10 158

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["10 158"];

In[4]:=

PD[K]

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

Out[4]=

X6271 X3,10,4,11 X14,8,15,7 X8,14,9,13 X9,2,10,3 X11,18,12,19 X5,17,6,16 X17,5,18,4 X20,16,1,15 X19,12,20,13

In[5]:=

GaussCode[K]

Out[5]=

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

In[6]:=

DTCode[K]

Out[6]=

6 -10 -16 14 -2 -18 8 20 -4 -12

(The path below may be different on your system)

In[7]:=

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

In[8]:=

ConwayNotation[K]

Out[8]=

[-30:2:2]

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,−2,1,1,3,2,−1,2,3})

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]]

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

In[14]:=

Draw[ap]

Out[14]=

-Graphics-

Further quantum knot invariants for 10_158.

A1 Invariants.

Weight	Invariant
1	q9−2q7 + 3q5−q3 + q−2q−3 + 2q−5−2q−7 + 2q−9
2	q26−2q24 + 7q20−7q18−7q16 + 16q14−3q12−14q10 + 13q8 + 3q6−12q4 + 3q2 + 7−2q−2−8q−4 + 8q−6 + 8q−8−15q−10 + 4q−12 + 14q−14−13q−16−4q−18 + 11q−20−4q−22−5q−24 + 3q−26 + q−28
3	q51−2q49 + 4q45 + q43−9q41−8q39 + 19q37 + 21q35−24q33−41q31 + 16q29 + 68q27−2q25−83q23−23q21 + 84q19 + 48q17−73q15−63q13 + 52q11 + 67q9−26q7−59q5 + q3 + 52q + 20q−1−38q−3−41q−5 + 28q−7 + 57q−9−14q−11−74q−13−q−15 + 85q−17 + 18q−19−83q−21−43q−23 + 73q−25 + 59q−27−50q−29−65q−31 + 23q−33 + 60q−35 + 3q−37−44q−39−14q−41 + 22q−43 + 16q−45−6q−47−12q−49 + 2q−53 + 2q−55

A2 Invariants.

Weight	Invariant
1,0	q12−q10 + 2q8 + q6−q4 + 2q2−2 + q−2−2q−4−q−6 + q−8−q−10 + 2q−12 + q−14
1,1	q36−4q34 + 10q32−20q30 + 40q28−68q26 + 108q24−154q22 + 201q20−240q18 + 256q16−236q14 + 179q12−80q10−42q8 + 178q6−313q4 + 412q2−480 + 494q−2−462q−4 + 394q−6−278q−8 + 154q−10−12q−12−106q−14 + 200q−16−260q−18 + 274q−20−256q−22 + 206q−24−150q−26 + 94q−28−52q−30 + 22q−32−4q−34 + 2q−38
2,0	q32−q30 + 4q26−4q22−q20 + 6q18 + 2q16−9q14 + 2q12 + 6q10−3q8−6q6 + 2q4 + 2q2−3 + 2q−2 + 3q−4−q−6−2q−8 + 8q−10 + q−12−5q−14 + 5q−16 + 6q−18−3q−20−7q−22 + 2q−26−4q−28−4q−30 + 2q−32 + 3q−34 + 2q−36

A3 Invariants.

Weight	Invariant
0,1,0	q28−2q26 + 5q22−6q20 + 11q16−10q14 + q12 + 12q10−8q8−q6 + 6q4−3q2−4−2q−2 + 3q−4−q−6−7q−8 + 10q−10 + 4q−12−11q−14 + 9q−16 + 2q−18−10q−20 + 6q−22 + q−24−3q−26 + 3q−28
1,0,0	q15−q13 + 3q11 + 2q7−q5 + q3−q−q−1−2q−5−2q−9 + 2q−11−q−13 + 2q−15 + q−17 + q−19

A4 Invariants.

Weight	Invariant
0,1,0,0	q34−q32−q30 + 4q28−q26−5q24 + 6q22 + 8q20−5q18−2q16 + 10q14 + 2q12−11q10−q8 + 9q6−9q4−8q2 + 9−q−2−11q−4 + 7q−6 + 8q−8−6q−10 + q−12 + 12q−14 + 2q−16−10q−18 + q−20 + 5q−22−7q−24−6q−26 + 4q−28 + 2q−30−q−32 + 2q−34 + 2q−36 + q−38
1,0,0,0	q18−q16 + 3q14 + q12 + q10 + 2q8−q6 + q4−2q2−2q−2−2q−6−q−10−q−12 + 2q−14−q−16 + 2q−18 + q−20 + q−22 + q−24

B2 Invariants.

