10 72

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10_71

10_73

Contents

Image:10 72.gif
(KnotPlot image)

See the full Rolfsen Knot Table.

Visit 10 72's page at the Knot Server (KnotPlot driven, includes 3D interactive images!)

Visit 10_72's page at Knotilus!

Visit 10 72's page at the original Knot Atlas!

[edit 10 72 Quick Notes]
[edit 10 72 Further Notes and Views]


[edit] Knot presentations

Planar diagram presentation X4251 X10,6,11,5 X8394 X2,9,3,10 X16,8,17,7 X18,12,19,11 X20,14,1,13 X12,20,13,19 X14,18,15,17 X6,16,7,15
Gauss code 1, -4, 3, -1, 2, -10, 5, -3, 4, -2, 6, -8, 7, -9, 10, -5, 9, -6, 8, -7
Dowker-Thistlethwaite code 4 8 10 16 2 18 20 6 14 12
Conway Notation [211,3,2+]


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

Length is 11, width is 4,

Braid index is 4

Image:10 72_ML.gif Image:10 72_AP.gif
[{9, 12}, {11, 3}, {12, 10}, {6, 11}, {5, 7}, {4, 6}, {8, 5}, {7, 2}, {3, 1}, {2, 9}, {1, 8}, {10, 4}]

[edit Notes on presentations of 10 72]


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 72"];
In[4]:= PD[K]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= X4251 X10,6,11,5 X8394 X2,9,3,10 X16,8,17,7 X18,12,19,11 X20,14,1,13 X12,20,13,19 X14,18,15,17 X6,16,7,15
In[5]:= GaussCode[K]
Out[5]= 1, -4, 3, -1, 2, -10, 5, -3, 4, -2, 6, -8, 7, -9, 10, -5, 9, -6, 8, -7
In[6]:= DTCode[K]
Out[6]= 4 8 10 16 2 18 20 6 14 12

(The path below may be different on your system)

In[7]:= AppendTo[$Path, "C:/bin/LinKnot/"];
In[8]:= ConwayNotation[K]
Out[8]= [211,3,2+]
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(4,{1,1,1,1,2,2,−1,2,−3,2,−3})
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]= { 4, 11, 4 }
In[11]:= Show[BraidPlot[br]]
Image:BraidPart1.gifImage:BraidPart1.gifImage:BraidPart1.gifImage:BraidPart1.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart3.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gif
Image:BraidPart2.gifImage:BraidPart2.gifImage:BraidPart2.gifImage:BraidPart2.gifImage:BraidPart1.gifImage:BraidPart1.gifImage:BraidPart4.gifImage:BraidPart1.gifImage:BraidPart0.gifImage:BraidPart1.gifImage:BraidPart0.gif
Image:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart2.gifImage:BraidPart2.gifImage:BraidPart0.gifImage:BraidPart2.gifImage:BraidPart3.gifImage:BraidPart2.gifImage:BraidPart3.gif
Image:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart4.gifImage:BraidPart0.gifImage:BraidPart4.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 72_ML.gif
Out[12]= -Graphics-
In[13]:= ap = ArcPresentation[K]
Out[13]= ArcPresentation[{9, 12}, {11, 3}, {12, 10}, {6, 11}, {5, 7}, {4, 6}, {8, 5}, {7, 2}, {3, 1}, {2, 9}, {1, 8}, {10, 4}]
In[14]:= Draw[ap]
Image:10 72_AP.gif
Out[14]= -Graphics-

[edit] Three dimensional invariants

Symmetry type Reversible
Unknotting number 2
3-genus 3
Bridge index 3
Super bridge index Missing
Nakanishi index 1
Maximal Thurston-Bennequin number [1][-13]
Hyperbolic Volume 12.9296
A-Polynomial See Data:10 72/A-polynomial

[edit Notes for 10 72's three dimensional invariants]

[edit] Four dimensional invariants

Smooth 4 genus 2
Topological 4 genus 2
Concordance genus 3
Rasmussen s-Invariant -4

