10 112

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10_111

10_113

Contents

Image:10 112.gif
(KnotPlot image)

See the full Rolfsen Knot Table.

Visit 10 112's page at the Knot Server (KnotPlot driven, includes 3D interactive images!)

Visit 10_112's page at Knotilus!

Visit 10 112's page at the original Knot Atlas!

[edit 10 112 Quick Notes]
[edit 10 112 Further Notes and Views]


[edit] Knot presentations

Planar diagram presentation X6271 X8394 X18,11,19,12 X20,13,1,14 X2,16,3,15 X4,17,5,18 X12,19,13,20 X10,6,11,5 X14,7,15,8 X16,10,17,9
Gauss code 1, -5, 2, -6, 8, -1, 9, -2, 10, -8, 3, -7, 4, -9, 5, -10, 6, -3, 7, -4
Dowker-Thistlethwaite code 6 8 10 14 16 18 20 2 4 12
Conway Notation [8*3]


Minimum Braid Representative A Morse Link Presentation An Arc Presentation
Image:BraidPart3.gifImage:BraidPart3.gifImage:BraidPart3.gifImage:BraidPart0.gifImage:BraidPart3.gifImage:BraidPart0.gifImage:BraidPart3.gifImage:BraidPart0.gifImage:BraidPart3.gifImage:BraidPart0.gif
Image:BraidPart4.gifImage:BraidPart4.gifImage:BraidPart4.gifImage:BraidPart1.gifImage:BraidPart4.gifImage:BraidPart1.gifImage:BraidPart4.gifImage:BraidPart1.gifImage:BraidPart4.gifImage:BraidPart1.gif
Image:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart2.gifImage:BraidPart0.gifImage:BraidPart2.gifImage:BraidPart0.gifImage:BraidPart2.gifImage:BraidPart0.gifImage:BraidPart2.gif

Length is 10, width is 3,

Braid index is 3

Image:10 112_ML.gif Image:10 112_AP.gif
[{3, 12}, {2, 10}, {4, 11}, {5, 3}, {6, 4}, {7, 5}, {1, 6}, {9, 2}, {10, 8}, {12, 9}, {11, 7}, {8, 1}]

[edit Notes on presentations of 10 112]


Computer Talk
The above data is available with the Mathematica package KnotTheory`. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:= AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`
Loading KnotTheory` version of May 31, 2006, 14:15:20.091.

Read more at http://katlas.org/wiki/KnotTheory.

In[3]:= K = Knot["10 112"];
In[4]:= PD[K]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= X6271 X8394 X18,11,19,12 X20,13,1,14 X2,16,3,15 X4,17,5,18 X12,19,13,20 X10,6,11,5 X14,7,15,8 X16,10,17,9
In[5]:= GaussCode[K]
Out[5]= 1, -5, 2, -6, 8, -1, 9, -2, 10, -8, 3, -7, 4, -9, 5, -10, 6, -3, 7, -4
In[6]:= DTCode[K]
Out[6]= 6 8 10 14 16 18 20 2 4 12

(The path below may be different on your system)

In[7]:= AppendTo[$Path, "C:/bin/LinKnot/"];
In[8]:= ConwayNotation[K]
Out[8]= [8*3]
In[9]:= br = BR[K]
KnotTheory::credits: The minimum braids representing the knots with up to 10 crossings were provided by Thomas Gittings. See arXiv:math.GT/0401051.
Out[9]= BR(3,{−1,−1,−1,2,−1,2,−1,2,−1,2})
In[10]:= {First[br], Crossings[br], BraidIndex[K]}
KnotTheory::credits: The braid index data known to KnotTheory` is taken from Charles Livingston's http://www.indiana.edu/~knotinfo/.
KnotTheory::loading: Loading precomputed data in IndianaData`.
Out[10]= { 3, 10, 3 }
In[11]:= Show[BraidPlot[br]]
Image:BraidPart3.gifImage:BraidPart3.gifImage:BraidPart3.gifImage:BraidPart0.gifImage:BraidPart3.gifImage:BraidPart0.gifImage:BraidPart3.gifImage:BraidPart0.gifImage:BraidPart3.gifImage:BraidPart0.gif
Image:BraidPart4.gifImage:BraidPart4.gifImage:BraidPart4.gifImage:BraidPart1.gifImage:BraidPart4.gifImage:BraidPart1.gifImage:BraidPart4.gifImage:BraidPart1.gifImage:BraidPart4.gifImage:BraidPart1.gif
Image:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart2.gifImage:BraidPart0.gifImage:BraidPart2.gifImage:BraidPart0.gifImage:BraidPart2.gifImage:BraidPart0.gifImage:BraidPart2.gif
Out[11]= -Graphics-
In[12]:= Show[DrawMorseLink[K]]
KnotTheory::credits: "MorseLink was added to KnotTheory` by Siddarth Sankaran at the University of Toronto in the summer of 2005."
KnotTheory::credits: "DrawMorseLink was written by Siddarth Sankaran at the University of Toronto in the summer of 2005."
Image:10 112_ML.gif
Out[12]= -Graphics-
In[13]:= ap = ArcPresentation[K]
Out[13]= ArcPresentation[{3, 12}, {2, 10}, {4, 11}, {5, 3}, {6, 4}, {7, 5}, {1, 6}, {9, 2}, {10, 8}, {12, 9}, {11, 7}, {8, 1}]
In[14]:= Draw[ap]
Image:10 112_AP.gif
Out[14]= -Graphics-

