10 107

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["10 107"];

In[4]:=

PD[K]

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

Out[4]=

X1425 X3,12,4,13 X7,14,8,15 X9,19,10,18 X19,7,20,6 X5,17,6,16 X17,11,18,10 X13,8,14,9 X15,1,16,20 X11,2,12,3

In[5]:=

GaussCode[K]

Out[5]=

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

In[6]:=

DTCode[K]

Out[6]=

4 12 16 14 18 2 8 20 10 6

(The path below may be different on your system)

In[7]:=

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

In[8]:=

ConwayNotation[K]

Out[8]=

[210:2:20]

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

In[14]:=

Draw[ap]

Out[14]=

-Graphics-

Further quantum knot invariants for 10_107.

A1 Invariants.

Weight	Invariant
1	−q11 + 3q9−4q7 + 4q5−3q3 + q + 2q−1−2q−3 + 5q−5−4q−7 + 2q−9−q−11
2	q32−3q30 + 11q26−13q24−11q22 + 34q20−13q18−36q16 + 44q14 + 6q12−46q10 + 26q8 + 22q6−29q4−6q2 + 24 + 4q−2−34q−4 + 15q−6 + 36q−8−43q−10−5q−12 + 47q−14−27q−16−19q−18 + 28q−20−5q−22−11q−24 + 7q−26−2q−30 + q−32
3	−q63 + 3q61−7q57−2q55 + 17q53 + 14q51−40q49−38q47 + 59q45 + 92q43−62q41−174q39 + 34q37 + 257q35 + 44q33−315q31−159q29 + 327q27 + 272q25−281q23−358q21 + 188q19 + 399q17−82q15−384q13−18q11 + 328q9 + 115q7−253q5−188q3 + 159q + 257q−1−64q−3−308q−5−47q−7 + 342q−9 + 162q−11−341q−13−267q−15 + 293q−17 + 351q−19−201q−21−384q−23 + 85q−25 + 365q−27 + 11q−29−282q−31−86q−33 + 188q−35 + 104q−37−100q−39−83q−41 + 37q−43 + 52q−45−10q−47−26q−49 + 3q−51 + 10q−53−q−55−3q−57 + 2q−61−q−63

A2 Invariants.

Weight	Invariant
1,0	−q16 + q14 + 2q12−3q10 + 2q8−q6−2q4 + 3q2−2 + 4q−2−q−4 + q−6 + 3q−8−3q−10 + q−12−q−16
2,0	q42−q40−3q38 + q36 + 7q34 + 2q32−13q30−5q28 + 17q26 + 5q24−20q22−6q20 + 22q18 + 8q16−24q14 + q12 + 21q10−6q8−11q6 + 5q4 + 3q2−10 + 6q−2 + 8q−4−13q−6−3q−8 + 24q−10 + 5q−12−26q−14 + 6q−16 + 22q−18−4q−20−20q−22 + 15q−26−3q−28−9q−30 + q−32 + 4q−34−2q−38 + q−42

A3 Invariants.

Weight	Invariant
0,1,0	q34−3q32 + q30 + 7q28−13q26 + 4q24 + 19q22−28q20 + 6q18 + 28q16−36q14 + 4q12 + 28q10−24q8−4q6 + 16q4−q2−9−4q−2 + 21q−4−4q−6−23q−8 + 34q−10 + 2q−12−33q−14 + 30q−16−25q−20 + 16q−22 + q−24−10q−26 + 5q−28 + q−30−2q−32 + q−34
1,0,0	−q21 + q19 + 2q15−3q13 + 3q11−3q9 + q7−2q5 + 2q3 + 3q−3−q−5 + 3q−7−q−9 + 4q−11−3q−13 + q−15−q−17−q−21

B2 Invariants.

Weight	Invariant
0,1	−q34 + 3q32−7q30 + 13q28−21q26 + 30q24−37q22 + 42q20−42q18 + 36q16−24q14 + 8q12 + 12q10−34q8 + 54q6−70q4 + 79q2−81 + 74q−2−59q−4 + 42q−6−19q−8 + 20q−12−31q−14 + 40q−16−42q−18 + 39q−20−32q−22 + 23q−24−16q−26 + 9q−28−5q−30 + 2q−32−q−34
1,0	q56−3q52−3q50 + 4q48 + 10q46−17q42−12q40 + 18q38 + 28q36−6q34−39q32−15q30 + 37q28 + 34q26−22q24−44q22 + q20 + 42q18 + 15q16−32q14−22q12 + 22q10 + 25q8−14q6−27q4 + 7q2 + 29−q−2−30q−4−5q−6 + 31q−8 + 17q−10−29q−12−27q−14 + 24q−16 + 43q−18−7q−20−45q−22−14q−24 + 37q−26 + 30q−28−19q−30−35q−32−2q−34 + 24q−36 + 12q−38−10q−40−13q−42 + 7q−46 + 3q−48−2q−50−2q−52 + q−56

G2 Invariants.

