10 122

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["10 122"];

In[4]:=

PD[K]

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

Out[4]=

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

In[5]:=

GaussCode[K]

Out[5]=

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

In[6]:=

DTCode[K]

Out[6]=

6 10 12 14 18 16 20 2 4 8

(The path below may be different on your system)

In[7]:=

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

In[8]:=

ConwayNotation[K]

Out[8]=

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

In[14]:=

Draw[ap]

Out[14]=

-Graphics-

Further quantum knot invariants for 10_122.

A1 Invariants.

Weight	Invariant
1	q9−3q7 + 4q5−5q3 + 4q + 4q−5−4q−7 + 4q−9−4q−11 + q−13
2	q26−3q24 + q22 + 7q20−14q18 + 7q16 + 23q14−40q12 + q10 + 52q8−43q6−24q4 + 54q2−9−35q−2 + 20q−4 + 27q−6−25q−8−20q−10 + 45q−12−3q−14−51q−16 + 40q−18 + 24q−20−56q−22 + 12q−24 + 36q−26−28q−28−10q−30 + 18q−32−q−34−4q−36 + q−38
3	q51−3q49 + q47 + 4q45−2q43−5q41 + 6q39 + 13q37−32q35−24q33 + 72q31 + 68q29−125q27−165q25 + 162q23 + 308q21−132q19−453q17 + 19q15 + 544q13 + 147q11−543q9−308q7 + 419q5 + 433q3−247q−463q−1 + 53q−3 + 437q−5 + 119q−7−367q−9−253q−11 + 290q−13 + 360q−15−208q−17−441q−19 + 111q−21 + 511q−23 + 5q−25−542q−27−156q−29 + 525q−31 + 300q−33−420q−35−431q−37 + 258q−39 + 476q−41−67q−43−426q−45−97q−47 + 300q−49 + 184q−51−153q−53−177q−55 + 35q−57 + 118q−59 + 26q−61−57q−63−25q−65 + 12q−67 + 13q−69−q−71−4q−73 + q−75

A2 Invariants.

Weight	Invariant
1,0	q12−2q10 + q8 + q6−4q4 + 3q2−2 + 2q−2 + 3q−4 + 5q−8−3q−10−3q−16 + q−18
1,1	q36−6q34 + 18q32−38q30 + 71q28−128q26 + 212q24−320q22 + 460q20−650q18 + 896q16−1186q14 + 1517q12−1834q10 + 2064q8−2134q6 + 1924q4−1414q2 + 586 + 492q−2−1670q−4 + 2834q−6−3788q−8 + 4442q−10−4690q−12 + 4518q−14−3944q−16 + 3034q−18−1890q−20 + 652q−22 + 538q−24−1540q−26 + 2230q−28−2566q−30 + 2544q−32−2260q−34 + 1804q−36−1300q−38 + 842q−40−480q−42 + 244q−44−104q−46 + 34q−48−8q−50 + q−52
2,0	q32−2q30−q28 + 5q26−2q24−8q22 + 6q20 + 14q18−8q16−20q14 + 10q12 + 26q10−20q8−20q6 + 18q4 + 10q2−13−11q−2 + 17q−4−2q−6−6q−8 + 13q−10 + 5q−12−10q−14 + 12q−16 + 21q−18−17q−20−14q−22 + 11q−24 + 10q−26−21q−28−18q−30 + 19q−32 + 9q−34−12q−36−7q−38 + 7q−40 + 9q−42−3q−44−3q−46 + q−48

A3 Invariants.

Weight	Invariant
0,1,0	q28−3q26 + 9q22−10q20−6q18 + 24q16−15q14−20q12 + 40q10−15q8−29q6 + 40q4−13q2−24 + 18q−2 + 4q−4−4q−6−4q−8 + 22q−10 + 17q−12−28q−14 + 11q−16 + 23q−18−43q−20 + 3q−22 + 26q−24−34q−26 + 10q−28 + 16q−30−17q−32 + 7q−34 + 3q−36−4q−38 + q−40
1,0,0	q15−2q13 + 2q11−2q9 + 2q7−4q5 + 3q3−3q + q−1 + q−3 + 2q−5 + 4q−7 + q−9 + 5q−11−3q−13 + q−15−4q−17 + q−19−3q−21 + q−23

A4 Invariants.

Weight	Invariant
0,1,0,0	q34−2q32−2q30 + 6q28 + q26−9q24−q22 + 14q20 + q18−22q16 + 4q14 + 30q12−13q10−29q8 + 26q6 + 17q4−36q2−12 + 27q−2−8q−4−30q−6 + 25q−8 + 31q−10−24q−12 + 10q−14 + 53q−16−11q−18−32q−20 + 28q−22 + 11q−24−42q−26−19q−28 + 23q−30−2q−32−25q−34 + 9q−36 + 19q−38−8q−40−5q−42 + 9q−44−q−46−3q−48 + q−50
1,0,0,0	q18−2q16 + 2q14−q12−q10 + 2q8−4q6 + 3q4−3q2 + q−4 + 2q−6 + 3q−8 + 5q−10 + q−12 + 5q−14−3q−16 + q−18−3q−20−3q−22 + q−24−3q−26 + q−28

B2 Invariants.

