10 108

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["10 108"];

In[4]:=

PD[K]

KnotTheory::loading: Loading precomputed data in PD4Knots`.

Out[4]=

X6271 X16,4,17,3 X20,13,1,14 X14,7,15,8 X8,19,9,20 X18,9,19,10 X10,17,11,18 X12,6,13,5 X4,12,5,11 X2,16,3,15

In[5]:=

GaussCode[K]

Out[5]=

1, -10, 2, -9, 8, -1, 4, -5, 6, -7, 9, -8, 3, -4, 10, -2, 7, -6, 5, -3

In[6]:=

DTCode[K]

Out[6]=

6 16 12 14 18 4 20 2 10 8

(The path below may be different on your system)

In[7]:=

AppendTo[$Path, "C:/bin/LinKnot/"];

In[8]:=

ConwayNotation[K]

Out[8]=

[30:20:20]

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,−2,1,1,3,−2,1,−2,−3,−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]]

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."

Out[12]=

-Graphics-

In[13]:=

ap = ArcPresentation[K]

Out[13]=

ArcPresentation[{5, 11}, {7, 12}, {10, 6}, {11, 9}, {8, 10}, {4, 7}, {3, 5}, {9, 4}, {2, 8}, {1, 3}, {12, 2}, {6, 1}]

In[14]:=

Draw[ap]

Out[14]=

-Graphics-

Further quantum knot invariants for 10_108.

A1 Invariants.

Weight	Invariant
1	−q9 + 2q7−2q5 + 3q3−q + q−1−2q−5 + 3q−7−2q−9 + 2q−11−q−13
2	q28−2q26−3q24 + 7q22 + q20−12q18 + 7q16 + 11q14−14q12−q10 + 15q8−8q6−9q4 + 12q2 + 1−11q−2 + 4q−4 + 10q−6−7q−8−7q−10 + 13q−12 + 2q−14−15q−16 + 7q−18 + 8q−20−11q−22 + 3q−24 + 4q−26−5q−28 + 3q−30−2q−34 + q−36
3	−q57 + 2q55 + 3q53−2q51−10q49−4q47 + 18q45 + 17q43−16q41−37q39 + q37 + 50q35 + 26q33−46q31−54q29 + 24q27 + 74q25 + 7q23−71q21−42q19 + 56q17 + 64q15−33q13−74q11 + 8q9 + 74q7 + 13q5−70q3−26q + 68q−1 + 38q−3−58q−5−51q−7 + 50q−9 + 62q−11−30q−13−73q−15 + 69q−19 + 36q−21−51q−23−69q−25 + 25q−27 + 82q−29 + 10q−31−76q−33−33q−35 + 56q−37 + 39q−39−31q−41−31q−43 + 12q−45 + 17q−47−3q−49−8q−51 + 2q−53 + 2q−57−q−61−q−63 + 2q−67−q−69
4	q96−2q94−3q92 + 2q90 + 5q88 + 13q86−3q84−23q82−24q80−9q78 + 55q76 + 60q74 + 7q72−71q70−131q68−17q66 + 112q64 + 179q62 + 81q60−177q58−240q56−119q54 + 180q52 + 358q50 + 134q48−188q46−425q44−215q42 + 262q40 + 454q38 + 265q36−288q34−548q32−222q30 + 304q28 + 596q26 + 194q24−407q22−555q20−131q18 + 493q16 + 510q14−40q12−514q10−390q8 + 218q6 + 505q4 + 167q2−353−402q−2 + 79q−4 + 433q−6 + 199q−8−302q−10−397q−12 + 18q−14 + 442q−16 + 306q−18−215q−20−473q−22−228q−24 + 326q−26 + 519q−28 + 156q−30−357q−32−592q−34−154q−36 + 466q−38 + 620q−40 + 147q−42−592q−44−648q−46−q−48 + 631q−50 + 591q−52−165q−54−627q−56−363q−58 + 240q−60 + 520q−62 + 138q−64−266q−66−296q−68−22q−70 + 225q−72 + 117q−74−49q−76−106q−78−45q−80 + 62q−82 + 30q−84−q−86−18q−88−18q−90 + 19q−92 + q−94−2q−98−6q−100 + 6q−102−q−104 + q−106−2q−110 + q−112

A2 Invariants.