Weight

Invariant

1,0

q80−3q78 + 7q76−13q74 + 15q72−14q70 + 3q68 + 22q66−52q64 + 86q62−103q60 + 81q58−21q56−81q54 + 193q52−265q50 + 263q48−160q46−20q44 + 222q42−364q40 + 386q38−262q36 + 37q34 + 184q32−320q30 + 303q28−144q26−75q24 + 258q22−309q20 + 196q18 + 30q16−286q14 + 447q12−447q10 + 279q8−290q4 + 500q2−540 + 409q−2−151q−4−144q−6 + 361q−8−422q−10 + 318q−12−95q−14−130q−16 + 276q−18−268q−20 + 115q−22 + 106q−24−293q−26 + 355q−28−262q−30 + 54q−32 + 177q−34−333q−36 + 372q−38−279q−40 + 115q−42 + 54q−44−183q−46 + 223q−48−191q−50 + 117q−52−32q−54−29q−56 + 60q−58−67q−60 + 53q−62−33q−64 + 13q−66 + q−68−9q−70 + 9q−72−8q−74 + 5q−76−2q−78 + q−80

.

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["10 107"];

In[4]:=

Alexander[K][t]

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

Out[4]=

−t3 + 8t2−22t + 31−22t−1 + 8t−2−t−3

In[5]:=

Conway[K][z]

Out[5]=

−z6 + 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]=

−q5 + 3q4−7q3 + 12q2−14q + 16−15q−1 + 12q−2−8q−3 + 4q−4−q−5

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

−z6 + 2a2z4 + 2z4a−2−2z4−a4z2 + 2a2z2 + 3z2a−2−z2a−4−2z2 + 2a−2−a−4

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

2az9 + 2z9a−1 + 6a2z8 + 5z8a−2 + 11z8 + 7a3z7 + 11az7 + 9z7a−1 + 5z7a−3 + 4a4z6−5a2z6−4z6a−2 + 3z6a−4−16z6 + a5z5−12a3z5−27az5−22z5a−1−7z5a−3 + z5a−5−6a4z4−4a2z4−2z4a−2−5z4a−4 + 5z4−a5z3 + 6a3z3 + 17az3 + 15z3a−1 + 3z3a−3−2z3a−5 + 2a4z2 + 2a2z2 + 3z2a−2 + 3z2a−4−a3z−3az−3za−1 + za−5−2a−2−a−4

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["10 107"];

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 + 8t2−22t + 31−22t−1 + 8t−2−t−3, −q5 + 3q4−7q3 + 12q2−14q + 16−15q−1 + 12q−2−8q−3 + 4q−4−q−5 }

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	q15−3q14 + 2q13 + 8q12−21q11 + 8q10 + 41q9−68q8 + 115q6−120q5−38q4 + 194q3−141q2−87q + 232−121q−1−117q−2 + 209q−3−70q−4−113q−5 + 137q−6−18q−7−75q−8 + 57q−9 + 5q−10−28q−11 + 12q−12 + 3q−13−4q−14 + q−15
3	−q30 + 3q29−2q28−3q27 + q26 + 14q25−9q24−32q23 + 17q22 + 76q21−24q20−152q19 + 280q17 + 60q16−426q15−196q14 + 573q13 + 414q12−706q11−665q10 + 756q9 + 966q8−764q7−1225q6 + 682q5 + 1469q4−584q3−1614q2 + 421q + 1713−263q−1−1712q−2 + 74q−3 + 1648q−4 + 105q−5−1499q−6−272q−7 + 1282q−8 + 407q−9−1018q−10−483q−11 + 736q−12 + 484q−13−465q−14−428q−15 + 250q−16 + 328q−17−106q−18−215q−19 + 27q−20 + 120q−21 + 6q−22−61q−23−6q−24 + 23q−25 + 4q−26−7q−27−3q−28 + 4q−29−q−30
4	q50−3q49 + 2q48 + 3q47−6q46 + 6q45−13q44 + 15q43 + 21q42−42q41−q40−45q39 + 96q38 + 138q37−149q36−151q35−260q34 + 330q33 + 694q32−90q31−609q30−1281q29 + 319q28 + 2052q27 + 1043q26−782q25−3636q24−1189q23 + 3436q22 + 3898q21 + 884q20−6369q19−4885q18 + 3065q17 + 7369q16 + 4989q15−7525q14−9427q13 + 286q12 + 9469q11 + 10039q10−6397q9−12745q8−3593q7 + 9462q6 + 14012q5−3925q4−13971q3−7011q2 + 7925q + 16069−1101q−1−13328q−2−9408q−3 + 5386q−4 + 16197q−5 + 1818q−6−10982q−7−10666q−8 + 1973q−9 + 14263q−10 + 4470q−11−7006q−12−10208q−13−1651q−14 + 10202q−15 + 5740q−16−2377q−17−7599q−18−3896q−19 + 5182q−20 + 4758q−21 + 904q−22−3850q−23−3702q−24 + 1377q−25 + 2416q−26 + 1706q−27−1013q−28−2035q−29−111q−30 + 597q−31 + 996q−32 + 39q−33−661q−34−181q−35−17q−36 + 305q−37 + 101q−38−133q−39−34q−40−45q−41 + 54q−42 + 26q−43−22q−44 + q−45−9q−46 + 7q−47 + 3q−48−4q−49 + q−50