Weight	Invariant
0,1	q28−2q26 + 4q24−7q22 + 10q20−12q18 + 13q16−10q14 + 9q12−2q10−2q8 + 9q6−14q4 + 19q2−24 + 22q−2−21q−4 + 15q−6−11q−8 + 4q−10 + 2q−12−7q−14 + 11q−16−12q−18 + 12q−20−10q−22 + 9q−24−5q−26 + 3q−28
1,0	q46−2q42−2q40 + 2q38 + 6q36 + q34−8q32−6q30 + 7q28 + 12q26−2q24−13q22−4q20 + 13q18 + 8q16−7q14−9q12 + 4q10 + 9q8−q6−10q4−2q2 + 9 + 2q−2−8q−4−6q−6 + 6q−8 + 7q−10−4q−12−9q−14 + 4q−16 + 13q−18 + 2q−20−12q−22−7q−24 + 10q−26 + 11q−28−5q−30−12q−32−2q−34 + 8q−36 + 5q−38−4q−40−4q−42 + q−44 + 3q−46

D4 Invariants.

Weight	Invariant
1,0,0,0	q38−2q36 + 2q34−3q32 + 6q30−8q28 + 8q26−8q24 + 12q22−9q20 + 8q18−4q16 + 6q14 + 3q12−5q10 + 6q8−9q6 + 14q4−18q2 + 14−20q−2 + 16q−4−15q−6 + 11q−8−12q−10 + 8q−12 + q−16 + 3q−18−6q−20 + 10q−22−9q−24 + 9q−26−10q−28 + 9q−30−6q−32 + 6q−34−4q−36 + 3q−38

G2 Invariants.

Weight

Invariant

1,0

q66−2q64 + 4q62−6q60 + 5q58−3q56−2q54 + 13q52−22q50 + 30q48−30q46 + 13q44 + 9q42−37q40 + 62q38−64q36 + 48q34−7q32−35q30 + 66q28−67q26 + 42q24 + q22−39q20 + 53q18−36q16 + q14 + 47q12−75q10 + 70q8−35q6−22q4 + 69q2−99 + 94q−2−61q−4 + 11q−6 + 39q−8−77q−10 + 85q−12−65q−14 + 20q−16 + 22q−18−52q−20 + 52q−22−24q−24−13q−26 + 50q−28−63q−30 + 45q−32−4q−34−45q−36 + 77q−38−77q−40 + 50q−42−6q−44−31q−46 + 52q−48−51q−50 + 37q−52−11q−54−8q−56 + 15q−58−17q−60 + 11q−62−2q−64 + q−68

.

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["10 158"];

In[4]:=

Alexander[K][t]

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

Out[4]=

−t3 + 4t2−10t + 15−10t−1 + 4t−2−t−3

In[5]:=

Conway[K][z]

Out[5]=

−z6−2z4−3z2 + 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]=

{ 45, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

2q4−4q3 + 6q2−8q + 8−7q−1 + 6q−2−3q−3 + q−4

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

−z6 + a2z4 + z4a−2−4z4 + 2a2z2 + z2a−2−6z2 + 2a2 + a−4−2

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

z8a−2 + z8 + 4az7 + 5z7a−1 + z7a−3 + 5a2z6 + z6a−2 + 6z6 + 3a3z5−5az5−7z5a−1 + z5a−3 + a4z4−8a2z4−z4a−2 + 3z4a−4−13z4−3a3z3 + 3az3 + 2z3a−1−4z3a−3−a4z2 + 5a2z2−2z2a−2−5z2a−4 + 9z2−az + za−1 + 2za−3−2a2 + a−4−2

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["10 158"];

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]=

{ −t3 + 4t2−10t + 15−10t−1 + 4t−2−t−3, 2q4−4q3 + 6q2−8q + 8−7q−1 + 6q−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]=

