8 12

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["8 12"];

In[4]:=

PD[K]

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

Out[4]=

X4251 X10,8,11,7 X8394 X2,9,3,10 X14,6,15,5 X16,11,1,12 X12,15,13,16 X6,14,7,13

In[5]:=

GaussCode[K]

Out[5]=

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

In[6]:=

DTCode[K]

Out[6]=

4 8 14 10 2 16 6 12

(The path below may be different on your system)

In[7]:=

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

In[8]:=

ConwayNotation[K]

Out[8]=

[2222]

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(5,{−1,2,−1,−3,2,4,−3,4})

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]=

{ 5, 8, 5 }

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

In[14]:=

Draw[ap]

Out[14]=

-Graphics-

Further quantum knot invariants for 8_12.

A1 Invariants.

Weight	Invariant
1	q9−q7 + 2q5−q3−q−3 + 2q−5−q−7 + q−9
2	q26−q24−q22 + 4q20−2q18−5q16 + 7q14−7q10 + 5q8 + 2q6−4q4 + q2 + 3 + q−2−4q−4 + 2q−6 + 5q−8−7q−10 + 7q−14−5q−16−2q−18 + 4q−20−q−22−q−24 + q−26
3	q51−q49−q47 + q45 + 3q43−2q41−7q39 + 2q37 + 12q35 + q33−16q31−7q29 + 20q27 + 12q25−20q23−17q21 + 16q19 + 22q17−12q15−20q13 + 7q11 + 18q9−2q7−13q5−4q3 + 9q + 9q−1−4q−3−13q−5−2q−7 + 18q−9 + 7q−11−20q−13−12q−15 + 22q−17 + 16q−19−17q−21−20q−23 + 12q−25 + 20q−27−7q−29−16q−31 + q−33 + 12q−35 + 2q−37−7q−39−2q−41 + 3q−43 + q−45−q−47−q−49 + q−51
4	q84−q82−q80 + q78 + 3q74−4q72−5q70 + 3q68 + 5q66 + 14q64−9q62−22q60−7q58 + 11q56 + 44q54 + 4q52−40q50−42q48−7q46 + 77q44 + 44q42−32q40−76q38−49q36 + 76q34 + 80q32 + 5q30−78q28−82q26 + 44q24 + 81q22 + 33q20−50q18−75q16 + 8q14 + 52q12 + 42q10−15q8−49q6−20q4 + 18q2 + 41 + 18q−2−20q−4−49q−6−15q−8 + 42q−10 + 52q−12 + 8q−14−75q−16−50q−18 + 33q−20 + 81q−22 + 44q−24−82q−26−78q−28 + 5q−30 + 80q−32 + 76q−34−49q−36−76q−38−32q−40 + 44q−42 + 77q−44−7q−46−42q−48−40q−50 + 4q−52 + 44q−54 + 11q−56−7q−58−22q−60−9q−62 + 14q−64 + 5q−66 + 3q−68−5q−70−4q−72 + 3q−74 + q−78−q−80−q−82 + q−84
5	q125−q123−q121 + q119 + q113−2q111−4q109 + 3q107 + 7q105 + 5q103−13q99−19q97−6q95 + 23q93 + 39q91 + 22q89−23q87−67q85−59q83 + 8q81 + 95q79 + 114q77 + 28q75−105q73−174q71−96q69 + 88q67 + 233q65 + 180q63−46q61−258q59−266q57−31q55 + 253q53 + 337q51 + 116q49−221q47−367q45−192q43 + 157q41 + 366q39 + 250q37−93q35−333q33−262q31 + 25q29 + 273q27 + 260q25 + 20q23−207q21−230q19−56q17 + 140q15 + 196q13 + 81q11−80q9−157q7−104q5 + 27q3 + 127q + 127q−1 + 27q−3−104q−5−157q−7−80q−9 + 81q−11 + 196q−13 + 140q−15−56q−17−230q−19−207q−21 + 20q−23 + 260q−25 + 273q−27 + 25q−29−262q−31−333q−33−93q−35 + 250q−37 + 366q−39 + 157q−41−192q−43−367q−45−221q−47 + 116q−49 + 337q−51 + 253q−53−31q−55−266q−57−258q−59−46q−61 + 180q−63 + 233q−65 + 88q−67−96q−69−174q−71−105q−73 + 28q−75 + 114q−77 + 95q−79 + 8q−81−59q−83−67q−85−23q−87 + 22q−89 + 39q−91 + 23q−93−6q−95−19q−97−13q−99 + 5q−103 + 7q−105 + 3q−107−4q−109−2q−111 + q−113 + q−119−q−121−q−123 + q−125

A2 Invariants.