Weight	Invariant
1,0	−q12 + q10 + 2q4−q2 + 2−q−4 + q−6−2q−8 + 2q−10 + q−16−q−18
1,1	q36−4q34 + 14q32−38q30 + 74q28−132q26 + 196q24−256q22 + 308q20−318q18 + 286q16−204q14 + 95q12 + 44q10−196q8 + 322q6−425q4 + 492q2−524 + 502q−2−427q−4 + 330q−6−200q−8 + 74q−10 + 48q−12−134q−14 + 180q−16−198q−18 + 178q−20−152q−22 + 130q−24−104q−26 + 83q−28−70q−30 + 70q−32−64q−34 + 48q−36−42q−38 + 38q−40−28q−42 + 18q−44−12q−46 + 8q−48−4q−50 + q−52
2,0	q34−q32−3q30 + 3q26 + q24−4q22−q20 + 7q18 + 3q16−5q14 + 4q10 + 3q8−6q6−2q4 + 4q2−3−q−2 + 2q−4−2q−8 + 5q−10 + 3q−12−4q−14−q−16 + 7q−18 + 3q−20−10q−22 + q−24 + 6q−26−2q−28−4q−30 + 3q−34−q−38 + q−40−q−42−q−44 + q−46

A3 Invariants.

Weight	Invariant
0,1,0	q26−2q24 + 2q22 + q20−6q18 + 6q16−2q14−6q12 + 9q10−2q8−5q6 + 10q4−q2−6 + 8q−2 + q−4−4q−6 + q−8 + 3q−10 + 2q−12−8q−14 + 3q−16 + 6q−18−11q−20 + 3q−22 + 9q−24−9q−26 + 8q−30−5q−32−3q−34 + 5q−36−q−38−2q−40 + q−42
1,0,0	−q15 + q13−q11 + 2q9−q7 + 2q5−q3 + 2q + q−3−q−7 + q−9−2q−11 + 2q−13−q−15 + 2q−17−q−19 + q−21−q−23

A4 Invariants.

Weight	Invariant
0,1,0,0	q32−q30 + 3q26−2q24−2q22 + 4q20−2q18−7q16 + 3q14 + 3q12−6q10−3q8 + 12q6 + 5q4−9q2 + 6 + 15q−2−7q−4−7q−6 + 11q−8−q−10−9q−12 + 2q−14 + 5q−16−6q−18−5q−20 + 8q−22 + q−24−9q−26 + 3q−28 + 9q−30−5q−32−4q−34 + 5q−36 + 3q−38−3q−40−3q−42 + 2q−44 + 2q−46−2q−48−q−50 + q−52
1,0,0,0	−q18 + q16−q14 + q12 + q10−q8 + 2q6−q4 + 2q2 + q−2 + q−4−q−10 + q−12−2q−14 + 2q−16−q−18 + q−20 + q−22−q−24 + q−26−q−28

B2 Invariants.

Weight	Invariant
0,1	−q26 + 2q24−6q22 + 9q20−12q18 + 16q16−16q14 + 18q12−15q10 + 12q8−5q6−2q4 + 11q2−20 + 26q−2−31q−4 + 32q−6−31q−8 + 27q−10−20q−12 + 14q−14−5q−16−2q−18 + 9q−20−13q−22 + 15q−24−15q−26 + 14q−28−12q−30 + 9q−32−7q−34 + 5q−36−3q−38 + 2q−40−q−42
1,0	q44−2q40−2q38 + 4q36 + 5q34−5q32−9q30 + q28 + 13q26 + 4q24−14q22−11q20 + 11q18 + 17q16−2q14−18q12−5q10 + 15q8 + 13q6−8q4−14q2 + 2 + 12q−2 + 2q−4−10q−6−3q−8 + 10q−10 + 5q−12−10q−14−7q−16 + 10q−18 + 12q−20−6q−22−15q−24 + q−26 + 15q−28 + 4q−30−14q−32−10q−34 + 8q−36 + 14q−38−q−40−12q−42−6q−44 + 7q−46 + 10q−48−7q−52−5q−54 + 2q−56 + 5q−58 + q−60−2q−62−2q−64 + q−68

D4 Invariants.