[edit] Three dimensional invariants

Symmetry type Reversible
Unknotting number 2
3-genus 4
Bridge index 3
Super bridge index Missing
Nakanishi index 1
Maximal Thurston-Bennequin number [-9][-3]
Hyperbolic Volume 14.7559
A-Polynomial See Data:10 112/A-polynomial

[edit Notes for 10 112's three dimensional invariants]

[edit] Four dimensional invariants

Smooth 4 genus 1
Topological 4 genus 1
Concordance genus 4
Rasmussen s-Invariant 2

[edit Notes for 10 112's four dimensional invariants]

[edit] Polynomial invariants

Alexander polynomial −t4 + 5t3−11t2 + 17t−19 + 17t−1−11t−2 + 5t−3−t−4
Conway polynomial −z8−3z6−z4 + 2z2 + 1
2nd Alexander ideal (db, data sources) {1}
Determinant and Signature { 87, -2 }
Jones polynomial q3−4q2 + 7q−10 + 14q−1−14q−2 + 14q−3−11q−4 + 7q−5−4q−6 + q−7
HOMFLY-PT polynomial (db, data sources) −a2z8 + a4z6−5a2z6 + z6 + 3a4z4−7a2z4 + 3z4 + a4z2 + z2−2a4 + 4a2−1
Kauffman polynomial (db, data sources) 3a3z9 + 3az9 + 7a4z8 + 13a2z8 + 6z8 + 8a5z7 + 4a3z7 + 4z7a−1 + 7a6z6−9a4z6−35a2z6 + z6a−2−18z6 + 4a7z5−8a5z5−17a3z5−16az5−11z5a−1 + a8z4−7a6z4 + 3a4z4 + 28a2z4−2z4a−2 + 15z4−3a7z3−a5z3 + 9a3z3 + 13az3 + 6z3a−1 + a6z2 + a4z2−3a2z2−3z2 + 2a5z + 2a3z−2a4−4a2−1
The A2 invariant q20−2q18 + q16−3q14−q12 + 2q10−q8 + 6q6−q4 + 3q2−2q−2 + q−4−2q−6 + q−8
The G2 invariant q114−3q112 + 6q110−10q108 + 9q106−6q104−2q102 + 18q100−32q98 + 47q96−50q94 + 34q92−6q90−34q88 + 75q86−104q84 + 117q82−101q80 + 50q78 + 25q76−110q74 + 182q72−205q70 + 161q68−63q66−68q64 + 177q62−218q60 + 166q58−45q56−99q54 + 184q52−176q50 + 58q48 + 108q46−239q44 + 275q42−192q40 + 16q38 + 182q36−322q34 + 358q32−271q30 + 102q28 + 103q26−251q24 + 316q22−264q20 + 137q18 + 29q16−166q14 + 221q12−170q10 + 47q8 + 109q6−214q4 + 212q2−106−64q−2 + 214q−4−286q−6 + 244q−8−112q−10−54q−12 + 184q−14−238q−16 + 206q−18−112q−20 + 3q−22 + 71q−24−104q−26 + 94q−28−58q−30 + 25q−32 + 5q−34−17q−36 + 17q−38−14q−40 + 7q−42−3q−44 + q−46
Further Quantum Invariants
Further quantum knot invariants for 10_112.

A1 Invariants.