{}

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	q13 + 2q12−8q11 + 2q10 + 17q9−23q8−7q7 + 44q6−33q5−26q4 + 67q3−33q2−42q + 73−24q−1−46q−2 + 58q−3−9q−4−36q−5 + 31q−6 + 2q−7−17q−8 + 8q−9 + 2q−10−3q−11 + q−12
3	2q26−2q24−12q23 + 8q22 + 22q21 + 4q20−48q19−22q18 + 69q17 + 61q16−85q15−110q14 + 84q13 + 170q12−71q11−226q10 + 44q9 + 271q8−4q7−312q6−29q5 + 331q4 + 67q3−341q2−98q + 334 + 125q−1−309q−2−149q−3 + 274q−4 + 158q−5−216q−6−164q−7 + 159q−8 + 148q−9−95q−10−128q−11 + 52q−12 + 88q−13−14q−14−58q−15 + 31q−17 + 3q−18−13q−19−2q−20 + 4q−21 + 2q−22−3q−23 + q−24
4	q44 + 2q43−8q41−6q40−6q39 + 23q38 + 39q37−11q36−41q35−97q34 + 13q33 + 156q32 + 105q31 + 2q30−315q29−201q28 + 199q27 + 369q26 + 337q25−455q24−626q23−75q22 + 542q21 + 948q20−274q19−995q18−635q17 + 406q16 + 1549q15 + 177q14−1101q13−1211q12 + 39q11 + 1927q10 + 652q9−993q8−1608q7−352q6 + 2060q5 + 1007q4−781q3−1803q2−679q + 1980 + 1232q−1−482q−2−1785q−3−951q−4 + 1639q−5 + 1300q−6−65q−7−1485q−8−1121q−9 + 1039q−10 + 1112q−11 + 341q−12−911q−13−1030q−14 + 389q−15 + 666q−16 + 491q−17−318q−18−658q−19 + 10q−20 + 211q−21 + 338q−22−q−23−263q−24−53q−25−2q−26 + 127q−27 + 41q−28−64q−29−11q−30−23q−31 + 26q−32 + 14q−33−12q−34 + 2q−35−6q−36 + 4q−37 + 2q−38−3q−39 + q−40
5	2q66 + 2q64−4q63−12q62−12q61 + 10q60 + 18q59 + 46q58 + 40q57−44q56−112q55−110q54−32q53 + 157q52 + 327q51 + 199q50−151q49−495q48−580q47−124q46 + 663q45 + 1082q44 + 644q43−502q42−1585q41−1535q40−20q39 + 1879q38 + 2563q37 + 1060q36−1760q35−3571q34−2475q33 + 1081q32 + 4293q31 + 4125q30 + 114q29−4574q28−5714q27−1729q26 + 4346q25 + 7080q24 + 3532q23−3682q22−8103q21−5279q20 + 2711q19 + 8717q18 + 6878q17−1604q16−9057q15−8176q14 + 547q13 + 9079q12 + 9223q11 + 468q10−9020q9−9999q8−1322q7 + 8795q6 + 10574q5 + 2135q4−8488q3−10991q2−2896q + 8024 + 11225q−1 + 3678q−2−7312q−3−11259q−4−4511q−5 + 6352q−6 + 10977q−7 + 5295q−8−5021q−9−10328q−10−5983q−11 + 3490q−12 + 9194q−13 + 6359q−14−1780q−15−7664q−16−6351q−17 + 259q−18 + 5791q−19 + 5805q−20 + 1034q−21−3918q−22−4867q−23−1719q−24 + 2186q−25 + 3616q−26 + 1971q−27−890q−28−2419q−29−1710q−30 + 80q−31 + 1350q−32 + 1281q−33 + 290q−34−636q−35−805q−36−337q−37 + 213q−38 + 429q−39 + 252q−40−25q−41−189q−42−159q−43−18q−44 + 75q−45 + 68q−46 + 18q−47−10q−48−36q−49−17q−50 + 14q−51 + 9q−52−q−53 + 3q−54−2q−55−6q−56 + 4q−57 + 2q−58−3q−59 + q−60
6	q93 + 2q92−6q89−8q88−16q87−6q86 + 23q85 + 49q84 + 53q83 + 27q82−13q81−149q80−201q79−151q78 + 72q77 + 294q76 + 462q75 + 520q74−12q73−624q72−1148q71−961q70−286q69 + 958q68 + 2323q67 + 2127q66 + 724q65−1827q64−3549q63−4097q62−1793q61 + 2869q60 + 6130q59 + 6668q58 + 2524q57−3377q56−9745q55−10545q54−3860q53 + 5739q52 + 14062q51 + 14064q50 + 6345q49−9153q48−20164q47−18853q46−5949q45 + 13461q44 + 25817q43 + 24981q42 + 4011q41−20582q40−33606q39−26563q38−281q37 + 27898q36 + 42963q35 + 25729q34−8209q33−38908q32−46125q31−21794q30 + 18315q29 + 52054q28 + 46125q27 + 10739q26−33987q25−57558q24−41701q23 + 3463q22 + 52198q21 + 59237q20 + 27904q19−24708q18−61232q17−55134q16−9724q15 + 48293q14 + 65607q13 + 39708q12−16293q11−61087q10−62882q9−19072q8 + 43936q7 + 68412q6 + 47482q5−9520q4−59520q3−67754q2−26585q + 38770 + 69206q−1 + 53985q−2−1702q−3−55314q−4−70716q−5−35066q−6 + 29559q−7 + 66002q−8 + 59592q−9 + 9892q−10−44730q−11−68900q−12−44099q−13 + 13888q−14 + 54540q−15 + 60275q−16 + 23623q−17−25874q−18−57448q−19−48179q−20−4933q−21 + 33670q−22 + 50518q−23 + 32154q−24−3966q−25−36090q−26−41141q−27−17708q−28 + 10460q−29 + 31009q−30 + 28894q−31 + 10489q−32−13435q−33−24716q−34−18027q−35−4160q−36 + 11338q−37 + 16738q−38 + 12231q−39 + 65q−40−8917q−41−10062q−42−6722q−43 + 603q−44 + 5509q−45 + 6757q−46 + 2925q−47−1027q−48−2934q−49−3501q−50−1529q−51 + 529q−52 + 2083q−53 + 1423q−54 + 510q−55−182q−56−934q−57−729q−58−283q−59 + 371q−60 + 290q−61 + 217q−62 + 155q−63−124q−64−165q−65−125q−66 + 59q−67 + 14q−68 + 28q−69 + 59q−70−8q−71−23q−72−31q−73 + 20q−74−3q−75−6q−76 + 14q−77−q−78−2q−79−6q−80 + 4q−81 + 2q−82−3q−83 + q−84