Weight	Invariant
1,0	q14 + q12−q10 + q8−q4 + q2−1 + q−2−q−4 + q−8−q−10 + q−12 + q−14
1,1	q36−2q34 + 6q32−10q30 + 19q28−30q26 + 42q24−54q22 + 64q20−70q18 + 62q16−46q14 + 23q12 + 10q10−44q8 + 80q6−106q4 + 124q2−130 + 124q−2−106q−4 + 80q−6−44q−8 + 10q−10 + 23q−12−46q−14 + 62q−16−70q−18 + 64q−20−54q−22 + 42q−24−30q−26 + 19q−28−10q−30 + 6q−32−2q−34 + q−36
2,0	q36 + q34−2q30 + 3q26−4q22−q20 + 4q18 + 2q16−5q14 + 4q10−q8−2q6 + q4 + 2q2 + 2q−2 + q−4−2q−6−q−8 + 4q−10−5q−14 + 2q−16 + 4q−18−q−20−4q−22 + 3q−26−2q−30 + q−34 + q−36

A3 Invariants.

Weight	Invariant
0,1,0	q28−q26 + q24 + 3q22−3q20 + q18 + 5q16−6q14 + 4q10−5q8−q6 + 3q4 + q2 + q−2 + 3q−4−q−6−5q−8 + 4q−10−6q−14 + 5q−16 + q−18−3q−20 + 3q−22 + q−24−q−26 + q−28
1,0,0	q19 + q17 + q15−q13 + q11−q9−q5 + q3 + q−3−q−5−q−9 + q−11−q−13 + q−15 + q−17 + q−19

B2 Invariants.

Weight	Invariant
0,1	q28−q26 + 3q24−3q22 + 5q20−5q18 + 5q16−4q14 + 2q12−3q8 + 5q6−7q4 + 9q2−10 + 9q−2−7q−4 + 5q−6−3q−8 + 2q−12−4q−14 + 5q−16−5q−18 + 5q−20−3q−22 + 3q−24−q−26 + q−28
1,0	q46−q42−q40 + 2q38 + 3q36−4q32−2q30 + 4q28 + 5q26−2q24−6q22−q20 + 5q18 + 2q16−4q14−3q12 + 2q10 + 3q8−q6−3q4 + q2 + 5 + q−2−3q−4−q−6 + 3q−8 + 2q−10−3q−12−4q−14 + 2q−16 + 5q−18−q−20−6q−22−2q−24 + 5q−26 + 4q−28−2q−30−4q−32 + 3q−36 + 2q−38−q−40−q−42 + q−46

G2 Invariants.

Weight

Invariant

1,0

q66−q64 + 3q62−3q60 + 2q58−3q54 + 9q52−11q50 + 12q48−8q46 + 10q42−17q40 + 23q38−18q36 + 8q34 + 4q32−16q30 + 17q28−13q26 + 3q24 + 6q22−12q20 + 9q18−12q14 + 21q12−23q10 + 15q8−q6−14q4 + 27q2−29 + 27q−2−14q−4−q−6 + 15q−8−23q−10 + 21q−12−12q−14 + 9q−18−12q−20 + 6q−22 + 3q−24−13q−26 + 17q−28−16q−30 + 4q−32 + 8q−34−18q−36 + 23q−38−17q−40 + 10q−42−8q−46 + 12q−48−11q−50 + 9q−52−3q−54 + 2q−58−3q−60 + 3q−62−q−64 + q−66

.

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["8 12"];

In[4]:=

Alexander[K][t]

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

Out[4]=

t2−7t + 13−7t−1 + t−2

In[5]:=

Conway[K][z]

Out[5]=

z4−3z2 + 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]=

{ 29, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

q4−2q3 + 4q2−5q + 5−5q−1 + 4q−2−2q−3 + q−4

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

a4−2z2a2−a2 + z4 + z2 + 1−2z2a−2−a−2 + a−4

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

az7 + z7a−1 + 2a2z6 + 2z6a−2 + 4z6 + 2a3z5 + 2az5 + 2z5a−1 + 2z5a−3 + a4z4−a2z4−z4a−2 + z4a−4−4z4−3a3z3−3az3−3z3a−1−3z3a−3−2a4z2−2a2z2−2z2a−2−2z2a−4 + a3z + za−3 + a4 + a2 + a−2 + a−4 + 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["8 12"];

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]=

{ t2−7t + 13−7t−1 + t−2, q4−2q3 + 4q2−5q + 5−5q−1 + 4q−2−2q−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]=