Weight Invariant
1 q15−3q13 + 3q11−4q9 + 3q7 + 4q−3q−1 + 3q−3−3q−5 + q−7
2 q42−3q40 + 8q36−10q34 + q32 + 19q30−24q28−5q26 + 34q24−24q22−19q20 + 32q18−3q16−22q14 + 9q12 + 19q10−12q8−17q6 + 28q4 + 4q2−33 + 23q−2 + 18q−4−34q−6 + 5q−8 + 23q−10−17q−12−7q−14 + 12q−16−q−18−3q−20 + q−22
3 q81−3q79 + 5q75 + 2q73−6q71−6q69 + 12q67−5q65−17q63 + 17q61 + 41q59−34q57−85q55 + 41q53 + 149q51−16q49−209q47−40q45 + 238q43 + 117q41−225q39−182q37 + 154q35 + 224q33−74q31−218q29−21q27 + 189q25 + 96q23−142q21−152q19 + 96q17 + 191q15−48q13−219q11 + q9 + 241q7 + 53q5−238q3−119q + 221q−1 + 179q−3−157q−5−228q−7 + 78q−9 + 231q−11 + 10q−13−196q−15−81q−17 + 129q−19 + 109q−21−58q−23−95q−25 + 5q−27 + 62q−29 + 19q−31−30q−33−15q−35 + 7q−37 + 8q−39−q−41−3q−43 + q−45
4 q132−3q130 + 5q126−q124 + 6q122−13q120−7q118 + 5q116−8q114 + 43q112 + 4q110−5q108−39q106−115q104 + 56q102 + 140q100 + 191q98−44q96−479q94−272q92 + 239q90 + 819q88 + 495q86−795q84−1234q82−420q80 + 1352q78 + 1806q76−107q74−1950q72−1927q70 + 623q68 + 2626q66 + 1469q64−1162q62−2728q60−1001q58 + 1753q56 + 2269q54 + 494q52−1872q50−1863q48 + 88q46 + 1660q44 + 1462q42−416q40−1605q38−1000q36 + 713q34 + 1600q32 + 521q30−1187q28−1500q26 + 174q24 + 1659q22 + 1136q20−980q18−2018q16−376q14 + 1728q12 + 1941q10−397q8−2381q6−1386q4 + 1078q2 + 2541 + 905q−2−1742q−4−2211q−6−471q−8 + 1943q−10 + 1957q−12−71q−14−1709q−16−1670q−18 + 288q−20 + 1549q−22 + 1148q−24−202q−26−1332q−28−801q−30 + 243q−32 + 887q−34 + 646q−36−264q−38−566q−40−377q−42 + 116q−44 + 400q−46 + 164q−48−55q−50−195q−52−101q−54 + 59q−56 + 62q−58 + 41q−60−20q−62−29q−64−q−66 + 3q−68 + 8q−70−q−72−3q−74 + q−76
5 q195−3q193 + 5q189−q187 + 3q185−q183−14q181−14q179 + 8q177 + 26q175 + 39q173 + 27q171−44q169−119q167−111q165 + 46q163 + 233q161 + 297q159 + 89q157−396q155−732q153−406q151 + 522q149 + 1377q147 + 1244q145−325q143−2358q141−2807q139−538q137 + 3226q135 + 5261q133 + 2790q131−3383q129−8363q127−6776q125 + 1819q123 + 11098q121 + 12328q119 + 2363q117−12050q115−18353q113−9116q111 + 9844q109 + 22819q107 + 17308q105−3982q103−23854q101−24722q99−4508q97 + 20466q95 + 29011q93 + 13444q91−13220q89−28708q87−20448q85 + 4095q83 + 24112q81 + 23630q79 + 4488q77−16577q75−22887q73−10738q71 + 8556q69 + 19145q67 + 13802q65−1675q63−14248q61−14346q59−2957q57 + 9853q55 + 13501q53 + 5484q51−6920q49−12671q47−6700q45 + 5592q43 + 12784q41 + 7707q39−5350q37−14066q35−9442q33 + 5142q31 + 16192q29 + 12420q27−3966q25−18267q23−16515q21 + 1026q19 + 19110q17 + 21012q15 + 3968q13−17673q11−24717q9−10372q7 + 13293q5 + 26080q3 + 17093q−6169q−1−24159q−3−22158q−5−2447q−7 + 18441q−9 + 24012q−11 + 10687q−13−10037q−15−21692q−17−16237q−19 + 756q−21 + 15621q−23 + 17721q−25 + 6925q−27−7576q−29−15018q−31−11105q−33−52q−35 + 9478q−37 + 11239q−39 + 5213q−41−3369q−43−8312q−45−6957q−47−1275q−49 + 4158q−51 + 5888q−53 + 3506q−55−666q−57−3504q−59−3462q−61−1239q−63 + 1192q−65 + 2281q−67 + 1653q−69 + 133q−71−1007q−73−1162q−75−552q−77 + 181q−79 + 572q−81 + 454q−83 + 82q−85−172q−87−215q−89−108q−91 + 16q−93 + 82q−95 + 55q−97 + q−99−19q−101−15q−103−5q−105 + 3q−107 + 8q−109−q−111−3q−113 + q−115
6 q270−3q268 + 5q264−q262 + 3q260−4q258−2q256−21q254−11q252 + 42q250 + 28q248 + 36q246−3q244−55q242−157q240−127q238 + 107q236 + 225q234 + 326q232 + 190q230−160q228−734q226−821q224−93q222 + 758q220 + 1583q218 + 1449q216 + 33q214−2441q212−3745q210−2161q208 + 1426q206 + 5785q204 + 7359q202 + 3530q200−5400q198−13415q196−13431q194−3589q192 + 13408q190 + 26609q188 + 24074q186 + 1478q184−29679q182−48096q180−37772q178 + 4186q176 + 55610q174 + 81157q172 + 54349q170−17776q168−93662q166−122736q164−71743q162 + 39507q160 + 145090q158 + 169711q156 + 83551q154−71025q152−202254q150−215345q148−87509q146 + 112381q144 + 257605q142 + 248375q140 + 79265q138−154324q136−301417q134−263819q132−57139q130 + 189978q128 + 322701q126 + 256764q124 + 29559q122−212564q120−320019q118−227969q116−1781q114 + 216152q112 + 293743q110 + 188732q108−21342q106−204151q104−250290q102−145830q100 + 36003q98 + 180150q96 + 202898q94 + 104568q92−45655q90−151559q88−157267q86−67675q84 + 52872q82 + 127501q80 + 115629q78 + 31904q76−62677q74−108267q72−76318q70 + 5389q68 + 78347q66 + 90646q64 + 34071q62−48065q60−95505q58−68472q56 + 14805q54 + 95245q52 + 107948q50 + 36161q48−72932q46−140329q44−108342q42 + 8598q40 + 135384q38 + 176663q36 + 93826q34−65353q32−193987q30−197546q28−65014q26 + 124891q24 + 241972q22 + 202838q20 + 25026q18−178074q16−272696q14−194117q12 + 14895q10 + 218361q8 + 286782q6 + 174267q4−47186q2−239543−284538q−2−153010q−4 + 67717q−6 + 243003q−8 + 267322q−10 + 134083q−12−72869q−14−229033q−16−243101q−18−118385q−20 + 66287q−22 + 199949q−24 + 213825q−26 + 107454q−28−49879q−30−164671q−32−180650q−34−97700q−36 + 27443q−38 + 126436q−40 + 147194q−42 + 88688q−44−7493q−46−88419q−48−113621q−50−79365q−52−8326q−54 + 55890q−56 + 83122q−58 + 66022q−60 + 19031q−62−29776q−64−57423q−66−51503q−68−22737q−70 + 12115q−72 + 35303q−74 + 37728q−76 + 21839q−78−2141q−80−19462q−82−24867q−84−17354q−86−3304q−88 + 9548q−90 + 15028q−92 + 11770q−94 + 4397q−96−3622q−98−7855q−100−7459q−102−3406q−104 + 1041q−106 + 3498q−108 + 3919q−110 + 2199q−112 + 41q−114−1561q−116−1718q−118−1042q−120−227q−122 + 523q−124 + 687q−126 + 471q−128 + 48q−130−147q−132−200q−134−164q−136−25q−138 + 53q−140 + 74q−142 + 15q−144 + 2q−146−5q−148−19q−150−5q−152 + 3q−154 + 8q−156−q−158−3q−160 + q−162

