10 98

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 98"];

In[4]:=

PD[K]

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

Out[4]=

X1627 X3,10,4,11 X7,18,8,19 X17,8,18,9 X9,2,10,3 X11,16,12,17 X5,15,6,14 X15,5,16,4 X13,20,14,1 X19,12,20,13

In[5]:=

GaussCode[K]

Out[5]=

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

In[6]:=

DTCode[K]

Out[6]=

6 10 14 18 2 16 20 4 8 12

(The path below may be different on your system)

In[7]:=

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

In[8]:=

ConwayNotation[K]

Out[8]=

[.2.2.2.20]

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,−2,3,−2,1,−2,−2,3,−2})

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

In[14]:=

Draw[ap]

Out[14]=

-Graphics-

Further quantum knot invariants for 10_98.

A1 Invariants.

Weight	Invariant
1	q21−2q19 + 4q17−4q15 + q13−2q11−q9 + 4q7−2q5 + 4q3−2q + q−1
2	q58−2q56 + q54 + 5q52−11q50 + 2q48 + 19q46−24q44−7q42 + 34q40−20q38−16q36 + 31q34−20q30 + 6q28 + 16q26−14q24−20q22 + 24q20 + 4q18−32q16 + 21q14 + 20q12−29q10 + 4q8 + 21q6−13q4−5q2 + 9−q−2−2q−4 + q−6
3	q111−2q109 + q107 + 2q105−2q103−5q101 + 5q99 + 13q97−15q95−30q93 + 27q91 + 60q89−27q87−110q85 + 13q83 + 166q81 + 23q79−198q77−87q75 + 213q73 + 145q71−179q69−197q67 + 115q65 + 206q63−39q61−194q59−40q57 + 164q55 + 103q53−111q51−152q49 + 75q47 + 186q45−16q43−218q41−28q39 + 217q37 + 84q35−213q33−146q31 + 174q29 + 189q27−111q25−212q23 + 41q21 + 199q19 + 31q17−155q15−71q13 + 94q11 + 86q9−42q7−66q5 + 4q3 + 42q + 10q−1−19q−3−8q−5 + 6q−7 + 4q−9−q−11−2q−13 + q−15
4	q180−2q178 + q176 + 2q174−5q172 + 4q170−2q168 + 5q166 + 2q164−27q162 + 10q160 + 16q158 + 42q156 + 6q154−128q152−54q150 + 70q148 + 250q146 + 141q144−353q142−426q140−73q138 + 675q136 + 776q134−314q132−1120q130−884q128 + 743q126 + 1767q124 + 546q122−1303q120−2018q118−161q116 + 2024q114 + 1719q112−374q110−2249q108−1343q106 + 1040q104 + 1980q102 + 854q100−1289q98−1704q96−275q94 + 1237q92 + 1411q90−77q88−1307q86−1085q84 + 381q82 + 1433q80 + 736q78−864q76−1504q74−192q72 + 1394q70 + 1357q68−486q66−1871q64−819q62 + 1198q60 + 1976q58 + 222q56−1893q54−1621q52 + 376q50 + 2145q48 + 1267q46−1044q44−1950q42−877q40 + 1316q38 + 1772q36 + 314q34−1199q32−1498q30−13q28 + 1127q26 + 986q24−8q22−967q20−634q18 + 119q16 + 606q14 + 465q12−173q10−361q8−235q6 + 74q4 + 245q2 + 83−35q−2−108q−4−49q−6 + 40q−8 + 28q−10 + 18q−12−13q−14−14q−16 + 3q−18 + q−20 + 4q−22−q−24−2q−26 + q−28

A2 Invariants.

Weight	Invariant
1,0	q30−q28 + 2q26 + 2q24−3q22−5q18−q16 + q14 + 5q10−q8 + 2q6 + q4−q2 + 1
1,1	q84−4q82 + 10q80−20q78 + 40q76−72q74 + 114q72−174q70 + 266q68−376q66 + 494q64−632q62 + 766q60−860q58 + 866q56−786q54 + 593q52−272q50−140q48 + 616q46−1070q44 + 1472q42−1744q40 + 1876q38−1845q36 + 1644q34−1308q32 + 856q30−396q28−82q26 + 494q24−800q22 + 968q20−996q18 + 934q16−780q14 + 598q12−416q10 + 268q8−150q6 + 80q4−36q2 + 14−4q−2 + q−4
2,0	q76−q74 + q72 + 2q70−q68−4q66 + 2q64 + 7q62−6q60−13q58 + q56 + 8q54−11q52−5q50 + 19q48 + 18q46−2q44 + 10q40−8q38−15q36−q34−6q32−14q30 + 3q28 + 10q26−6q24−6q22 + 14q20 + 9q18−10q16−4q14 + 13q12 + 3q10−7q8−q6 + 5q4 + 3q2−2−q−2 + q−4

A3 Invariants.