{}

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	q12−2q11 + 6q9−8q8−3q7 + 18q6−15q5−10q4 + 30q3−18q2−16q + 35−16q−1−18q−2 + 30q−3−10q−4−15q−5 + 18q−6−3q−7−8q−8 + 6q−9−2q−11 + q−12
3	q24−2q23 + 2q21 + 3q20−7q19−5q18 + 11q17 + 13q16−18q15−22q14 + 20q13 + 40q12−26q11−54q10 + 23q9 + 73q8−20q7−88q6 + 15q5 + 100q4−9q3−108q2 + 4q + 109 + 4q−1−108q−2−9q−3 + 100q−4 + 15q−5−88q−6−20q−7 + 73q−8 + 23q−9−54q−10−26q−11 + 40q−12 + 20q−13−22q−14−18q−15 + 13q−16 + 11q−17−5q−18−7q−19 + 3q−20 + 2q−21−2q−23 + q−24
4	q40−2q39 + 2q37−q36 + 4q35−9q34−q33 + 10q32 + q31 + 13q30−32q29−14q28 + 25q27 + 19q26 + 46q25−72q24−58q23 + 23q22 + 54q21 + 130q20−105q19−134q18−21q17 + 81q16 + 255q15−101q14−209q13−104q12 + 77q11 + 381q10−64q9−257q8−187q7 + 52q6 + 464q5−20q4−267q3−244q2 + 18q + 493 + 18q−1−244q−2−267q−3−20q−4 + 464q−5 + 52q−6−187q−7−257q−8−64q−9 + 381q−10 + 77q−11−104q−12−209q−13−101q−14 + 255q−15 + 81q−16−21q−17−134q−18−105q−19 + 130q−20 + 54q−21 + 23q−22−58q−23−72q−24 + 46q−25 + 19q−26 + 25q−27−14q−28−32q−29 + 13q−30 + q−31 + 10q−32−q−33−9q−34 + 4q−35−q−36 + 2q−37−2q−39 + q−40
5	q60−2q59 + 2q57−q56 + 2q54−5q53−2q52 + 9q51 + 3q50−2q49−3q48−18q47−8q46 + 22q45 + 32q44 + 14q43−20q42−63q41−52q40 + 30q39 + 99q38 + 101q37−q36−149q35−185q34−39q33 + 177q32 + 285q31 + 144q30−202q29−411q28−251q27 + 169q26 + 520q25 + 428q24−118q23−632q22−588q21 + 23q20 + 695q19 + 777q18 + 91q17−748q16−931q15−217q14 + 766q13 + 1064q12 + 339q11−761q10−1171q9−444q8 + 743q7 + 1238q6 + 535q5−705q4−1286q3−605q2 + 666q + 1291 + 666q−1−605q−2−1286q−3−705q−4 + 535q−5 + 1238q−6 + 743q−7−444q−8−1171q−9−761q−10 + 339q−11 + 1064q−12 + 766q−13−217q−14−931q−15−748q−16 + 91q−17 + 777q−18 + 695q−19 + 23q−20−588q−21−632q−22−118q−23 + 428q−24 + 520q−25 + 169q−26−251q−27−411q−28−202q−29 + 144q−30 + 285q−31 + 177q−32−39q−33−185q−34−149q−35−q−36 + 101q−37 + 99q−38 + 30q−39−52q−40−63q−41−20q−42 + 14q−43 + 32q−44 + 22q−45−8q−46−18q−47−3q−48−2q−49 + 3q−50 + 9q−51−2q−52−5q−53 + 2q−54−q−56 + 2q−57−2q−59 + q−60
6	q84−2q83 + 2q81−q80−2q78 + 6q77−6q76−3q75 + 11q74−q73−2q72−12q71 + 11q70−16q69−7q68 + 38q67 + 16q66 + 3q65−41q64 + q63−68q62−33q61 + 98q60 + 93q59 + 77q58−58q57−37q56−237q55−183q54 + 124q53 + 255q52 + 331q51 + 93q50 + 5q49−556q48−604q47−102q46 + 356q45 + 782q44 + 595q43 + 402q42−827q41−1294q40−786q39 + 85q38 + 1183q37 + 1419q36 + 1336q35−712q34−1957q33−1851q32−715q31 + 1194q30 + 2244q29 + 2655q28−93q27−2262q26−2933q25−1845q24 + 746q23 + 2752q22 + 3943q21 + 806q20−2153q19−3708q18−2911q17 + 70q16 + 2891q15 + 4876q14 + 1634q13−1812q12−4100q11−3656q10−556q9 + 2791q8 + 5381q7 + 2212q6−1428q5−4191q4−4055q3−1036q2 + 2568q + 5533 + 