A2 Invariants.

Weight Invariant
1,0 q20−2q18 + q16−3q14−q12 + 2q10−q8 + 6q6−q4 + 3q2−2q−2 + q−4−2q−6 + q−8
1,1 q60−6q58 + 18q56−38q54 + 69q52−120q50 + 190q48−268q46 + 357q44−460q42 + 576q40−686q38 + 790q36−878q34 + 906q32−846q30 + 671q28−372q26−56q24 + 558q22−1086q20 + 1554q18−1926q16 + 2140q14−2158q12 + 2014q10−1680q8 + 1236q6−688q4 + 132q2 + 388−822q−2 + 1101q−4−1220q−6 + 1182q−8−1040q−10 + 827q−12−590q−14 + 384q−16−226q−18 + 117q−20−52q−22 + 20q−24−6q−26 + q−28
2,0 q52−2q50−q48 + 3q46−3q44 + 2q42 + 6q40−q38−3q36 + 2q34 + 8q32−11q30−15q28 + 7q26 + q24−15q22 + 2q20 + 17q18−q16−2q14 + 9q12 + 6q10−7q8 + q6 + 14q4−9q2−6 + 11q−2−13q−6−q−8 + 9q−10−q−12−7q−14 + 2q−16 + 6q−18−2q−20−2q−22 + q−24

A3 Invariants.

Weight Invariant
0,1,0 q48−3q46 + 8q42−9q40−3q38 + 18q36−13q34−11q32 + 28q30−12q28−18q26 + 24q24−11q22−18q20 + 7q18 + 3q16−q14−3q12 + 19q10 + 14q8−19q6 + 11q4 + 14q2−28 + 4q−2 + 15q−4−21q−6 + 7q−8 + 9q−10−10q−12 + 5q−14 + q−16−3q−18 + q−20
1,0,0 q25−2q23 + 2q21−5q19 + q17−4q15 + 2q13 + q11 + 4q9 + 4q7 + q5 + 3q3−3q + 2q−1−4q−3 + 2q−5−2q−7 + q−9
1,0,1 q78−6q76 + 15q74−17q72−3q70 + 45q68−86q66 + 84q64−10q62−112q60 + 210q58−204q56 + 69q54 + 154q52−343q50 + 362q48−147q46−237q44 + 591q42−689q40 + 425q38 + 131q36−695q34 + 991q32−892q30 + 417q28 + 108q26−527q24 + 596q22−431q20 + 217q18−87q16 + 239q14−424q12 + 582q10−413q8−31q6 + 577q4−987q2 + 1018−706q−2 + 167q−4 + 339q−6−639q−8 + 659q−10−445q−12 + 140q−14 + 113q−16−223q−18 + 197q−20−104q−22 + 17q−24 + 26q−26−29q−28 + 17q−30−6q−32 + q−34

A4 Invariants.

Weight Invariant
0,1,0,0 q58−2q56−2q54 + 6q52−2q50−9q48 + 11q46 + 10q44−17q42 + q40 + 20q38−5q36−19q34 + 9q32 + 7q30−28q28−15q26 + 17q24−12q22−20q20 + 34q18 + 19q16−11q14 + 20q12 + 30q10−13q8−14q6 + 8q4−21−4q−2 + 13q−4−5q−6−3q−8 + 9q−10−5q−14 + 4q−16−q−18−2q−20 + q−22
1,0,0,0 q30−2q28 + 2q26−4q24−q22−2q20−3q18 + 2q16 + 6q12 + 2q10 + 6q8 + 3q4−2q2−1−3q−4 + 2q−6−2q−8 + q−10

B2 Invariants.

