10 56

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10_55

10_57

Contents

Image:10 56.gif
(KnotPlot image)

See the full Rolfsen Knot Table.

Visit 10 56's page at the Knot Server (KnotPlot driven, includes 3D interactive images!)

Visit 10_56's page at Knotilus!

Visit 10 56's page at the original Knot Atlas!

[edit 10 56 Quick Notes]
[edit 10 56 Further Notes and Views]


[edit] Knot presentations

Planar diagram presentation X4251 X10,4,11,3 X12,6,13,5 X18,14,19,13 X16,7,17,8 X6,17,7,18 X20,16,1,15 X14,20,15,19 X8,12,9,11 X2,10,3,9
Gauss code 1, -10, 2, -1, 3, -6, 5, -9, 10, -2, 9, -3, 4, -8, 7, -5, 6, -4, 8, -7
Dowker-Thistlethwaite code 4 10 12 16 2 8 18 20 6 14
Conway Notation [221,3,2]


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

Length is 11, width is 4,

Braid index is 4

Image:10 56_ML.gif Image:10 56_AP.gif
[{3, 12}, {2, 5}, {1, 3}, {9, 4}, {10, 8}, {7, 9}, {8, 2}, {6, 11}, {5, 7}, {4, 6}, {12, 10}, {11, 1}]

[edit Notes on presentations of 10 56]


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 56"];
In[4]:= PD[K]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= X4251 X10,4,11,3 X12,6,13,5 X18,14,19,13 X16,7,17,8 X6,17,7,18 X20,16,1,15 X14,20,15,19 X8,12,9,11 X2,10,3,9
In[5]:= GaussCode[K]
Out[5]= 1, -10, 2, -1, 3, -6, 5, -9, 10, -2, 9, -3, 4, -8, 7, -5, 6, -4, 8, -7
In[6]:= DTCode[K]
Out[6]= 4 10 12 16 2 8 18 20 6 14

(The path below may be different on your system)

In[7]:= AppendTo[$Path, "C:/bin/LinKnot/"];
In[8]:= ConwayNotation[K]
Out[8]= [221,3,2]
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,{1,1,1,2,−1,2,−3,2,2,2,−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:BraidPart0.gifImage:BraidPart3.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gif
Image:BraidPart2.gifImage:BraidPart2.gifImage:BraidPart2.gifImage:BraidPart1.gifImage:BraidPart4.gifImage:BraidPart1.gifImage:BraidPart0.gifImage:BraidPart1.gifImage:BraidPart1.gifImage:BraidPart1.gifImage:BraidPart0.gif
Image:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart2.gifImage:BraidPart0.gifImage:BraidPart2.gifImage:BraidPart3.gifImage:BraidPart2.gifImage:BraidPart2.gifImage:BraidPart2.gifImage:BraidPart3.gif
Image:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart4.gifImage:BraidPart0.gifImage:BraidPart0.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 56_ML.gif
Out[12]= -Graphics-
In[13]:= ap = ArcPresentation[K]
Out[13]= ArcPresentation[{3, 12}, {2, 5}, {1, 3}, {9, 4}, {10, 8}, {7, 9}, {8, 2}, {6, 11}, {5, 7}, {4, 6}, {12, 10}, {11, 1}]
In[14]:= Draw[ap]
Image:10 56_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 12.3988
A-Polynomial See Data:10 56/A-polynomial

[edit Notes for 10 56'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 56's four dimensional invariants]