[edit Notes for 10 72's four dimensional invariants]

[edit] Polynomial invariants

Alexander polynomial −2t3 + 9t2−16t + 19−16t−1 + 9t−2−2t−3
Conway polynomial −2z6−3z4 + 2z2 + 1
2nd Alexander ideal (db, data sources) {1}
Determinant and Signature { 73, 4 }
Jones polynomial q10−4q9 + 7q8−10q7 + 12q6−12q5 + 11q4−8q3 + 5q2−2q + 1
HOMFLY-PT polynomial (db, data sources) −z6a−4−z6a−6 + z4a−2−3z4a−4−2z4a−6 + z4a−8 + 3z2a−2−3z2a−4 + z2a−6 + z2a−8 + 2a−2−2a−4 + 2a−6−a−8
Kauffman polynomial (db, data sources) z9a−5 + z9a−7 + 2z8a−4 + 6z8a−6 + 4z8a−8 + 2z7a−3 + 4z7a−5 + 9z7a−7 + 7z7a−9 + z6a−2−2z6a−4−8z6a−6 + 2z6a−8 + 7z6a−10−6z5a−3−14z5a−5−19z5a−7−7z5a−9 + 4z5a−11−4z4a−2−6z4a−4−4z4a−6−11z4a−8−8z4a−10 + z4a−12 + 5z3a−3 + 11z3a−5 + 9z3a−7−3z3a−11 + 5z2a−2 + 8z2a−4 + 7z2a−6 + 6z2a−8 + 2z2a−10−za−3−3za−5−za−7 + za−9−2a−2−2a−4−2a−6−a−8
The A2 invariant 1 + q−4 + 2q−6−2q−8 + 2q−10−2q−12 + 2q−16−q−18 + 3q−20−2q−22−2q−28 + q−30
The G2 invariant q−2−q−4 + 4q−6−5q−8 + 6q−10−5q−12 + 12q−16−22q−18 + 35q−20−37q−22 + 28q−24−q−26−37q−28 + 79q−30−104q−32 + 101q−34−61q−36−13q−38 + 92q−40−150q−42 + 165q−44−120q−46 + 34q−48 + 60q−50−132q−52 + 142q−54−98q−56 + 16q−58 + 68q−60−110q−62 + 91q−64−19q−66−73q−68 + 150q−70−169q−72 + 122q−74−18q−76−106q−78 + 206q−80−240q−82 + 200q−84−92q−86−41q−88 + 152q−90−206q−92 + 185q−94−103q−96−5q−98 + 89q−100−120q−102 + 85q−104−9q−106−71q−108 + 117q−110−105q−112 + 40q−114 + 45q−116−124q−118 + 161q−120−142q−122 + 80q−124 + 3q−126−77q−128 + 118q−130−121q−132 + 93q−134−45q−136−2q−138 + 34q−140−53q−142 + 50q−144−34q−146 + 19q−148−2q−150−6q−152 + 9q−154−10q−156 + 6q−158−3q−160 + q−162
Further Quantum Invariants
Further quantum knot invariants for 10_72.

A1 Invariants.