Weight Invariant
0,1 q48−3q46 + 6q44−10q42 + 15q40−21q38 + 26q36−31q34 + 33q32−32q30 + 24q28−14q26−2q24 + 17q22−34q20 + 49q18−59q16 + 67q14−63q12 + 59q10−44q8 + 31q6−11q4−4q2 + 18−28q−2 + 33q−4−35q−6 + 31q−8−27q−10 + 20q−12−13q−14 + 7q−16−3q−18 + q−20
1,0 q78−3q74−3q72 + 3q70 + 9q68 + 2q66−12q64−12q62 + 8q60 + 22q58 + 6q56−23q54−22q52 + 13q50 + 34q48 + 3q46−33q44−19q42 + 23q40 + 25q38−15q36−31q34 + q32 + 26q30 + 3q28−24q26−9q24 + 22q22 + 15q20−13q18−13q16 + 19q14 + 24q12−8q10−29q8 + q6 + 33q4 + 15q2−30−32q−2 + 13q−4 + 35q−6 + 3q−8−31q−10−17q−12 + 18q−14 + 21q−16−5q−18−15q−20−2q−22 + 9q−24 + 4q−26−3q−28−3q−30 + q−34

D4 Invariants.

Weight Invariant
1,0,0,0 q66−3q64 + 3q62−4q60 + 9q58−12q56 + 12q54−15q52 + 22q50−23q48 + 21q46−25q44 + 29q42−21q40 + 15q38−13q36 + 3q34 + 6q32−20q30 + 18q28−39q26 + 41q24−46q22 + 51q20−48q18 + 56q16−35q14 + 45q12−27q10 + 23q8−9q6 + q4 + 3q2−18 + 20q−2−26q−4 + 25q−6−28q−8 + 28q−10−23q−12 + 20q−14−16q−16 + 12q−18−7q−20 + 4q−22−3q−24 + q−26

G2 Invariants.

Weight Invariant
1,0 q114−3q112 + 6q110−10q108 + 9q106−6q104−2q102 + 18q100−32q98 + 47q96−50q94 + 34q92−6q90−34q88 + 75q86−104q84 + 117q82−101q80 + 50q78 + 25q76−110q74 + 182q72−205q70 + 161q68−63q66−68q64 + 177q62−218q60 + 166q58−45q56−99q54 + 184q52−176q50 + 58q48 + 108q46−239q44 + 275q42−192q40 + 16q38 + 182q36−322q34 + 358q32−271q30 + 102q28 + 103q26−251q24 + 316q22−264q20 + 137q18 + 29q16−166q14 + 221q12−170q10 + 47q8 + 109q6−214q4 + 212q2−106−64q−2 + 214q−4−286q−6 + 244q−8−112q−10−54q−12 + 184q−14−238q−16 + 206q−18−112q−20 + 3q−22 + 71q−24−104q−26 + 94q−28−58q−30 + 25q−32 + 5q−34−17q−36 + 17q−38−14q−40 + 7q−42−3q−44 + q−46

.

Computer Talk
The above data is available with the Mathematica package KnotTheory`, as shown in the (simulated) Mathematica session below. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting. This Mathematica session is also available (albeit only for the knot 5_2) as the notebook PolynomialInvariantsSession.nb.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:= AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`
Loading KnotTheory` version of August 31, 2006, 11:25:27.5625.

Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:= K = Knot["10 112"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= −t4 + 5t3−11t2 + 17t−19 + 17t−1−11t−2 + 5t−3−t−4
In[5]:= Conway[K][z]
Out[5]= −z8−3z6−z4 + 2z2 + 1
In[6]:= Alexander[K, 2][t]
KnotTheory::credits: The program Alexander[K, r] to compute Alexander ideals was written by Jana Archibald at the University of Toronto in the summer of 2005.
Out[6]= {1}
In[7]:= {KnotDet[K], KnotSignature[K]}
Out[7]= { 87, -2 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= q3−4q2 + 7q−10 + 14q−1−14q−2 + 14q−3−11q−4 + 7q−5−4q−6 + q−7
In[9]:= HOMFLYPT[K][a, z]
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
Out[9]= −a2z8 + a4z6−5a2z6 + z6 + 3a4z4−7a2z4 + 3z4 + a4z2 + z2−2a4 + 4a2−1
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= 3a3z9 + 3az9 + 7a4z8 + 13a2z8 + 6z8 + 8a5z7 + 4a3z7 + 4z7a−1 + 7a6z6−9a4z6−35a2z6 + z6a−2−18z6 + 4a7z5−8a5z5−17a3z5−16az5−11z5a−1 + a8z4−7a6z4 + 3a4z4 + 28a2z4−2z4a−2 + 15z4−3a7z3−a5z3 + 9a3z3 + 13az3 + 6z3a−1 + a6z2 + a4z2−3a2z2−3z2 + 2a5z + 2a3z−2a4−4a2−1

[edit] "Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a184,}

Same Jones Polynomial (up to mirroring, q\leftrightarrow q^{-1}): {}

Computer Talk
The above data is available with the Mathematica package KnotTheory`. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:= AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`
Loading KnotTheory` version of May 31, 2006, 14:15:20.091.

Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:= K = Knot["10 112"];
In[4]:= {A = Alexander[K][t], J = Jones[K][q]}
KnotTheory::loading: Loading precomputed data in PD4Knots`.
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[4]= { −t4 + 5t3−11t2 + 17t−19 + 17t−1−11t−2 + 5t−3−t−4, q3−4q2 + 7q−10 + 14q−1−14q−2 + 14q−3−11q−4 + 7q−5−4q−6 + q−7 }
In[5]:= DeleteCases[Select[AllKnots[], (A === Alexander[#][t]) &], K]
KnotTheory::loading: Loading precomputed data in DTCode4KnotsTo11`.
KnotTheory::credits: The GaussCode to PD conversion was written by Siddarth Sankaran at the University of Toronto in the summer of 2005.
Out[5]= {K11a184,}
In[6]:= DeleteCases[ Select[ AllKnots[], (J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) & ], K ]
KnotTheory::loading: Loading precomputed data in Jones4Knots11`.
Out[6]= {}

[edit] Vassiliev invariants

V2 and V3: (2, -2)
V2,1 through V6,9:
V2,1 V3,1 V4,1 V4,2 V4,3 V5,1 V5,2 V5,3 V5,4 V6,1 V6,2 V6,3 V6,4 V6,5 V6,6 V6,7 V6,8 V6,9
8 −16 32 \frac{220}{3} \frac{68}{3} −128 -\frac{736}{3} -\frac{160}{3} −48 \frac{256}{3} 128 \frac{1760}{3} \frac{544}{3} \frac{14431}{15} -\frac{604}{15} \frac{21724}{45} \frac{545}{9} \frac{1471}{15}

V2,1 through V6,9 were provided by Petr Dunin-Barkowski <barkovs@itep.ru>, Andrey Smirnov <asmirnov@itep.ru>, and Alexei Sleptsov <sleptsov@itep.ru> and uploaded on October 2010 by User:Drorbn. Note that they are normalized differently than V2 and V3.

[edit] Khovanov Homology

The coefficients of the monomials trqj are shown, along with their alternating sums χ (fixed j, alternation over r). The squares with yellow highlighting are those on the "critical diagonals", where j−2r = s + 1 or j−2r = s−1, where s = -2 is the signature of 10 112. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
 -6-5-4-3-2-101234χ
7          11
5         3 -3
3        41 3
1       63  -3
-1      84   4
-3     77    0
-5    77     0
-7   47      3
-9  37       -4
-11 14        3
-13 3         -3
-151          1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i = −3 i = −1
r = −6 {\mathbb Z}
r = −5 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r = −4 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r = −3 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r = −2 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r = −1 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r = 0 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{8}
r = 1 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r = 2 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r = 3 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r = 4 {\mathbb Z}_2 {\mathbb Z}

[edit] The Coloured Jones Polynomials

The Coloured Jones Polynomials (in the n + 1-dimensional representation of sl(2))

