10 114

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["10 114"];

In[4]:=

PD[K]

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

Out[4]=

X6271 X8394 X18,13,19,14 X20,11,1,12 X12,19,13,20 X2,16,3,15 X4,17,5,18 X10,6,11,5 X14,7,15,8 X16,10,17,9

In[5]:=

GaussCode[K]

Out[5]=

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

In[6]:=

DTCode[K]

Out[6]=

6 8 10 14 16 20 18 2 4 12

(The path below may be different on your system)

In[7]:=

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

In[8]:=

ConwayNotation[K]

Out[8]=

[8*30]

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,−2,1,3,−2,3,−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]]

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, 13}, {2, 11}, {4, 10}, {9, 3}, {10, 8}, {5, 9}, {1, 4}, {7, 2}, {8, 12}, {11, 6}, {13, 7}, {12, 5}, {6, 1}]

In[14]:=

Draw[ap]

Out[14]=

-Graphics-

Further quantum knot invariants for 10_114.

A1 Invariants.

Weight	Invariant
1	q13−3q11 + 3q9−4q7 + 4q5 + 3q−1−4q−3 + 4q−5−3q−7 + q−9
2	q38−3q36−q34 + 12q32−8q30−17q28 + 27q26 + 4q24−39q22 + 24q20 + 24q18−43q16 + 6q14 + 33q12−23q10−14q8 + 24q6 + 9q4−26q2 + 38q−2−24q−4−26q−6 + 43q−8−8q−10−30q−12 + 25q−14 + 2q−16−14q−18 + 8q−20 + q−22−3q−24 + q−26
3	q75−3q73−q71 + 8q69 + 7q67−16q65−30q63 + 23q61 + 66q59−2q57−112q55−54q53 + 140q51 + 142q49−124q47−235q45 + 52q43 + 303q41 + 54q39−318q37−163q35 + 279q33 + 257q31−217q29−302q27 + 125q25 + 321q23−50q21−307q19−25q17 + 276q15 + 96q13−226q11−167q9 + 162q7 + 241q5−70q3−291q−46q−1 + 317q−3 + 160q−5−286q−7−261q−9 + 216q−11 + 304q−13−116q−15−291q−17 + 27q−19 + 230q−21 + 25q−23−151q−25−37q−27 + 81q−29 + 30q−31−41q−33−15q−35 + 23q−37 + 2q−39−9q−41−2q−43 + 5q−45 + q−47−3q−49 + q−51

A2 Invariants.

Weight	Invariant
1,0	q18−2q16−3q10 + 4q8 + 2q4 + 2q2−2 + 3q−2−3q−4 + q−6 + q−8−2q−10 + q−12
1,1	q52−6q50 + 20q48−52q46 + 119q44−240q42 + 424q40−670q38 + 971q36−1274q34 + 1510q32−1624q30 + 1560q28−1284q26 + 786q24−128q22−620q20 + 1380q18−2070q16 + 2604q14−2922q12 + 2992q10−2792q8 + 2358q6−1732q4 + 1018q2−276−380q−2 + 881q−4−1196q−6 + 1332q−8−1326q−10 + 1202q−12−1022q−14 + 830q−16−640q−18 + 468q−20−328q−22 + 220q−24−134q−26 + 73q−28−38q−30 + 18q−32−6q−34 + q−36
2,0	q48−2q46−2q44 + 5q42 + 3q40−4q38−7q36 + 7q34 + 12q32−13q30−12q28 + 10q26 + 9q24−12q22−12q20 + 16q18 + 8q16−14q14 + 11q10−6q8−q6 + 14q4−2q2−7 + 8q−2 + 13q−4−17q−6−15q−8 + 19q−10 + 6q−12−18q−14−2q−16 + 14q−18 + 3q−20−9q−22−2q−24 + 6q−26−q−28−2q−30 + q−32

A3 Invariants.

Weight	Invariant
0,1,0	q40−3q38 + q36 + 5q34−11q32 + 10q30 + 9q28−24q26 + 17q24 + 7q22−34q20 + 17q18 + 12q16−26q14 + 13q12 + 17q10−8q8−3q6 + 5q4 + 12q2−15−9q−2 + 31q−4−19q−6−16q−8 + 33q−10−14q−12−16q−14 + 22q−16−5q−18−10q−20 + 9q−22−3q−26 + q−28
1,0,0	q23−2q21 + q19−3q17 + q15−3q13 + 4q11 + 3q7 + q5 + q3 + q−2q−1 + 3q−3−3q−5 + 2q−7−2q−9 + 2q−11−2q−13 + q−15

A4 Invariants.

Weight	Invariant
0,1,0,0	q50−2q48−2q46 + 6q44−3q42−8q40 + 12q38 + 9q36−15q34−2q32 + 19q30−7q28−31q26 + 3q24 + 23q22−21q20−16q18 + 36q16 + 12q14−21q12 + 15q10 + 20q8−19q6−12q4 + 20q2 + 1−25q−2 + 10q−4 + 26q−6−18q−8−15q−10 + 20q−12 + 6q−14−18q−16−3q−18 + 12q−20 + 2q−22−8q−24 + 6q−28−2q−30−2q−32 + q−34
1,0,0,0	q28−2q26 + q24−2q22−2q20 + q18−3q16 + 4q14 + 3q10 + 2q8 + q6 + q4 + 1−2q−2 + 3q−4−3q−6 + 2q−8−q−10−q−12 + 2q−14−2q−16 + q−18

B2 Invariants.