Weight Invariant
1 q−q−1 + 3q−3−3q−5 + 3q−7−q−9 + 2q−13−3q−15 + 3q−17−3q−19 + q−21
2 q6−q4−q2 + 5−3q−2−7q−4 + 14q−6−q−8−19q−10 + 20q−12 + 9q−14−28q−16 + 13q−18 + 17q−20−21q−22−q−24 + 15q−26−3q−28−14q−30 + 4q−32 + 18q−34−19q−36−9q−38 + 29q−40−14q−42−16q−44 + 23q−46−4q−48−11q−50 + 9q−52−3q−56 + q−58
3 q15−q13−q11 + q9 + 4q7−3q5−8q3 + 3q + 18q−1−q−3−30q−5−9q−7 + 48q−9 + 28q−11−58q−13−59q−15 + 61q−17 + 98q−19−48q−21−129q−23 + 19q−25 + 153q−27 + 20q−29−159q−31−57q−33 + 145q−35 + 88q−37−119q−39−111q−41 + 85q−43 + 117q−45−45q−47−114q−49 + q−51 + 104q−53 + 44q−55−85q−57−88q−59 + 54q−61 + 127q−63−21q−65−152q−67−19q−69 + 163q−71 + 53q−73−147q−75−79q−77 + 120q−79 + 89q−81−85q−83−82q−85 + 51q−87 + 63q−89−25q−91−43q−93 + 10q−95 + 27q−97−7q−99−11q−101 + q−103 + 6q−105−3q−109 + q−111
4 q28−q26−q24 + q22 + 4q18−5q16−6q14 + 5q12 + 5q10 + 18q8−15q6−33q4−3q2 + 23 + 78q−2−5q−4−98q−6−84q−8 + 3q−10 + 215q−12 + 128q−14−118q−16−279q−18−209q−20 + 297q−22 + 428q−24 + 119q−26−401q−28−644q−30 + 61q−32 + 639q−34 + 620q−36−156q−38−974q−40−476q−42 + 449q−44 + 1022q−46 + 387q−48−883q−50−917q−52−38q−54 + 1021q−56 + 828q−58−472q−60−999q−62−461q−64 + 719q−66 + 942q−68−44q−70−804q−72−656q−74 + 331q−76 + 821q−78 + 317q−80−491q−82−723q−84−102q−86 + 580q−88 + 670q−90−56q−92−698q−94−617q−96 + 175q−98 + 946q−100 + 498q−102−459q−104−1030q−106−387q−108 + 899q−110 + 937q−112 + 26q−114−1038q−116−828q−118 + 488q−120 + 934q−122 + 449q−124−625q−126−836q−128 + 46q−130 + 542q−132 + 507q−134−184q−136−502q−138−106q−140 + 160q−142 + 296q−144 + 4q−146−191q−148−59q−150 + 9q−152 + 107q−154 + 14q−156−54q−158−7q−160−8q−162 + 27q−164 + 3q−166−14q−168 + q−170−2q−172 + 6q−174−3q−178 + q−180
5 q45−q43−q41 + q39 + 2q33−3q31−5q29 + 5q27 + 8q25 + 3q23−17q19−26q17 + 2q15 + 41q13 + 51q11 + 21q9−57q7−124q5−82q3 + 67q + 218q−1 + 222q−3−3q−5−328q−7−457q−9−197q−11 + 358q−13 + 790q−15 + 605q−17−209q−19−1082q−21−1234q−23−299q−25 + 1194q−27 + 1998q−29 + 1189q−31−865q−33−2625q−35−2445q−37−56q−39 + 2858q−41 + 3794q−43 + 1550q−45−2392q−47−4857q−49−3454q−51 + 1144q−53 + 5326q−55 + 5348q−57 + 728q−59−4937q−61−6824q−63−2941q−65 + 3758q−67 + 7597q−69 + 5017q−71−2030q−73−7521q−75−6611q−77 + 82q−79 + 6752q−81 + 7517q−83 + 1680q−85−5546q−87−7698q−89−3024q−91 + 4151q−93 + 7346q−95 + 3877q−97−2855q−99−6657q−101−4284q−103 + 1736q−105 + 5822q−107 + 4459q−109−785q−111−5006q−113−4552q−115−104q−117 + 4201q−119 + 4693q−121 + 1118q−123−3345q−125−4922q−127−2337q−129 + 2293q−131 + 5139q−133 + 3769q−135−889q−137−5149q−139−5329q−141−879q−143 + 4753q−145 + 6710q−147 + 2931q−149−3746q−151−7655q−153−5034q−155 + 2215q−157 + 7831q−159 + 6774q−161−248q−163−7158q−165−7847q−167−1729q−169 + 5726q−171 + 7985q−173 + 3349q−175−3837q−177−7234q−179−4287q−181 + 1912q−183 + 5836q−185 + 4447q−187−369q−189−4139q−191−3938q−193−624q−195 + 2551q−197 + 3065q−199 + 1021q−201−1342q−203−2076q−205−1000q−207 + 555q−209 + 1241q−211 + 786q−213−163q−215−675q−217−480q−219 + 4q−221 + 300q−223 + 273q−225 + 43q−227−150q−229−123q−231−12q−233 + 47q−235 + 51q−237 + 11q−239−21q−241−25q−243 + 2q−245 + 15q−247 + 3q−249−4q−251−2q−253−2q−255−2q−257 + 6q−259−3q−263 + q−265

A2 Invariants.