Weight	Invariant
1,0,0,0	q34−2q32 + 4q30−6q28 + 9q26−11q24 + 11q22−14q20 + 15q18−15q16 + 11q14−10q12 + 9q10−3q8−q6 + 7q4−8q2 + 18−18q−2 + 23q−4−23q−6 + 26q−8−25q−10 + 23q−12−21q−14 + 16q−16−14q−18 + 8q−20−4q−22−q−24 + 5q−26−8q−28 + 10q−30−10q−32 + 13q−34−12q−36 + 9q−38−9q−40 + 10q−42−7q−44 + 4q−46−5q−48 + 5q−50−2q−52 + q−54−2q−56 + q−58

G2 Invariants.

Weight

Invariant

1,0

q60−2q58 + 6q56−11q54 + 13q52−12q50−2q48 + 25q46−46q44 + 58q42−47q40 + 11q38 + 36q36−81q34 + 98q32−77q30 + 24q28 + 37q26−81q24 + 89q22−54q20 + 3q18 + 48q16−68q14 + 53q12−11q10−39q8 + 74q6−77q4 + 59q2−12−42q−2 + 88q−4−108q−6 + 93q−8−49q−10−18q−12 + 74q−14−102q−16 + 93q−18−49q−20−10q−22 + 61q−24−75q−26 + 47q−28−q−30−45q−32 + 67q−34−50q−36 + 12q−38 + 31q−40−57q−42 + 63q−44−46q−46 + 16q−48 + 13q−50−36q−52 + 41q−54−36q−56 + 27q−58−12q−60 + 2q−62 + 9q−64−21q−66 + 24q−68−23q−70 + 16q−72−7q−74 + 7q−78−12q−80 + 13q−82−10q−84 + 7q−86−2q−88−q−90 + 2q−92−4q−94 + 3q−96−2q−98 + q−100

.

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["10 108"];

In[4]:=

Alexander[K][t]

KnotTheory::loading: Loading precomputed data in PD4Knots`.

Out[4]=

2t3−8t2 + 14t−15 + 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]=

{ 63, 2 }

In[8]:=

Jones[K][q]

KnotTheory::loading: Loading precomputed data in Jones4Knots`.

Out[8]=

−q6 + 3q5−5q4 + 8q3−10q2 + 10q−9 + 8q−1−5q−2 + 3q−3−q−4

In[9]:=

HOMFLYPT[K][a, z]

KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.

Out[9]=

z6a−2 + z6−a2z4 + 3z4a−2−z4a−4 + 3z4−2a2z2 + 2z2a−2−2z2a−4 + 2z2 + 1

In[10]:=

Kauffman[K][a, z]

KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.

Out[10]=

2az9 + 2z9a−1 + 3a2z8 + 6z8a−2 + 9z8 + a3z7−3az7 + 4z7a−1 + 8z7a−3−13a2z6−13z6a−2 + 7z6a−4−33z6−4a3z5−11az5−29z5a−1−17z5a−3 + 5z5a−5 + 17a2z4 + 4z4a−2−9z4a−4 + 3z4a−6 + 33z4 + 5a3z3 + 19az3 + 28z3a−1 + 10z3a−3−3z3a−5 + z3a−7−7a2z2 + 2z2a−4−z2a−6−10z2−2a3z−6az−6za−1−2za−3 + 1

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["10 108"];

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−15 + 14t−1−8t−2 + 2t−3, −q6 + 3q5−5q4 + 8q3−10q2 + 10q−9 + 8q−1−5q−2 + 3q−3−q−4 }

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]=

{K11n161,}

In[6]:=

DeleteCases[ Select[ AllKnots[], (J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) & ], K ]

KnotTheory::loading: Loading precomputed data in Jones4Knots11`.