[edit] Polynomial invariants

Alexander polynomial −2t3 + 8t2−14t + 17−14t−1 + 8t−2−2t−3
Conway polynomial −2z6−4z4 + 1
2nd Alexander ideal (db, data sources) {1}
Determinant and Signature { 65, 4 }
Jones polynomial q10−3q9 + 6q8−9q7 + 10q6−11q5 + 10q4−7q3 + 5q2−2q + 1
HOMFLY-PT polynomial (db, data sources) −z6a−4−z6a−6 + z4a−2−3z4a−4−3z4a−6 + z4a−8 + 3z2a−2−2z2a−4−3z2a−6 + 2z2a−8 + 2a−2−2a−6 + a−8
Kauffman polynomial (db, data sources) z9a−5 + z9a−7 + 2z8a−4 + 6z8a−6 + 4z8a−8 + 2z7a−3 + 3z7a−5 + 7z7a−7 + 6z7a−9 + z6a−2−3z6a−4−14z6a−6−5z6a−8 + 5z6a−10−6z5a−3−13z5a−5−21z5a−7−11z5a−9 + 3z5a−11−4z4a−2−3z4a−4 + 12z4a−6 + 4z4a−8−6z4a−10 + z4a−12 + 4z3a−3 + 11z3a−5 + 21z3a−7 + 11z3a−9−3z3a−11 + 5z2a−2 + 3z2a−4−7z2a−6−2z2a−8 + 2z2a−10−z2a−12−4za−5−8za−7−4za−9−2a−2 + 2a−6 + a−8
The A2 invariant 1 + q−4 + 2q−6−q−8 + 3q−10−q−12−3q−18 + q−20−2q−22 + q−24 + q−26−q−28 + q−30
The G2 invariant q−2−q−4 + 4q−6−5q−8 + 6q−10−4q−12−q−14 + 12q−16−20q−18 + 30q−20−30q−22 + 20q−24 + 2q−26−32q−28 + 65q−30−79q−32 + 73q−34−37q−36−17q−38 + 73q−40−108q−42 + 110q−44−70q−46 + 6q−48 + 57q−50−93q−52 + 86q−54−39q−56−18q−58 + 67q−60−80q−62 + 48q−64 + 9q−66−79q−68 + 121q−70−120q−72 + 71q−74 + 7q−76−92q−78 + 146q−80−158q−82 + 116q−84−44q−86−46q−88 + 109q−90−130q−92 + 103q−94−38q−96−28q−98 + 71q−100−73q−102 + 35q−104 + 21q−106−68q−108 + 89q−110−64q−112 + 13q−114 + 50q−116−94q−118 + 107q−120−83q−122 + 37q−124 + 12q−126−54q−128 + 72q−130−68q−132 + 50q−134−20q−136−4q−138 + 19q−140−29q−142 + 26q−144−19q−146 + 11q−148−2q−150−3q−152 + 5q−154−6q−156 + 4q−158−2q−160 + q−162
Further Quantum Invariants
Further quantum knot invariants for 10_56.

A1 Invariants.

Weight Invariant
1 q−q−1 + 3q−3−2q−5 + 3q−7−q−9−q−11 + q−13−3q−15 + 3q−17−2q−19 + q−21
2 q6−q4−q2 + 5−2q−2−6q−4 + 12q−6−16q−10 + 14q−12 + 9q−14−21q−16 + 7q−18 + 14q−20−15q−22−4q−24 + 11q−26−11q−30 + 2q−32 + 16q−34−13q−36−9q−38 + 22q−40−9q−42−13q−44 + 16q−46−q−48−8q−50 + 5q−52−2q−56 + q−58
3 q15−q13−q11 + q9 + 4q7−2q5−7q3 + 2q + 15q−1−23q−5−9q−7 + 36q−9 + 24q−11−39q−13−48q−15 + 36q−17 + 74q−19−18q−21−93q−23−9q−25 + 101q−27 + 40q−29−96q−31−66q−33 + 79q−35 + 81q−37−55q−39−93q−41 + 34q−43 + 86q−45−5q−47−79q−49−19q−51 + 63q−53 + 47q−55−44q−57−70q−59 + 18q−61 + 91q−63 + 11q−65−103q−67−39q−69 + 99q−71 + 63q−73−83q−75−72q−77 + 56q−79 + 73q−81−33q−83−59q−85 + 12q−87 + 39q−89−23q−93−2q−95 + 12q−97−4q−101 + 2q−105−2q−109 + q−111
4 q28−q26−q24 + q22 + 4q18−4q16−5q14 + 4q12 + 3q10 + 15q8−12q6−25q4−q2 + 16 + 59q−2−4q−4−70q−6−60q−8−6q−10 + 150q−12 + 97q−14−64q−16−183q−18−171q−20 + 163q−22 + 285q−24 + 140q−26−196q−28−447q−30−76q−32 + 330q−34 + 472q−36 + 77q−38−549q−40−451q−42 + 66q−44 + 622q−46 + 471q−48−333q−50−639q−52−307q−54 + 470q−56 + 670q−58−5q−60−555q−62−503q−64 + 215q−66 + 613q−68 + 210q−70−357q−72−501q−74 + q−76 + 442q−78 + 336q−80−148q−82−439q−84−211q−86 + 226q−88 + 456q−90 + 128q−92−320q−94−472q−96−95q−98 + 515q−100 + 467q−102−58q−104−630q−106−481q−108 + 349q−110 + 659q−112 + 306q−114−489q−116−683q−118 + 15q−120 + 508q−122 + 499q−124−142q−126−532q−128−199q−130 + 172q−132 + 379q−134 + 80q−136−222q−138−156q−140−33q−142 + 150q−144 + 84q−146−40q−148−42q−150−47q−152 + 31q−154 + 23q−156−5q−158 + 4q−160−16q−162 + 5q−164 + 3q−166−3q−168 + 4q−170−3q−172 + 2q−174−2q−178 + q−180
5 q45−q43−q41 + q39 + 2q33−2q31−4q29 + 4q27 + 6q25 + q23−q21−13q19−18q17 + 4q15 + 34q13 + 37q11 + 10q9−47q7−91q5−55q3 + 55q + 158q−1 + 156q−3−4q−5−225q−7−314q−9−149q−11 + 214q−13 + 519q−15 + 435q−17−62q−19−637q−21−827q−23−347q−25 + 578q−27 + 1225q−29 + 958q−31−169q−33−1401q−35−1708q−37−617q−39 + 1206q−41 + 2328q−43 + 1674q−45−490q−47−2576q−49−2787q−51−672q−53 + 2279q−55 + 3643q−57 + 2059q−59−1415q−61−3997q−63−3386q−65 + 148q−67 + 3776q−69 + 4349q−71 + 1221q−73−3045q−75−4789q−77−2428q−79 + 2023q−81 + 4709q−83 + 3267q−85−997q−87−4217q−89−3627q−91 + 95q−93 + 3529q−95 + 3648q−97 + 510q−99−2830q−101−3367q−103−897q−105 + 2183q−107 + 3071q−109 + 1118q−111−1684q−113−2771q−115−1338q−117 + 1199q−119 + 2594q−121 + 1672q−123−685q−125−2452q−127−2175q−129−16q−131 + 2274q−133 + 2790q−135 + 959q−137−1893q−139−3410q−141−2108q−143 + 1204q−145 + 3818q−147 + 3325q−149−161q−151−3825q−153−4394q−155−1116q−157 + 3324q−159 + 5039q−161 + 2407q−163−2327q−165−5079q−167−3463q−169 + 1071q−171 + 4497q−173 + 3972q−175 + 208q−177−3418q−179−3921q−181−1180q−183 + 2171q−185 + 3312q−187 + 1679q−189−998q−191−2423q−193−1712q−195 + 161q−197 + 1511q−199 + 1397q−201 + 280q−203−752q−205−956q−207−412q−209 + 269q−211 + 557q−213 + 340q−215−36q−217−252q−219−220q−221−52q−223 + 94q−225 + 121q−227 + 44q−229−23q−231−45q−233−31q−235−3q−237 + 19q−239 + 19q−241−2q−243−7q−245−2q−247−4q−249 + 5q−253 + q−255−3q−257 + 2q−259−2q−263 + q−265