Weight Invariant
1,0 1 + q−4 + 2q−6−2q−8 + 2q−10−2q−12 + 2q−16−q−18 + 3q−20−2q−22−2q−28 + q−30
1,1 q4−2q2 + 8−16q−2 + 37q−4−68q−6 + 120q−8−194q−10 + 293q−12−410q−14 + 534q−16−642q−18 + 719q−20−726q−22 + 654q−24−488q−26 + 243q−28 + 78q−30−436q−32 + 784q−34−1095q−36 + 1324q−38−1450q−40 + 1446q−42−1320q−44 + 1088q−46−774q−48 + 418q−50−66q−52−248q−54 + 490q−56−644q−58 + 709q−60−698q−62 + 632q−64−528q−66 + 409q−68−298q−70 + 206q−72−128q−74 + 71q−76−38q−78 + 18q−80−6q−82 + q−84
2,0 q4−1 + q−2 + 4q−4−5q−8 + q−10 + 9q−12−3q−14−10q−16 + 6q−18 + 9q−20−7q−22−6q−24 + 10q−26 + 6q−28−7q−30 + 3q−32 + 9q−34−8q−36−6q−38 + 7q−40−5q−42−11q−44 + 4q−46 + 10q−48−7q−50−6q−52 + 10q−54 + 5q−56−10q−58−3q−60 + 7q−62−2q−66 + 3q−70−q−72−2q−74 + q−76

A3 Invariants.

Weight Invariant
0,1,0 1−q−2 + 2q−4 + 3q−6−4q−8 + 5q−10 + 6q−12−13q−14 + 9q−16 + 9q−18−21q−20 + 11q−22 + 12q−24−21q−26 + 6q−28 + 12q−30−10q−32−3q−34 + 6q−36 + 5q−38−9q−40−6q−42 + 18q−44−10q−46−14q−48 + 23q−50−7q−52−15q−54 + 17q−56−2q−58−9q−60 + 8q−62−3q−66 + q−68
1,0,0 q−1 + 2q−5 + 3q−9−2q−11 + 2q−13−3q−15−q−19 + q−21 + 2q−23 + 3q−27−2q−29 + q−31−3q−33 + q−35−2q−37 + q−39

A4 Invariants.

Weight Invariant
0,1,0,0 q−2 + q−6 + 3q−8 + 2q−10 + 3q−14 + 3q−16−3q−18−4q−20 + 6q−22 + 3q−24−9q−26 + 4q−28 + 17q−30−6q−32−13q−34 + 10q−36 + 5q−38−19q−40−7q−42 + 15q−44−q−46−11q−48 + 14q−50 + 12q−52−15q−54 + 14q−58−11q−60−14q−62 + 10q−64 + 6q−66−13q−68−3q−70 + 13q−72 + q−74−9q−76 + 3q−78 + 6q−80−3q−82−2q−84 + q−86
1,0,0,0 q−2 + 2q−6 + q−8 + q−10 + 3q−12−2q−14 + 2q−16−3q−18−q−20−q−22−q−24 + q−26 + q−28 + 3q−30 + 3q−34−2q−36 + q−38−2q−40−2q−42 + q−44−2q−46 + q−48

B2 Invariants.