Out[6]=

{}

n	Jn
2	q17−3q16 + 2q15 + 4q14−11q13 + 11q12 + 3q11−25q10 + 30q9 + 2q8−47q7 + 47q6 + 13q5−67q4 + 47q3 + 30q2−73q + 32 + 42q−1−62q−2 + 11q−3 + 43q−4−39q−5−5q−6 + 30q−7−14q−8−9q−9 + 11q−10−q−11−3q−12 + q−13
3	−q33 + 3q32−2q31−q30−q29 + 4q28−3q26 + q25−6q24 + 5q23 + 17q22−4q21−49q20 + 5q19 + 87q18 + 13q17−138q16−38q15 + 173q14 + 85q13−195q12−132q11 + 191q10 + 172q9−162q8−201q7 + 118q6 + 215q5−70q4−213q3 + 17q2 + 208q + 26−183q−1−77q−2 + 164q−3 + 109q−4−122q−5−143q−6 + 82q−7 + 150q−8−25q−9−151q−10−16q−11 + 121q−12 + 53q−13−84q−14−66q−15 + 43q−16 + 61q−17−12q−18−42q−19−6q−20 + 23q−21 + 9q−22−9q−23−5q−24 + q−25 + 3q−26−q−27
4	q54−3q53 + 2q52 + q51−2q50 + 8q49−15q48 + 6q47 + 3q46−q45 + 26q44−52q43 + 6q42 + 20q41 + 30q40 + 58q39−159q38−55q37 + 77q36 + 196q35 + 166q34−406q33−329q32 + 107q31 + 600q30 + 548q29−686q28−932q27−157q26 + 1062q25 + 1304q24−646q23−1564q22−804q21 + 1118q20 + 2043q19−173q18−1718q17−1424q16 + 680q15 + 2278q14 + 340q13−1355q12−1617q11 + 126q10 + 2033q9 + 598q8−834q7−1481q6−298q5 + 1618q4 + 693q3−333q2−1247q−652 + 1137q−1 + 742q−2 + 187q−3−909q−4−939q−5 + 529q−6 + 618q−7 + 661q−8−359q−9−956q−10−95q−11 + 194q−12 + 809q−13 + 242q−14−554q−15−387q−16−332q−17 + 483q−18 + 502q−19−q−20−198q−21−524q−22 + 6q−23 + 292q−24 + 236q−25 + 124q−26−300q−27−172q−28−7q−29 + 115q−30 + 187q−31−42q−32−74q−33−74q−34−14q−35 + 73q−36 + 18q−37 + 4q−38−21q−39−19q−40 + 9q−41 + 3q−42 + 5q−43−q−44−3q−45 + q−46
5	−q80 + 3q79−2q78−q77 + 2q76−5q75 + 3q74 + 9q73−6q72−9q71 + 5q70−3q69 + 12q68 + 15q67−28q66−37q65 + 13q64 + 65q63 + 65q62−26q61−158q60−168q59 + 71q58 + 378q57 + 350q56−152q55−729q54−724q53 + 169q52 + 1349q51 + 1442q50−140q49−2183q48−2577q47−246q46 + 3200q45 + 4288q44 + 1094q43−4137q42−6440q41−2673q40 + 4693q39 + 8806q38 + 4912q37−4481q36−10951q35−7654q34 + 3439q33 + 12407q32 + 10374q31−1604q30−12858q29−12675q28−612q27 + 12362q26 + 14075q25 + 2765q24−11063q23−14553q22−4520q21 + 9456q20 + 14225q19 + 5636q18−7799q17−13395q16−6263q15 + 6350q14 + 12398q13 + 6554q12−5111q11−11423q10−6748q9 + 3968q8 + 10498q7 + 7043q6−2763q5−9620q4−7381q3 + 1392q2 + 8515q + 7784 + 201q−1−7191q−2−7951q−3−1886q−4 + 5414q−5 + 7779q−6 + 3519q−7−3373q−8−7002q−9−4775q−10 + 1077q−11 + 5663q−12 + 5432q−13 + 1027q−14−3732q−15−5278q−16−2755q−17 + 1614q−18 + 4338q−19 + 3624q−20 + 437q−21−2757q−22−3680q−23−1900q−24 + 1000q−25 + 