Weight	Invariant
0,1,0	q68−2q66 + 6q62−7q60−4q58 + 17q56−13q54−13q52 + 24q50−14q48−14q46 + 27q44−5q40 + 16q38 + 4q36−8q34−17q32−q28−26q26 + 10q24 + 19q22−19q20 + 11q18 + 18q16−17q14 + 8q12 + 8q10−8q8 + 4q6 + 2q4−2q2 + 1
1,0,0	q39−q37 + 3q35 + 3q31−3q29−6q25−3q23−2q21 + 3q17 + q15 + 5q13−q11 + 3q9−q7 + 2q5−q3 + q

A4 Invariants.

Weight	Invariant
0,1,0,0	q86−q84−2q82 + 5q80 + 4q78−8q76−2q74 + 12q72−q70−20q68−3q66 + 10q64−15q62−17q60 + 22q58 + 22q56−3q54 + 21q52 + 33q50−3q48−18q46 + 4q44−15q42−43q40−15q38 + 10q36−16q34−15q32 + 25q30 + 16q28−8q26 + 4q24 + 18q22 + q20−6q18 + 6q16 + 7q14−3q12−q10 + 4q8−q4 + q2
1,0,0,0	q48−q46 + 3q44 + q42 + q40 + 3q38−3q36−6q32−4q30−4q28−2q26 + 2q22 + 4q20 + q18 + 5q16−q14 + 3q12 + 2q6−q4 + q2

B2 Invariants.

Weight	Invariant
0,1	q68−2q66 + 4q64−8q62 + 13q60−18q58 + 25q56−29q54 + 31q52−28q50 + 22q48−12q46−q44 + 16q42−33q40 + 44q38−56q36 + 58q34−59q32 + 50q30−39q28 + 24q26−8q24−5q22 + 19q20−25q18 + 32q16−29q14 + 28q12−22q10 + 16q8−10q6 + 6q4−2q2 + 1
1,0	q110−2q106−2q104 + 2q102 + 7q100 + 2q98−10q96−11q94 + 5q92 + 21q90 + 6q88−23q86−22q84 + 12q82 + 30q80 + q78−31q76−15q74 + 24q72 + 25q70−10q68−22q66 + 8q64 + 27q62 + 6q60−22q58−8q56 + 16q54 + 8q52−21q50−19q48 + 10q46 + 16q44−13q42−29q40 + 2q38 + 32q36 + 15q34−25q32−24q30 + 16q28 + 34q26 + 2q24−25q22−13q20 + 17q18 + 17q16−4q14−12q12−2q10 + 7q8 + 4q6−2q4−2q2 + q−2

D4 Invariants.

Weight	Invariant
1,0,0,0	q94−2q92 + 2q90−3q88 + 7q86−10q84 + 10q82−13q80 + 21q78−23q76 + 19q74−25q72 + 24q70−21q68 + 12q66−11q64 + 6q62 + 14q60−9q58 + 28q56−26q54 + 44q52−41q50 + 41q48−55q46 + 34q44−47q42 + 24q40−34q38 + 13q36−6q34 + 3q32 + 11q30−10q28 + 26q26−19q24 + 26q22−23q20 + 25q18−18q16 + 17q14−12q12 + 10q10−5q8 + 4q6−2q4 + q2

G2 Invariants.

Weight

Invariant

1,0

q162−2q160 + 4q158−6q156 + 6q154−5q152 + 10q148−21q146 + 31q144−39q142 + 33q140−18q138−12q136 + 58q134−94q132 + 117q130−107q128 + 56q126 + 17q124−112q122 + 185q120−203q118 + 154q116−41q114−84q112 + 183q110−192q108 + 134q106−19q104−110q102 + 175q100−143q98 + 34q96 + 116q94−230q92 + 256q90−161q88−7q86 + 163q84−297q82 + 322q80−242q78 + 78q76 + 92q74−232q72 + 288q70−230q68 + 90q66 + 43q64−158q62 + 191q60−129q58 + 8q56 + 128q54−194q52 + 181q50−68q48−83q46 + 204q44−244q42 + 197q40−81q38−51q36 + 157q34−188q32 + 162q30−84q28 + 3q26 + 51q24−78q22 + 69q20−44q18 + 20q16 + 2q14−11q12 + 13q10−10q8 + 6q6−2q4 + q2

.