Weight Invariant
0,1 1−q−2 + 4q−4−5q−6 + 10q−8−13q−10 + 18q−12−21q−14 + 23q−16−23q−18 + 19q−20−13q−22 + 2q−24 + 9q−26−20q−28 + 32q−30−40q−32 + 47q−34−46q−36 + 43q−38−35q−40 + 24q−42−12q−44 + 10q−48−19q−50 + 23q−52−25q−54 + 23q−56−20q−58 + 15q−60−10q−62 + 6q−64−3q−66 + q−68
1,0 q2−q−2−q−4 + 3q−6 + 4q−8−q−10−7q−12−2q−14 + 10q−16 + 11q−18−7q−20−18q−22−2q−24 + 22q−26 + 15q−28−16q−30−24q−32 + 4q−34 + 27q−36 + 8q−38−21q−40−15q−42 + 12q−44 + 17q−46−6q−48−16q−50 + 2q−52 + 16q−54−16q−58−4q−60 + 16q−62 + 9q−64−15q−66−15q−68 + 12q−70 + 20q−72−6q−74−25q−76−5q−78 + 24q−80 + 16q−82−15q−84−23q−86 + 2q−88 + 20q−90 + 9q−92−11q−94−12q−96 + 2q−98 + 9q−100 + 3q−102−3q−104−3q−106 + q−110

D4 Invariants.

Weight Invariant
1,0,0,0 q−2−q−4 + 3q−6−2q−8 + 7q−10−6q−12 + 11q−14−10q−16 + 16q−18−17q−20 + 17q−22−18q−24 + 18q−26−17q−28 + 10q−30−7q−32 + 3q−34 + 5q−36−14q−38 + 19q−40−21q−42 + 31q−44−35q−46 + 35q−48−34q−50 + 36q−52−32q−54 + 23q−56−22q−58 + 16q−60−6q−62−2q−64 + 3q−66−10q−68 + 18q−70−18q−72 + 16q−74−20q−76 + 20q−78−14q−80 + 12q−82−12q−84 + 9q−86−4q−88 + 3q−90−3q−92 + q−94

G2 Invariants.

Weight Invariant
1,0 q−2−q−4 + 4q−6−5q−8 + 6q−10−5q−12 + 12q−16−22q−18 + 35q−20−37q−22 + 28q−24−q−26−37q−28 + 79q−30−104q−32 + 101q−34−61q−36−13q−38 + 92q−40−150q−42 + 165q−44−120q−46 + 34q−48 + 60q−50−132q−52 + 142q−54−98q−56 + 16q−58 + 68q−60−110q−62 + 91q−64−19q−66−73q−68 + 150q−70−169q−72 + 122q−74−18q−76−106q−78 + 206q−80−240q−82 + 200q−84−92q−86−41q−88 + 152q−90−206q−92 + 185q−94−103q−96−5q−98 + 89q−100−120q−102 + 85q−104−9q−106−71q−108 + 117q−110−105q−112 + 40q−114 + 45q−116−124q−118 + 161q−120−142q−122 + 80q−124 + 3q−126−77q−128 + 118q−130−121q−132 + 93q−134−45q−136−2q−138 + 34q−140−53q−142 + 50q−144−34q−146 + 19q−148−2q−150−6q−152 + 9q−154−10q−156 + 6q−158−3q−160 + q−162

.

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 72"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= −2t3 + 9t2−16t + 19−16t−1 + 9t−2−2t−3
In[5]:= Conway[K][z]
Out[5]= −2z6−3z4 + 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]= { 73, 4 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= q10−4q9 + 7q8−10q7 + 12q6−12q5 + 11q4−8q3 + 5q2−2q + 1
In[9]:= HOMFLYPT[K][a, z]
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
Out[9]= −z6a−4−z6a−6 + z4a−2−3z4a−4−2z4a−6 + z4a−8 + 3z2a−2−3z2a−4 + z2a−6 + z2a−8 + 2a−2−2a−4 + 2a−6−a−8
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= z9a−5 + z9a−7 + 2z8a−4 + 6z8a−6 + 4z8a−8 + 2z7a−3 + 4z7a−5 + 9z7a−7 + 7z7a−9 + z6a−2−2z6a−4−8z6a−6 + 2z6a−8 + 7z6a−10−6z5a−3−14z5a−5−19z5a−7−7z5a−9 + 4z5a−11−4z4a−2−6z4a−4−4z4a−6−11z4a−8−8z4a−10 + z4a−12 + 5z3a−3 + 11z3a−5 + 9z3a−7−3z3a−11 + 5z2a−2 + 8z2a−4 + 7z2a−6 + 6z2a−8 + 2z2a−10−za−3−3za−5−za−7 + za−9−2a−2−2a−4−2a−6−a−8