2843q−26 + 2568q−27 + 587q−28−1580q−29−2404q−30−1523q−31 + 240q−32 + 1622q−33 + 1758q−34 + 727q−35−636q−36−1384q−37−1133q−38−177q−39 + 727q−40 + 1013q−41 + 611q−42−112q−43−634q−44−641q−45−233q−46 + 212q−47 + 433q−48 + 337q−49 + 42q−50−203q−51−241q−52−122q−53 + 24q−54 + 122q−55 + 112q−56 + 26q−57−40q−58−48q−59−31q−60−6q−61 + 24q−62 + 17q−63 + q−64−3q−65−3q−66−5q−67 + q−68 + 3q−69−q−70
6	q111−3q110 + 2q109 + q108−2q107 + 5q106−6q105 + 3q104−9q103 + 12q102 + 5q101−22q100 + 19q99−9q98 + 10q97−12q96 + 33q95−14q94−95q93 + 41q92 + 39q91 + 87q90 + 27q89 + 30q88−203q87−366q86 + 86q85 + 356q84 + 542q83 + 269q82−209q81−1128q80−1401q79 + 122q78 + 1687q77 + 2547q76 + 1501q75−1118q74−4487q73−5207q72−720q71 + 5346q70 + 9172q69 + 6757q68−1874q67−12759q66−16513q65−6740q64 + 10371q63 + 23970q62 + 22592q61 + 3544q60−24102q59−39357q58−26368q57 + 8130q56 + 42949q55 + 52636q54 + 25440q53−26293q52−66239q51−61777q50−13345q49 + 49816q48 + 84951q47 + 63530q46−6489q45−77209q44−96492q43−50565q42 + 32363q41 + 97465q40 + 97537q39 + 27996q38−62409q37−108472q36−80956q35 + 1832q34 + 84205q33 + 107686q32 + 54096q31−36325q30−96530q29−88864q28−20008q27 + 61840q26 + 97531q25 + 61125q24−17903q23−78153q22−81755q21−27420q20 + 45793q19 + 84006q18 + 58906q17−8820q16−64998q15−74653q14−30977q13 + 34853q12 + 75008q11 + 59566q10 + 948q9−53786q8−71787q7−39619q6 + 20066q5 + 65590q4 + 64028q3 + 17438q2−36353q−66601−51600q−1−2564q−2 + 47771q−3 + 64244q−4 + 36874q−5−9827q−6−50697q−7−57425q−8−27811q−9 + 19166q−10 + 51100q−11 + 48281q−12 + 19289q−13−21990q−14−47116q−15−43104q−16−12262q−17 + 23013q−18 + 40860q−19 + 36660q−20 + 9674q−21−20087q−22−37102q−23−30257q−24−7793q−25 + 15632q−26 + 30926q−27 + 26259q−28 + 8683q−29−13154q−30−24182q−31−21983q−32−9668q−33 + 8435q−34 + 18881q−35 + 19433q−36 + 8574q−37−3753q−38−13447q−39−16202q−40−9093q−41 + 653q−42 + 9795q−43 + 11599q−44 + 9093q−45 + 1870q−46−5920q−47−8909q−48−7892q−49−2526q−50 + 2047q−51 + 6356q−52 + 6486q−53 + 3254q−54−539q−55−3828q−56−4309q−57−3701q−58−444q−59 + 2026q−60 + 3043q−61 + 2648q−62 + 1015q−63−477q−64−2135q−65−1818q−66−1032q−67 + 74q−68 + 900q−69 + 1172q−70 + 997q−71 + 22q−72−356q−73−642q−74−520q−75−256q−76 + 110q−77 + 389q−78 + 225q−79 + 166q−80−15q−81−102q−82−160q−83−86q−84 + 24q−85 + 17q−86 + 54q−87 + 32q−88 + 18q−89−22q−90−20q−91 + q−92−7q−93 + 3q−94 + 3q−95 + 5q−96−q−97−3q−98 + q−99