Weight	Invariant
0,1	q40−3q38 + 7q36−13q34 + 21q32−30q30 + 35q28−40q26 + 39q24−35q22 + 24q20−9q18−10q16 + 32q14−49q12 + 67q10−74q8 + 79q6−71q4 + 60q2−43 + 23q−2−3q−4−15q−6 + 28q−8−37q−10 + 40q−12−38q−14 + 32q−16−25q−18 + 18q−20−11q−22 + 6q−24−3q−26 + q−28
1,0	q66−3q62−3q60 + 4q58 + 9q56−2q54−16q52−6q50 + 23q48 + 21q46−17q44−35q42 + q40 + 40q38 + 18q36−36q34−37q32 + 15q30 + 40q28 + 2q26−36q24−13q22 + 27q20 + 22q18−17q16−19q14 + 15q12 + 26q10−9q8−28q6 + 3q4 + 32q2 + 6−33q−2−18q−4 + 30q−6 + 30q−8−20q−10−40q−12 + 2q−14 + 40q−16 + 17q−18−27q−20−29q−22 + 8q−24 + 27q−26 + 8q−28−15q−30−14q−32 + 3q−34 + 10q−36 + 3q−38−3q−40−3q−42 + q−46

D4 Invariants.

Weight	Invariant
1,0,0,0	q54−3q52 + 4q50−7q48 + 12q46−17q44 + 22q42−25q40 + 31q38−32q36 + 30q34−31q32 + 26q30−23q28 + 7q26−2q24−8q22 + 22q20−33q18 + 46q16−45q14 + 60q12−60q10 + 59q8−54q6 + 53q4−43q2 + 29−21q−2 + 13q−4 + 4q−6−14q−8 + 17q−10−25q−12 + 33q−14−31q−16 + 26q−18−29q−20 + 27q−22−18q−24 + 14q−26−14q−28 + 10q−30−4q−32 + 3q−34−3q−36 + q−38

G2 Invariants.

Weight

Invariant

1,0

q94−3q92 + 7q90−14q88 + 17q86−18q84 + 7q82 + 23q80−59q78 + 102q76−118q74 + 87q72−8q70−116q68 + 233q66−288q64 + 240q62−90q60−114q58 + 292q56−369q54 + 304q52−124q50−102q48 + 262q46−301q44 + 192q42 + 8q40−188q38 + 283q36−240q34 + 79q32 + 135q30−323q28 + 407q26−345q24 + 160q22 + 105q20−340q18 + 471q16−440q14 + 265q12−9q10−238q8 + 371q6−346q4 + 186q2 + 42−216q−2 + 264q−4−170q−6−16q−8 + 194q−10−288q−12 + 259q−14−123q−16−57q−18 + 212q−20−287q−22 + 265q−24−164q−26 + 35q−28 + 77q−30−152q−32 + 168q−34−142q−36 + 92q−38−29q−40−22q−42 + 51q−44−63q−46 + 54q−48−34q−50 + 16q−52 + q−54−8q−56 + 10q−58−10q−60 + 6q−62−3q−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["10 114"];

In[4]:=

Alexander[K][t]

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

Out[4]=

−2t3 + 10t2−21t + 27−21t−1 + 10t−2−2t−3

In[5]:=

Conway[K][z]

Out[5]=

−2z6−2z4 + z2 + 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]=

{ 93, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

q4−4q3 + 8q2−12q + 15−15q−1 + 15q−2−11q−3 + 7q−4−4q−5 + q−6

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

−a2z6−z6 + a4z4−2a2z4 + z4a−2−2z4 + a4z2 + z2a−2−z2−a4 + 2a2

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

3a3z9 + 3az9 + 6a4z8 + 14a2z8 + 8z8 + 4a5z7 + 2a3z7 + 8az7 + 10z7a−1 + a6z6−17a4z6−35a2z6 + 8z6a−2−9z6−11a5z5−21a3z5−27az5−13z5a−1 + 4z5a−3−2a6z4 + 14a4z4 + 26a2z4−8z4a−2 + z4a−4 + z4 + 7a5z3 + 18a3z3 + 18az3 + 5z3a−1−2z3a−3−3a4z2−5a2z2 + 2z2a−2−2a3z−3az−za−1−a4−2a2

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["10 114"];

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 + 10t2−21t + 27−21t−1 + 10t−2−2t−3, q4−4q3 + 8q2−12q + 15−15q−1 + 15q−2−11q−3 + 7q−4−4q−5 + q−6 }