A2 Invariants.

Weight Invariant
1,0 1 + q−4 + 2q−6−q−8 + 3q−10−q−12−3q−18 + q−20−2q−22 + q−24 + q−26−q−28 + q−30
1,1 q4−2q2 + 8−16q−2 + 35q−4−58q−6 + 102q−8−152q−10 + 223q−12−294q−14 + 366q−16−418q−18 + 434q−20−408q−22 + 318q−24−180q−26−5q−28 + 210q−30−420q−32 + 610q−34−751q−36 + 832q−38−844q−40 + 784q−42−661q−44 + 488q−46−284q−48 + 80q−50 + 107q−52−252q−54 + 348q−56−396q−58 + 395q−60−360q−62 + 306q−64−246q−66 + 187q−68−134q−70 + 92q−72−60q−74 + 36q−76−20q−78 + 10q−80−4q−82 + q−84
2,0 q4−1 + q−2 + 4q−4 + q−6−3q−8 + 2q−10 + 9q−12−q−14−8q−16 + 3q−18 + 6q−20−7q−22−6q−24 + 5q−26 + 3q−28−6q−30 + q−32 + 6q−34−6q−36−3q−38 + 7q−40−2q−42−7q−44 + 5q−46 + 8q−48−5q−50−6q−52 + 6q−54 + 4q−56−6q−58−2q−60 + 5q−62−2q−66−q−68 + q−70−q−74 + q−76

A3 Invariants.

Weight Invariant
0,1,0 1−q−2 + 2q−4 + 3q−6−3q−8 + 5q−10 + 6q−12−9q−14 + 9q−16 + 7q−18−15q−20 + 9q−22 + 7q−24−18q−26 + q−28 + 4q−30−10q−32−4q−34 + 4q−36 + 7q−38−4q−40−q−42 + 16q−44−7q−46−10q−48 + 17q−50−7q−52−11q−54 + 12q−56−2q−58−6q−60 + 5q−62−2q−66 + q−68
1,0,0 q−1 + 2q−5 + 3q−9−q−11 + 3q−13−q−15 + q−17−q−19−q−21−q−23−3q−25 + q−27−2q−29 + 2q−31−q−33 + 2q−35−q−37 + q−39

A4 Invariants.