 n  Jn
2 q10−4q9 + 2q8 + 14q7−23q6−8q5 + 54q4−41q3−47q2 + 106q−36−103q−1 + 143q−2−12q−3−148q−4 + 148q−5 + 19q−6−158q−7 + 117q−8 + 38q−9−123q−10 + 66q−11 + 33q−12−65q−13 + 27q−14 + 14q−15−22q−16 + 9q−17 + 3q−18−4q−19 + q−20
3 q21−4q20 + 2q19 + 9q18−26q16−13q15 + 58q14 + 43q13−83q12−113q11 + 95q10 + 210q9−63q8−323q7−20q6 + 416q5 + 158q4−476q3−326q2 + 487q + 494−434q−1−666q−2 + 368q−3 + 785q−4−246q−5−906q−6 + 148q−7 + 956q−8−7q−9−1001q−10−100q−11 + 966q−12 + 231q−13−908q−14−310q−15 + 769q−16 + 375q−17−610q−18−380q−19 + 433q−20 + 332q−21−268q−22−259q−23 + 155q−24 + 163q−25−75q−26−94q−27 + 47q−28 + 37q−29−24q−30−19q−31 + 23q−32 + 3q−33−12q−34−2q−35 + 5q−36 + 3q−37−4q−38 + q−39
4 q36−4q35 + 2q34 + 9q33−5q32−3q31−32q30 + 11q29 + 70q28 + 16q27−6q26−192q25−83q24 + 210q23 + 235q22 + 230q21−476q20−576q19 + 21q18 + 537q17 + 1140q16−235q15−1220q14−1023q13 + 6q12 + 2270q11 + 1115q10−819q9−2284q8−1952q7 + 2231q6 + 2753q5 + 1209q4−2298q3−4366q2 + 491q + 3222 + 3856q−1−662q−2−5829q−3−1973q−4 + 2227q−5 + 5840q−6 + 1676q−7−6042q−8−4077q−9 + 585q−10 + 6878q−11 + 3792q−12−5519q−13−5562q−14−1089q−15 + 7191q−16 + 5500q−17−4440q−18−6449q−19−2802q−20 + 6586q−21 + 6689q−22−2562q−23−6251q−24−4374q−25 + 4635q−26 + 6680q−27−196q−28−4476q−29−4890q−30 + 1881q−31 + 4953q−32 + 1370q−33−1845q−34−3733q−35−122q−36 + 2403q−37 + 1347q−38−2q−39−1820q−40−576q−41 + 631q−42 + 533q−43 + 437q−44−530q−45−252q−46 + 51q−47 + 22q−48 + 230q−49−95q−50−17q−51−62q−53 + 59q−54−19q−55 + 17q−56 + 9q−57−23q−58 + 8q−59−6q−60 + 5q−61 + 3q−62−4q−63 + q−64
5 q55−4q54 + 2q53 + 9q52−5q51−8q50−9q49−8q48 + 22q47 + 63q46 + 22q45−74q44−133q43−115q42 + 65q41 + 317q40 + 394q39 + 44q38−524q37−848q36−545q35 + 472q34 + 1534q33 + 1564q32 + 104q31−1937q30−2976q29−1751q28 + 1492q27 + 4402q26 + 4276q25 + 445q24−4706q23−7184q22−4190q21 + 3047q20 + 9219q19 + 9027q18 + 1320q17−8945q16−13720q15−8006q14 + 5306q13 + 16469q12 + 15821q11 + 1851q10−15820q9−22871q8−11687q7 + 11014q6 + 27476q5 + 22575q4−2495q3−28442q2−32575q−8697 + 25475q−1 + 40565q−2 + 20767q−3−19455q−4−45362q−5−32362q−6 + 11130q−7 + 47609q−8 + 42408q−9−2411q−10−47264q−11−50446q−12−6411q−13 + 45857q−14 + 56709q−15 + 13975q−16−43492q−17−61496q−18−20995q−19 + 41233q−20 + 65425q−21 + 27032q−22−38415q−23−68688q−24−33287q−25 + 35262q−26 + 71176q−27 + 39436q−28−30398q−29−72336q−30−46097q−31 + 23873q−32 + 71274q−33 + 52009q−34−14921q−35−66993q−36−56686q−37 + 4579q−38 + 59125q−39 + 58319q−40 + 6144q−41−47851q−42−56204q−43−15438q−44 + 34582q−45 + 50059q−46 + 21632q−47−21187q−48−40637q−49−23983q−50 + 9608q−51 + 29845q−52 + 22500q−53−1315q−54−19347q−55−18472q−56−3367q−57 + 10885q−58 + 13263q−59 + 4988q−60−4934q−61−8507q−62−4597q−63 + 1606q−64 + 4668q−65 + 3401q−66 + 46q−67−2334q−68−2126q−69−429q−70 + 904q−71 + 1132q−72 + 495q−73−301q−74−557q−75−296q−76 + 49q−77 + 204q−78 + 169q−79 + 35q−80−72q−81−88q−82−15q−83 + 17q−84 + 12q−85 + 27q−86 + 3q−87−17q−88−3q−89 + 4q−90−6q−91 + 5q−92 + 3q−93−4q−94 + q−95
6 q78−4q77 + 2q76 + 9q75−5q74−8q73−14q72 + 15q71 + 3q70 + 15q69 + 68q68−26q67−86q66−153q65−21q64 + 56q63 + 210q62 + 491q61 + 190q60−250q59−903q58−836q57−620q56 + 367q55 + 2093q54 + 2348q53 + 1470q52−1324q51−3293q50−5067q49−3686q48 + 1697q47 + 6581q46 + 9489q45 + 6049q44−35q43−10547q42−16538q41−12353q40−932q39 + 14894q38 + 23370q37 + 23945q36 + 5342q35−18963q34−35541q33−35784q32−13872q31 + 17450q30 + 51592q29 + 54149q28 + 28028q27−18441q26−63016q25−78088q24−53589q23 + 17336q22 + 79351q21 + 108954q20 + 77740q19−4510q18−98846q17−152582q16−107807q15−3599q14 + 124933q13 + 192532q12 + 152823q11 + 7525q10−166458q9−241469q8−188271q7 + 217q6 + 206600q5 + 308987q4 + 214477q3−33219q2−265788q−363557−220510q−1 + 75072q−2 + 353982q−3 + 406834q−4 + 188234q−5−153273q−6−431978q−7−423976q−8−133940q−9 + 275403q−10 + 501456q−11 + 391334q−12 + 24539q−13−392844q−14−541057q−15−323845q−16 + 142871q−17 + 505015q−18 + 519968q−19 + 183718q−20−310007q−21−582336q−22−450631q−23 + 25931q−24 + 473028q−25 + 587365q−26 + 292811q−27−238220q−28−595039q−29−531071q−30−57346q−31 + 445995q−32 + 634805q−33 + 374947q−34−181645q−35−607338q−36−604029q−37−139053q−38 + 414046q−39 + 680395q−40 + 469528q−41−97920q−42−595466q−43−678658q−44−259600q−45 + 324454q−46 + 686103q−47 + 575432q−48 + 52303q−49−497136q−50−701406q−51−403747q−52 + 142621q−53 + 581643q−54 + 621571q−55 + 