2568q−1−1036q−2−4055q−3−4191q−4−1428q−5 + 2212q−6 + 5381q−7 + 2791q−8−556q−9−3656q−10−4100q−11−1812q−12 + 1634q−13 + 4876q−14 + 2891q−15 + 70q−16−2911q−17−3708q−18−2153q−19 + 806q−20 + 3943q−21 + 2752q−22 + 746q−23−1845q−24−2933q−25−2262q−26−93q−27 + 2655q−28 + 2244q−29 + 1194q−30−715q−31−1851q−32−1957q−33−712q−34 + 1336q−35 + 1419q−36 + 1183q−37 + 85q−38−786q−39−1294q−40−827q−41 + 402q−42 + 595q−43 + 782q−44 + 356q−45−102q−46−604q−47−556q−48 + 5q−49 + 93q−50 + 331q−51 + 255q−52 + 124q−53−183q−54−237q−55−37q−56−58q−57 + 77q−58 + 93q−59 + 98q−60−33q−61−68q−62 + q−63−41q−64 + 3q−65 + 16q−66 + 38q−67−7q−68−16q−69 + 11q−70−12q−71−2q−72−q−73 + 11q−74−3q−75−6q−76 + 6q−77−2q−78−q−80 + 2q−81−2q−83 + q−84
7	q112−2q111 + 2q109−q108−2q106 + 2q105 + 5q104−7q103−q102 + 7q101−q100−12q98−2q97 + 17q96−13q95 + 3q94 + 22q93 + 7q92 + 7q91−43q90−35q89 + 10q88−26q87 + 25q86 + 81q85 + 63q84 + 69q83−77q82−148q81−104q80−156q79 + 17q78 + 206q77 + 282q76 + 356q75 + 55q74−268q73−435q72−661q71−340q70 + 204q69 + 654q68 + 1130q67 + 788q66 + 49q65−750q64−1693q63−1545q62−598q61 + 680q60 + 2252q59 + 2499q58 + 1547q57−198q56−2717q55−3672q54−2860q53−655q52 + 2844q51 + 4768q50 + 4537q49 + 2104q48−2562q47−5828q46−6373q45−3891q44 + 1784q43 + 6441q42 + 8212q41 + 6132q40−511q39−6764q38−9928q37−8380q36−1108q35 + 6552q34 + 11294q33 + 10685q32 + 2988q31−6052q30−12328q29−12727q28−4870q27 + 5238q26 + 12969q25 + 14476q24 + 6663q23−4286q22−13308q21−15867q20−8241q19 + 3344q18 + 13388q17 + 16876q16 + 9543q15−2421q14−13285q13−17618q12−10578q11 + 1640q10 + 13082q9 + 18052q8 + 11361q7−915q6−12789q5−18322q4−11965q3 + 311q2 + 12426q + 18379 + 12426q−1 + 311q−2−11965q−3−18322q−4−12789q−5−915q−6 + 11361q−7 + 18052q−8 + 13082q−9 + 1640q−10−10578q−11−17618q−12−13285q−13−2421q−14 + 9543q−15 + 16876q−16 + 13388q−17 + 3344q−18−8241q−19−15867q−20−13308q−21−4286q−22 + 6663q−23 + 14476q−24 + 12969q−25 + 5238q−26−4870q−27−12727q−28−12328q−29−6052q−30 + 2988q−31 + 10685q−32 + 11294q−33 + 6552q−34−1108q−35−8380q−36−9928q−37−6764q−38−511q−39 + 6132q−40 + 8212q−41 + 6441q−42 + 1784q−43−3891q−44−6373q−45−5828q−46−2562q−47 + 2104q−48 + 4537q−49 + 4768q−50 + 2844q−51−655q−52−2860q−53−3672q−54−2717q−55−198q−56 + 1547q−57 + 2499q−58 + 2252q−59 + 680q−60−598q−61−1545q−62−1693q−63−750q−64 + 49q−65 + 788q−66 + 1130q−67 + 654q−68 + 204q−69−340q−70−661q−71−435q−72−268q−73 + 55q−74 + 356q−75 + 282q−76 + 206q−77 + 17q−78−156q−79−104q−80−148q−81−77q−82 + 69q−83 + 63q−84 + 81q−85 + 25q−86−26q−87 + 10q−88−35q−89−43q−90 + 7q−91 + 7q−92 + 22q−93 + 3q−94−13q−95 + 17q−96−2q−97−12q−98−q−100 + 7q−101−q−102−7q−103 + 5q−104 + 2q−105−2q−106−q−108 + 2q−109−2q−111 + q−112