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 98"];

In[4]:=

Alexander[K][t]

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

Out[4]=

−2t3 + 9t2−18t + 23−18t−1 + 9t−2−2t−3

In[5]:=

Conway[K][z]

Out[5]=

−2z6−3z4 + 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]=

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 81, -4 }

In[8]:=

Jones[K][q]

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

Out[8]=

1−3q−1 + 7q−2−9q−3 + 13q−4−14q−5 + 12q−6−11q−7 + 7q−8−3q−9 + q−10

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

z4a8 + 2z2a8 + 2a8−z6a6−3z4a6−5z2a6−5a6−z6a4−2z4a4 + z2a4 + 3a4 + z4a2 + 2z2a2 + a2

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

z4a12−z2a12 + 3z5a11−2z3a11 + 6z6a10−7z4a10 + 4z2a10 + 8z7a9−14z5a9 + 14z3a9−6za9 + 6z8a8−7z6a8 + 2z4a8 + 2a8 + 2z9a7 + 8z7a7−26z5a7 + 25z3a7−12za7 + 10z8a6−23z6a6 + 17z4a6−10z2a6 + 5a6 + 2z9a5 + 3z7a5−17z5a5 + 14z3a5−6za5 + 4z8a4−9z6a4 + 4z4a4−2z2a4 + 3a4 + 3z7a3−8z5a3 + 5z3a3 + z6a2−3z4a2 + 3z2a2−a2

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 98"];

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 + 9t2−18t + 23−18t−1 + 9t−2−2t−3, 1−3q−1 + 7q−2−9q−3 + 13q−4−14q−5 + 12q−6−11q−7 + 7q−8−3q−9 + q−10 }

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_87, K11a58, K11a165, K11n72,}

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	q2−3q + 1 + 11q−1−17q−2−7q−3 + 45q−4−34q−5−40q−6 + 94q−7−33q−8−93q−9 + 130q−10−13q−11−137q−12 + 136q−13 + 17q−14−147q−15 + 110q−16 + 37q−17−116q−18 + 63q−19 + 33q−20−62q−21 + 22q−22 + 16q−23−19q−24 + 5q−25 + 3q−26−3q−27 + q−28
3	q6−3q5 + q4 + 5q3 + 3q2−17q−10 + 34q−1 + 35q−2−55q−3−80q−4 + 58q−5 + 163q−6−47q−7−245q−8−26q−9 + 349q−10 + 121q−11−403q−12−279q−13 + 450q−14 + 421q−15−418q−16−599q−17 + 383q−18 + 718q−19−285q−20−844q−21 + 193q−22 + 920q−23−83q−24−955q−25−34q−26 + 961q−27 + 131q−28−894q−29−238q−30 + 807q−31 + 286q−32−649q−33−329q−34 + 495q−35 + 304q−36−325q−37−261q−38 + 195q−39 + 193q−40−104q−41−118q−42 + 42q−43 + 70q−44−21q−45−31q−46 + 9q−47 + 13q−48−6q−49−3q−50 + q−51 + 3q−52−3q−53 + q−54
4	q12−3q11 + q10 + 5q9−3q8 + 3q7−20q6 + 2q5 + 36q4 + 7q3 + 15q2−109q−57 + 109q−1 + 125q−2 + 177q−3−280q−4−366q−5−17q−6 + 313q−7 + 815q−8−139q−9−853q−10−770q−11−20q−12 + 1774q−13 + 855q−14−712q−15−1910q−16−1505q−17 + 2073q−18 + 2368q−19 + 746q−20−2366q−21−3698q−22 + 1000q−23 + 3274q−24 + 3057q−25−1488q−26−5467q−27−997q−28 + 3002q−29 + 5172q−30 + 266q−31−6245q−32−3014q−33 + 1950q−34 + 6557q−35 + 2109q−36−6208q−37−4600q−38 + 638q−39 + 7197q−40 + 3709q−41−5511q−42−5652q−43−828q−44 + 6975q−45 + 4939q−46−4023q−47−5826q−48−2340q−49 + 5546q−50 + 5354q−51−1880q−52−4700q−53−3280q−54 + 3163q−55 + 4448q−56−5q−57−2607q−58−2975q−59 + 978q−60 + 2591q−61 + 710q−62−758q−63−1754q−64−46q−65 + 964q−66 + 474q−67 + 48q−68−664q−69−147q−70 + 216q−71 + 121q−72 + 121q−73−170q−74−38q−75 + 36q−76−3q−77 + 47q−78−36q−79−2q−80 + 10q−81−9q−82 + 10q−83−7q−84 + q−85 + 3q−86−3q−87 + q−88