Weight Invariant
0,1,0,0 q−2 + q−6 + 3q−8 + 2q−10 + q−12 + 4q−14 + 4q−16 + 8q−22 + 3q−24−6q−26 + 4q−28 + 11q−30−10q−32−13q−34 + 4q−36−4q−38−21q−40−9q−42 + 8q−44−4q−46−5q−48 + 17q−50 + 14q−52−5q−54 + 8q−56 + 13q−58−9q−60−10q−62 + 7q−64 + q−66−12q−68−2q−70 + 9q−72−q−74−7q−76 + 3q−78 + 4q−80−2q−82−q−84 + q−86
1,0,0,0 q−2 + 2q−6 + q−8 + q−10 + 3q−12−q−14 + 3q−16−q−18 + q−20−q−24−q−26−2q−28−q−30−3q−32 + q−34−2q−36 + 2q−38 + 2q−44−q−46 + q−48

B2 Invariants.

Weight Invariant
0,1 1−q−2 + 4q−4−5q−6 + 9q−8−11q−10 + 16q−12−17q−14 + 19q−16−17q−18 + 13q−20−7q−22−q−24 + 10q−26−19q−28 + 28q−30−34q−32 + 36q−34−36q−36 + 31q−38−26q−40 + 15q−42−6q−44−3q−46 + 10q−48−15q−50 + 19q−52−19q−54 + 18q−56−14q−58 + 10q−60−7q−62 + 4q−64−2q−66 + q−68
1,0 q2−q−2−q−4 + 3q−6 + 4q−8−q−10−6q−12−q−14 + 9q−16 + 10q−18−6q−20−14q−22 + 19q−26 + 11q−28−14q−30−18q−32 + 5q−34 + 20q−36 + 4q−38−18q−40−11q−42 + 9q−44 + 10q−46−8q−48−13q−50 + 3q−52 + 11q−54−2q−56−13q−58 + 14q−62 + 7q−64−12q−66−9q−68 + 12q−70 + 16q−72−6q−74−19q−76−2q−78 + 19q−80 + 11q−82−13q−84−17q−86 + 2q−88 + 15q−90 + 5q−92−8q−94−8q−96 + q−98 + 6q−100 + 2q−102−2q−104−2q−106 + q−110

D4 Invariants.

Weight Invariant
1,0,0,0 q−2−q−4 + 3q−6−2q−8 + 7q−10−5q−12 + 10q−14−8q−16 + 15q−18−13q−20 + 15q−22−13q−24 + 15q−26−10q−28 + 7q−30−3q−32−q−34 + 4q−36−17q−38 + 13q−40−24q−42 + 21q−44−32q−46 + 26q−48−26q−50 + 29q−52−20q−54 + 20q−56−11q−58 + 13q−60−3q−64 + 4q−66−10q−68 + 14q−70−15q−72 + 12q−74−16q−76 + 15q−78−10q−80 + 8q−82−8q−84 + 6q−86−3q−88 + 2q−90−2q−92 + q−94

G2 Invariants.

Weight Invariant
1,0 q−2−q−4 + 4q−6−5q−8 + 6q−10−4q−12−q−14 + 12q−16−20q−18 + 30q−20−30q−22 + 20q−24 + 2q−26−32q−28 + 65q−30−79q−32 + 73q−34−37q−36−17q−38 + 73q−40−108q−42 + 110q−44−70q−46 + 6q−48 + 57q−50−93q−52 + 86q−54−39q−56−18q−58 + 67q−60−80q−62 + 48q−64 + 9q−66−79q−68 + 121q−70−120q−72 + 71q−74 + 7q−76−92q−78 + 146q−80−158q−82 + 116q−84−44q−86−46q−88 + 109q−90−130q−92 + 103q−94−38q−96−28q−98 + 71q−100−73q−102 + 35q−104 + 21q−106−68q−108 + 89q−110−64q−112 + 13q−114 + 50q−116−94q−118 + 107q−120−83q−122 + 37q−124 + 12q−126−54q−128 + 72q−130−68q−132 + 50q−134−20q−136−4q−138 + 19q−140−29q−142 + 26q−144−19q−146 + 11q−148−2q−150−3q−152 + 5q−154−6q−156 + 4q−158−2q−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 56"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= −2t3 + 8t2−14t + 17−14t−1 + 8t−2−2t−3
In[5]:= Conway[K][z]
Out[5]= −2z6−4z4 + 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]= { 65, 4 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= q10−3q9 + 6q8−9q7 + 10q6−11q5 + 10q4−7q3 + 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−4−3z4a−6 + z4a−8 + 3z2a−2−2z2a−4−3z2a−6 + 2z2a−8 + 2a−2−2a−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 + 3z7a−5 + 7z7a−7 + 6z7a−9 + z6a−2−3z6a−4−14z6a−6−5z6a−8 + 5z6a−10−6z5a−3−13z5a−5−21z5a−7−11z5a−9 + 3z5a−11−4z4a−2−3z4a−4 + 12z4a−6 + 4z4a−8−6z4a−10 + z4a−12 + 4z3a−3 + 11z3a−5 + 21z3a−7 + 11z3a−9−3z3a−11 + 5z2a−2 + 3z2a−4−7z2a−6−2z2a−8 + 2z2a−10−z2a−12−4za−5−8za−7−4za−9−2a−2 + 2a−6 + a−8