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

Same Alexander/Conway Polynomial: {}

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 72"];
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]= { −2t3 + 9t2−16t + 19−16t−1 + 9t−2−2t−3, q10−4q9 + 7q8−10q7 + 12q6−12q5 + 11q4−8q3 + 5q2−2q + 1 }
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]= {}
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, 4)

[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 = 4 is the signature of 10 72. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
 -2-1012345678χ
21          11
19         3 -3
17        41 3
15       63  -3
13      64   2
11     66    0
9    56     -1
7   36      3
5  25       -3
3 14        3
1 1         -1
-11          1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i = 3 i = 5
r = −2 {\mathbb Z}
r = −1 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r = 0 {\mathbb Z}^{4}\oplus{\mathbb Z}_2 {\mathbb Z}^{2}
r = 1 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r = 2 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r = 3 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r = 4 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r = 5 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r = 6 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r = 7 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r = 8 {\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 q28−4q27 + 3q26 + 10q25−24q24 + 10q23 + 37q22−63q21 + 12q20 + 80q19−101q18 + 2q17 + 117q16−115q15−16q14 + 128q13−97q12−32q11 + 108q10−59q9−36q8 + 67q7−22q6−25q5 + 28q4−4q3−10q2 + 7q−2q−1 + q−2
3 q54−4q53 + 3q52 + 6q51−4q50−16q49 + 7q48 + 40q47−21q46−69q45 + 25q44 + 128q43−33q42−202q41 + 22q40 + 302q39−2q38−401q37−46q36 + 502q35 + 108q34−583q33−179q32 + 633q31 + 256q30−656q29−321q28 + 636q27 + 385q26−596q25−424q24 + 521q23 + 454q22−434q21−456q20 + 325q19 + 446q18−227q17−399q16 + 123q15 + 344q14−48q13−266q12−11q11 + 196q10 + 33q9−120q8−48q7 + 76q6 + 34q5−34q4−28q3 + 19q2 + 13q−5−9q−1 + 4q−2 + 2q−3−2q−5 + q−6
4 q88−4q87 + 3q86 + 6q85−8q84 + 4q83−19q82 + 20q81 + 30q80−43q79 + 5q78−66q77 + 88q76 + 123q75−141q74−63q73−198q72 + 283q71 + 415q70−277q69−329q68−594q67 + 601q66 + 1106q65−242q64−825q63−1476q62 + 812q61 + 2180q60 + 243q59−1271q58−2792q57 + 602q56 + 3244q55 + 1154q54−1309q53−4078q52−41q51 + 3815q50 + 2111q49−861q48−4849q47−833q46 + 3734q45 + 2753q44−135q43−4939q42−1515q41 + 3113q40 + 2985q39 + 673q38−4435q37−2005q36 + 2126q35 + 2837q34 + 1433q33−3449q32−2228q31 + 946q30 + 2299q29 + 1960q28−2149q27−2035q26−113q25 + 1420q24 + 1994q23−879q22−1400q21−686q20 + 495q19 + 1496q18−61q17−624q16−667q15−83q14 + 791q13 + 182q12−104q11−358q10−214q9 + 285q8 + 112q7 + 57q6−112q5−127q4 + 73q3 + 25q2 + 43q−19−44q−1 + 18q−2−q−3 + 13q−4−q−5−11q−6 + 5q−7−q−8 + 2q−9−2q−11 + q−12

[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_71

10_73

Retrieved from "http://katlas.org/wiki/10_72"
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