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]=

{K11a93,}

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	q12−4q11 + 4q10 + 8q9−26q8 + 20q7 + 31q6−81q5 + 42q4 + 82q3−150q2 + 44q + 144−188q−1 + 18q−2 + 179q−3−173q−4−20q−5 + 170q−6−117q−7−47q−8 + 121q−9−50q−10−47q−11 + 58q−12−7q−13−24q−14 + 14q−15 + 2q−16−4q−17 + q−18
3	q24−4q23 + 4q22 + 4q21−6q20−11q19 + 15q18 + 25q17−44q16−37q15 + 86q14 + 76q13−162q12−151q11 + 262q10 + 281q9−365q8−469q7 + 437q6 + 701q5−453q4−946q3 + 412q2 + 1147q−296−1309q−1 + 167q−2 + 1368q−3 + 15q−4−1388q−5−162q−6 + 1309q−7 + 337q−8−1208q−9−463q−10 + 1027q−11 + 594q−12−837q−13−659q−14 + 600q−15 + 679q−16−363q−17−637q−18 + 158q−19 + 524q−20 + 9q−21−388q−22−93q−23 + 237q−24 + 120q−25−122q−26−95q−27 + 43q−28 + 62q−29−12q−30−27q−31 + 9q−33 + 2q−34−4q−35 + q−36
4	q40−4q39 + 4q38 + 4q37−10q36 + 9q35−16q34 + 19q33 + 11q32−52q31 + 54q30−25q29 + 58q28−30q27−239q26 + 184q25 + 155q24 + 303q23−242q22−1004q21 + 139q20 + 803q19 + 1465q18−229q17−2840q16−1111q15 + 1405q14 + 4180q13 + 1296q12−4984q11−4227q10 + 395q9 + 7366q8 + 4897q7−5521q6−7825q5−2743q4 + 8870q3 + 8925q2−3804q−9709−6410q−1 + 8019q−2 + 11304q−3−1105q−4−9327q−5−8912q−6 + 5828q−7 + 11623q−8 + 1385q−9−7529q−10−10036q−11 + 3156q−12 + 10543q−13 + 3532q−14−4917q−15−10112q−16 + 133q−17 + 8295q−18 + 5263q−19−1562q−20−8871q−21−2794q−22 + 4794q−23 + 5699q−24 + 1853q−25−5907q−26−4273q−27 + 864q−28 + 4074q−29 + 3720q−30−2150q−31−3397q−32−1602q−33 + 1349q−34 + 3137q−35 + 346q−36−1278q−37−1696q−38−416q−39 + 1365q−40 + 757q−41 + 83q−42−686q−43−586q−44 + 230q−45 + 275q−46 + 248q−47−84q−48−209q−49−12q−50 + 16q−51 + 74q−52 + 12q−53−33q−54−3q−55−5q−56 + 9q−57 + 2q−58−4q−59 + q−60
5	q60−4q59 + 4q58 + 4q57−10q56 + 5q55 + 4q54−12q53 + 5q52 + 13q51−12q50 + 14q49 + 25q48−47q47−76q46−35q45 + 88q44 + 235q43 + 221q42−139q41−681q40−717q39 + 142q38 + 1397q37 + 1861q36 + 430q35−2480q34−4230q33−2019q32 + 3533q31 + 7938q30 + 5819q29−3588q28−13168q27−12644q26 + 1323q25 + 18846q24 + 22958q23 + 5034q22−23319q21−36246q20−16501q19 + 24395q18 + 50614q17 + 32984q16−20091q15−63424q14−52999q13 + 9792q12 + 72019q11 + 73678q10 + 5716q9−74673q8−91983q7−24148q6 + 71094q5 + 105530q4 + 42813q3−62759q2−113105q−58967 + 51245q−1 + 115139q−2 + 71610q−3−39254q−4−112765q−5−79863q−6 + 27453q−7 + 107579q−8 + 85150q−9−17118q−10−100836q−11−87747q−12 + 7185q−13 + 93094q−14 + 89436q−15 + 2241q−16−84237q−17−89921q−18−12620q−19 + 73812q−20 + 89682q−21 + 23379q−22−61069q−23−87225q−24−34862q−25 + 45796q−26 + 82137q−27 + 45115q−28−28292q−29−72840q−30−53030q−31 + 9722q−32 + 59629q−33 + 56447q−34 + 7696q−35−42787q−36−54326q−37−21831q−38 + 24422q−39 + 46683q−40 + 30397q−41−7141q−42−34561q−43−32603q−44−6610q−45 + 20774q−46 + 28908q−47 + 14745q−48−7856q−49−21262q−50−17244q−51−1599q−52 + 12367q−53 + 15061q−54 + 6743q−55−4692q−56−10497q−57−7781q−58−384q−59 + 5600q−60 + 6372q−61 + 2581q−62−1991q−63−3928q−64−2730q−65−76q−66 + 1889q−67 + 1959q−68 + 674q−69−600q−70−1010q−71−639q−72 + 25q−73 + 438q−74 + 366q−75 + 78q−76−124q−77−144q−78−80q−79 + 22q−80 + 67q−81 + 23q−82−9q−83−9q−84−8q−85−5q−86 + 9q−87 + 2q−88−4q−89 + q−90