235112q−56−287062q−57−596395q−58−481338q−59−75312q−60 + 355455q−61 + 529521q−62 + 342567q−63−44939q−64−369190q−65−415401q−66−208035q−67 + 108338q−68 + 322841q−69 + 304969q−70 + 102154q−71−135601q−72−246291q−73−198805q−74−36886q−75 + 122951q−76 + 177133q−77 + 115245q−78−4372q−79−91715q−80−112645q−81−61507q−82 + 17368q−83 + 65883q−84 + 64252q−85 + 24702q−86−15829q−87−40520q−88−34699q−89−8179q−90 + 14459q−91 + 22294q−92 + 14378q−93 + 2588q−94−9363q−95−12103q−96−5644q−97 + 1258q−98 + 5297q−99 + 4536q−100 + 2604q−101−1348q−102−3173q−103−1815q−104−316q−105 + 938q−106 + 949q−107 + 1020q−108−44q−109−699q−110−399q−111−182q−112 + 113q−113 + 98q−114 + 292q−115 + 43q−116−125q−117−49q−118−46q−119 + 12q−120−20q−121 + 58q−122 + 13q−123−23q−124 + 3q−125−7q−126 + 4q−127−6q−128 + 5q−129 + 3q−130−4q−131 + q−132
7 q105−4q104 + 2q103 + 9q102−5q101−8q100−14q99 + 10q98 + 26q97−4q96 + 20q95 + 20q94−39q93−86q92−130q91−18q90 + 189q89 + 218q88 + 307q87 + 213q86−170q85−575q84−1114q83−937q82 + 19q81 + 1068q80 + 2323q79 + 2755q78 + 1680q77−476q76−4106q75−6650q74−6050q73−2726q72 + 4089q71 + 10866q70 + 14273q69 + 12544q68 + 2272q67−12339q66−24453q65−29328q64−19717q63 + 1745q62 + 27496q61 + 49490q60 + 51247q59 + 29082q58−11109q57−58147q56−86351q55−81674q54−40221q53 + 32573q52 + 103269q51 + 141541q50 + 125723q49 + 45576q48−66572q47−170667q46−221936q45−178291q44−51085q43 + 122230q42 + 276101q41 + 328024q40 + 248280q39 + 44859q38−219684q37−424559q36−479667q35−329249q34 + 1427q33 + 377537q32 + 650984q31 + 674467q30 + 385742q29−116839q28−651507q27−969702q26−876873q25−371829q24 + 388842q23 + 1079864q22 + 1348069q21 + 1026443q20 + 165462q19−892107q18−1646414q17−1716158q16−958013q15 + 358674q14 + 1638995q13 + 2276578q12 + 1862041q11 + 484452q10−1256298q9−2561391q8−2712791q7−1522784q6 + 513794q5 + 2479981q4 + 3353864q3 + 2601296q2 + 498263q−2022845−3681823q−1−3565159q−2−1639523q−3 + 1255163q−4 + 3659966q−5 + 4297269q−6 + 2766712q−7−288310q−8−3324924q−9−4744622q−10−3758569q−11−745625q−12 + 2758223q−13 + 4907479q−14 + 4544110q−15 + 1737138q−16−2065024q−17−4840920q−18−5101061q−19−2600896q−20 + 1345575q−21 + 4615074q−22 + 5448137q−23 + 3301323q−24−677090q−25−4314371q−26−5632970q−27−3829595q−28 + 110075q−29 + 4003226q−30 + 5712096q−31 + 4213556q−32 + 338862q−33−3735338q−34−5745173q−35−4491455q−36−677796q−37 + 3532924q−38 + 5780653q−39 + 4718685q−40 + 940126q−41−3399370q−42−5851981q−43−4945050q−44−1177198q−45 + 3304370q−46 + 5969456q−47 + 5218300q−48 + 1449833q−49−3198548q−50−6111765q−51−5558109q−52−1818171q−53 + 3006066q−54 + 6225844q−55 + 5956977q−56 + 2321108q−57−2654449q−58−6228347q−59−6356639q−60−2958913q−61 + 2081549q−62 + 6023339q−63 + 6666957q−64 + 3679256q−65−1277096q−66−5532402q−67−6768298q−68−4374923q−69 + 287316q−70 + 4722091q−71 + 6558152q−72 + 4910362q−73 + 771904q−74−3634733q−75−5980942q−76−5151849q−77−1743975q−78 + 2384254q−79 + 5059882q−80 + 5018303q−81 + 2473724q−82−1138645q−83−3903117q−84−4509092q−85−2849932q−86 + 70836q−87 + 2672182q−88 + 3709833q−89 + 2847577q−90 + 692290q−91−1543588q−92−2768946q−93−2524928q−94−1100025q−95 + 650464q−96 + 1846579q−97 + 2004985q−98 + 1188097q−99−55431q−100−1072392q−101−1427253q−102−1050542q−103−255713q−104 + 512977q−105 + 906860q−106 + 801291q−107 + 350733q−108−169154q−109−508679q−110−538782q−111−318086q−112−1268q−113 + 247773q−114 + 321322q−115 + 232766q−116 + 60507q−117−99750q−118−171068q−119−147523q−120−62536q−121 + 30928q−122 + 81618q−123 + 81789q−124 + 44758q−125−4126q−126−35039q−127−41684q−128−26819q−129−2055q−130 + 14365q−131 + 19478q−132 + 13821q−133 + 2242q−134−5527q−135−8897q−136−6990q−137−1259q−138 + 2497q−139 + 4247q−140 + 3300q−141 + 370q−142−1068q−143−1866q−144−1680q−145−298q−146 + 445q−147 + 1047q−148 + 895q−149 + 11q−150−214q−151−358q−152−390q−153−94q−154 + q−155 + 207q−156 + 237q−157−17q−158−23q−159−45q−160−59q−161−9q−162−25q−163 + 26q−164 + 44q−165−13q−166−3q−167−q−168−7q−169 + 4q−170−6q−171 + 5q−172 + 3q−173−4q−174 + q−175

[edit] Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session, or any of the Computer Talk sections above.

[edit] Modifying This Page

Read me first: Modifying Knot Pages

See/edit the Rolfsen Knot Page master template (intermediate).

See/edit the Rolfsen_Splice_Base (expert).

Back to the top.

10_111

10_113

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