651398588372694835825809965963109022795690300023180508816475564
comment: $10_{165}$; non-alternating
Definition
The volume $\mathrm{Vol}(S^3\setminus K)$ of the complete hyperbolic metric on the complement of a hyperbolic knot $K$ [8], for every hyperbolic prime knot $n_k$ with at most ten crossings, numbered as in the Rolfsen table [1] after Perko's correction [2].
Parameters
$n$
—   crossing number ($4\leq n\leq 10$)
$k$
—   index in the Rolfsen table ($1\leq k\leq N(n)$, where $N(n)$ is the number of prime knots with $n$ crossings, $1,2,3,7,21,49,165$ for $n=4,\ldots,10$, and $n_k$ is not one of the torus knots $5_1$, $7_1$, $9_1$, $8_{19}$, $10_{124}$)
Formulas
(1)
$\mathrm{Vol}(S^3\setminus K)=\sum_{i=1}^{m}D(z_i)$ for any ideal triangulation of the complement by $m$ positively oriented tetrahedra of shapes $z_1,\ldots,z_m$, $\operatorname{Im}z_i>0$, where $D(z)=\operatorname{Im}\mathrm{Li}_2(z)+\arg(1-z)\log|z|$ is the Bloch–Wigner dilogarithm, the volume of the ideal tetrahedron of shape $z$ [3]. The shapes satisfy the polynomial gluing equations of the triangulation, so they are algebraic numbers and the volume is a period.
(2)
$\mathrm{Vol}(4_1)=2D(e^{i\pi/3})=2\,\mathrm{Cl}_2(\pi/3)=3\,\mathrm{Cl}_2(2\pi/3) =-6\int_0^{\pi/3}\log|2\sin\theta|\,d\theta$, where $\mathrm{Cl}_2$ is the Clausen function [10]; it is twice the Gieseking constant [11], and it is the smallest volume of any orientable cusped hyperbolic 3-manifold [4], hence of any hyperbolic knot complement. Its digits are OEIS A091518 [17].
(3)
$\mathrm{Vol}(S^3\setminus\bar K)=\mathrm{Vol}(S^3\setminus K)$ for the mirror image $\bar K$, and $\mathrm{Vol}(S^3\setminus K')=\mathrm{Vol}(S^3\setminus K)$ for any mutant $K'$ of $K$ [5].
Comments
(4)
A prime knot is a torus knot, a satellite knot or hyperbolic [7], and no satellite knot has fewer than thirteen crossings [14]. Of the 249 prime knots with at most ten crossings, 243 are therefore hyperbolic and are listed here; the six torus knots $3_1=T(2,3)$, $5_1=T(2,5)$, $7_1=T(2,7)$, $9_1=T(2,9)$, $8_{19}=T(3,4)$ and $10_{124}=T(3,5)$ have Seifert fibred complements, carry no hyperbolic metric, and are left out rather than given the value $0$. The unknot is left out for the same reason.
(5)
The numbering is Rolfsen's after Perko's correction [2]: the knots Rolfsen listed as $10_{161}$ and $10_{162}$ are one knot, so there are $165$ prime knots with ten crossings rather than $166$ [16], and Rolfsen's $10_{163}$ to $10_{166}$ are $10_{162}$ to $10_{165}$ here [12]. KnotInfo [14], the Knot Atlas [13] and Sage's knot table use this numbering. SnapPy's own table of Rolfsen knots, Manifold('n_k') [15], does not: it keeps Rolfsen's 166 ten-crossing knots, with $10_{161}$ and $10_{162}$ isometric, so its $10_{k+1}$ is $10_k$ here for $k\geq 162$; and its $10_{83}$ and $10_{86}$ are each other's, neither being isometric to the exterior of KnotInfo's or the Knot Atlas's diagram of the knot of that name. The volumes here are of the knots as KnotInfo and the Knot Atlas name them.
(6)
The volume of the complement does not depend on the orientation of the knot or on the choice between a knot and its mirror image, so it is well defined for the name $n_k$, which names a knot only up to mirror image. It does not determine the knot: $\mathrm{Vol}(9_{42})$ and $\mathrm{Vol}(10_{132})$ agree to the hundred digits listed, and eleven of the volumes here agree to the hundred digits listed with the volume of a knot with eleven to thirteen crossings, among them $5_2$ and the $(-2,3,7)$ pretzel knot $12n_{242}$; each entry's comment names the other knot. Mutant knots have the same volume [5], but the first mutant pair, the Conway knot $11n_{34}$ and the Kinoshita–Terasaka knot $11n_{42}$, has eleven crossings.
(7)
Each entry's comment gives the knot's common name where it has one, the name of its complement in the SnapPy census of orientable cusped hyperbolic 3-manifolds where the complement has a triangulation with at most nine tetrahedra (67 of the 243 do; m004 is the figure-eight knot complement), whether the knot is alternating, and the knots with at most thirteen crossings whose volume agrees with the entry's to the hundred digits listed. The Chern–Simons invariant, the cusp volume and the cusp shape, which KnotInfo lists beside the volume, are not listed.
(8)
The Rolfsen table ends at ten crossings, and so do the names $n_k$. The 552 prime knots with eleven crossings are named $11a_1$ to $11a_{367}$ and $11n_1$ to $11n_{185}$ after Hoste and Thistlethwaite, alternating and non-alternating separately, which the two parameters here cannot express; KnotInfo carries the volumes of all prime knots to thirteen crossings, to ten decimals. The Alexander, Jones and Conway polynomials of the same knots are listed separately.
Programs
(P1)
Sage
from snappy import Link, Manifold       # sage -pip install snappy
Manifold('4_1').volume()                          # 2.02988321281931, a double
Manifold('4_1').high_precision().volume()         # 60 digits, not verified
Manifold('4_1').volume(verified=True, bits_prec=400)   # an interval, inside Sage only
# SnapPy's names are Rolfsen's before Perko, and its 10_83 and 10_86 are interchanged;
# the braid word of Sage's knot table names the knot as this table does:
Link(braid_closure=[-1, 2, -1, 2]).exterior().volume()   # 4_1 again
References
[1]
D. Rolfsen, Knots and Links, Publish or Perish, 1976, Appendix C: Table of knots and links.
[2]
K. A. Perko, On the classification of knots, Proc. Amer. Math. Soc. 45 (1974), 262–266.
[3]
W. D. Neumann and D. Zagier, Volumes of hyperbolic three-manifolds, Topology 24 (1985), 307–332.
[4]
C. Cao and G. R. Meyerhoff, The orientable cusped hyperbolic 3-manifolds of minimum volume, Invent. Math. 146 (2001), 451–478.
[5]
D. Ruberman, Mutation and volumes of knots in S^3, Invent. Math. 90 (1987), 189–215.
[6]
N. Hoffman, K. Ichihara, M. Kashiwagi, H. Masai, S. Oishi and A. Takayasu, Verified computations for hyperbolic 3-manifolds, Exp. Math. 25 (2016), 66–78.
[7]
W. P. Thurston, Three-dimensional manifolds, Kleinian groups and hyperbolic geometry, Bull. Amer. Math. Soc. (N.S.) 6 (1982), 357–381.
Links
Similar tables
Alexander polynomials of the prime knots with at most ten crossings —   the same knots; $\Delta(9_{42})\neq\Delta(10_{132})$ although the two volumes agree
Jones polynomials of the prime knots with at most ten crossings —   the same knots
Conway polynomials of the prime knots with at most ten crossings —   the same knots
Data properties
Entries are of type: real number
How well the digits are known: proven
Table is complete: yes
How they were obtained:

Each volume is a ball computed in arb at 397 bits from an ideal triangulation of the knot exterior that SnapPy [15] builds from the braid word of Sage's knot table, the closure of which is the knot $n_k$.

more

SnapPy supplies the gluing equations of the triangulation, which are integers, and a floating-point solution; the generator then proves, in ball arithmetic and without SnapPy's verify module, that a box of complex balls contains a solution of a square subset of the rectangular gluing equations (a Krawczyk test on $m-1$ edge equations and the meridian, the last edge equation being implied because the edge equations multiply to the identity), that every shape ball has positive imaginary part, and that every logarithmic gluing equation evaluated on the balls is within $0.1$ of $2\pi i$ (edges) or $0$ (meridian and longitude), which pins each to its exact value. That is the criterion of [6]: the triangulation is then a geometric triangulation of the complete hyperbolic structure, unique by Mostow–Prasad rigidity, and the volume is the sum of the Bloch–Wigner dilogarithms of the shape balls, arb's polylog; the digits written are those the ball supports (the widest ball, $10_{122}$, supports 113). Outside the generator all 243 balls were compared with SnapPy's quad-double volumes, which agree to 62 digits, and with KnotInfo's ten decimals, $\mathrm{Vol}(4_1)$ with $2\,\mathrm{Cl}_2(\pi/3)$ in arb and with OEIS A091518, and the exterior of each braid closure was checked to be isometric to the exterior of KnotInfo's diagram of the same knot. All agree, and the controls that must fail fail. The identification of the manifold with the complement of the knot named $n_k$ rests on those tables, not on a proof.



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