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

Same Alexander/Conway Polynomial: {10_25, K11a140,}

Same Jones Polynomial (up to mirroring, q\leftrightarrow q^{-1}): {10_25,}

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 56"];
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 + 8t2−14t + 17−14t−1 + 8t−2−2t−3, q10−3q9 + 6q8−9q7 + 10q6−11q5 + 10q4−7q3 + 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]= {10_25, K11a140,}
In[6]:= DeleteCases[ Select[ AllKnots[], (J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) & ], K ]
KnotTheory::loading: Loading precomputed data in Jones4Knots11`.
Out[6]= {10_25,}

[edit] Vassiliev invariants

V2 and V3: (0, -2)

[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 56. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
 -2-1012345678χ
21          11
19         2 -2
17        41 3
15       52  -3
13      54   1
11     65    -1
9    45     -1
7   36      3
5  24       -2
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}^{4}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r = 2 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r = 3 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r = 4 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r = 5 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r = 6 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r = 7 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
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 q28−3q27 + 2q26 + 6q25−16q24 + 9q23 + 23q22−45q21 + 13q20 + 54q19−76q18 + 9q17 + 83q16−90q15−4q14 + 94q13−79q12−19q11 + 83q10−50q9−26q8 + 55q7−20q6−21q5 + 25q4−4q3−9q2 + 7q−2q−1 + q−2
3 q54−3q53 + 2q52 + 2q51−q50−7q49 + 6q48 + 14q47−15q46−28q45 + 29q44 + 53q43−42q42−99q41 + 55q40 + 159q39−59q38−227q37 + 44q36 + 305q35−23q34−365q33−20q32 + 419q31 + 57q30−438q29−108q28 + 445q27 + 148q26−422q25−190q24 + 385q23 + 222q22−331q21−242q20 + 258q19 + 260q18−195q17−244q16 + 113q15 + 230q14−59q13−183q12 + 3q11 + 146q10 + 16q9−91q8−35q7 + 62q6 + 25q5−28q4−23q3 + 17q2 + 11q−5−8q−1 + 4q−2 + 2q−3−2q−5 + q−6
4 q88−3q87 + 2q86 + 2q85−5q84 + 8q83−10q82 + 8q81 + 4q80−26q79 + 28q78−19q77 + 36q76 + 12q75−104q74 + 33q73−17q72 + 160q71 + 78q70−287q69−90q68−83q67 + 462q66 + 377q65−494q64−461q63−416q62 + 852q61 + 1018q60−485q59−954q58−1114q57 + 1046q56 + 1813q55−132q54−1264q53−1944q52 + 897q51 + 2385q50 + 393q49−1216q48−2554q47 + 520q46 + 2537q45 + 841q44−888q43−2784q42 + 83q41 + 2309q40 + 1132q39−404q38−2678q37−358q36 + 1807q35 + 1276q34 + 163q33−2275q32−756q31 + 1089q30 + 1224q29 + 713q28−1600q27−956q26 + 312q25 + 892q24 + 1019q23−796q22−805q21−244q20 + 375q19 + 921q18−170q17−410q16−386q15−31q14 + 550q13 + 81q12−74q11−241q10−153q9 + 216q8 + 69q7 + 45q6−80q5−100q4 + 60q3 + 15q2 + 35q−14−37q−1 + 17q−2−2q−3 + 11q−4−q−5−10q−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_55

10_57

Retrieved from "http://katlas.org/wiki/10_56"
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