Weight	Invariant
0,1	q28−3q26 + 6q24−11q22 + 18q20−26q18 + 36q16−45q14 + 50q12−52q10 + 43q8−29q6 + 8q4 + 17q2−42 + 68q−2−86q−4 + 100q−6−100q−8 + 94q−10−75q−12 + 54q−14−25q−16−q−18 + 25q−20−41q−22 + 50q−24−54q−26 + 50q−28−44q−30 + 31q−32−21q−34 + 11q−36−4q−38 + q−40
1,0	q46−3q42−3q40 + 3q38 + 10q36 + 3q34−14q32−16q30 + 7q28 + 30q26 + 12q24−31q22−35q20 + 14q18 + 52q16 + 10q14−50q12−33q10 + 34q8 + 45q6−18q4−49q2−1 + 44q−2 + 11q−4−36q−6−18q−8 + 31q−10 + 27q−12−19q−14−26q−16 + 22q−18 + 38q−20−8q−22−45q−24−6q−26 + 48q−28 + 27q−30−44q−32−52q−34 + 18q−36 + 55q−38 + 6q−40−48q−42−28q−44 + 29q−46 + 34q−48−9q−50−24q−52−4q−54 + 14q−56 + 7q−58−4q−60−4q−62 + q−66

D4 Invariants.

Weight	Invariant
1,0,0,0	q38−3q36 + 3q34−4q32 + 10q30−14q28 + 14q26−19q24 + 30q22−33q20 + 31q18−39q16 + 45q14−37q12 + 29q10−26q8 + 13q6 + 4q4−18q2 + 26−50q−2 + 61q−4−66q−6 + 76q−8−76q−10 + 84q−12−62q−14 + 70q−16−49q−18 + 39q−20−21q−22 + 6q−24−26q−28 + 29q−30−40q−32 + 38q−34−44q−36 + 44q−38−37q−40 + 33q−42−25q−44 + 18q−46−11q−48 + 7q−50−4q−52 + q−54

G2 Invariants.

Weight

Invariant

1,0

q66−3q64 + 6q62−10q60 + 10q58−8q56 + q54 + 15q52−31q50 + 51q48−63q46 + 56q44−32q42−14q40 + 73q38−133q36 + 183q34−198q32 + 149q30−34q28−132q26 + 304q24−402q22 + 375q20−211q18−57q16 + 321q14−469q12 + 426q10−200q8−114q6 + 360q4−422q2 + 253 + 68q−2−380q−4 + 538q−6−459q−8 + 166q−10 + 226q−12−552q−14 + 696q−16−597q−18 + 301q−20 + 97q−22−438q−24 + 626q−26−590q−28 + 366q−30−28q−32−290q−34 + 464q−36−423q−38 + 194q−40 + 131q−42−392q−44 + 458q−46−299q−48−24q−50 + 353q−52−547q−54 + 517q−56−287q−58−43q−60 + 323q−62−459q−64 + 419q−66−246q−68 + 30q−70 + 130q−72−204q−74 + 186q−76−115q−78 + 45q−80 + 12q−82−35q−84 + 34q−86−24q−88 + 11q−90−4q−92 + q−94

.

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["10 122"];

In[4]:=

Alexander[K][t]

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

Out[4]=

−2t3 + 11t2−24t + 31−24t−1 + 11t−2−2t−3

In[5]:=

Conway[K][z]

Out[5]=

−2z6−z4 + 2z2 + 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]=

{ 105, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

q6−5q5 + 9q4−13q3 + 17q2−17q + 17−13q−1 + 8q−2−4q−3 + q−4

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

−z6a−2−z6 + a2z4−z4a−2 + z4a−4−2z4 + a2z2 + 3z2a−2−2z2 + 4a−2−2a−4−1

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

4z9a−1 + 4z9a−3 + 18z8a−2 + 8z8a−4 + 10z8 + 11az7 + 9z7a−1 + 3z7a−3 + 5z7a−5 + 8a2z6−42z6a−2−20z6a−4 + z6a−6−13z6 + 4a3z5−14az5−32z5a−1−25z5a−3−11z5a−5 + a4z4−7a2z4 + 24z4a−2 + 12z4a−4−z4a−6 + 3z4−2a3z3 + 6az3 + 18z3a−1 + 14z3a−3 + 4z3a−5 + 2a2z2 + 2z2 + 2za−3 + 2za−5−4a−2−2a−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["10 122"];

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 + 11t2−24t + 31−24t−1 + 11t−2−2t−3, q6−5q5 + 9q4−13q3 + 17q2−17q + 17−13q−1 + 8q−2−4q−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]=

{K11n185,}

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	q18−5q17 + 3q16 + 20q15−33q14−15q13 + 84q12−57q11−83q10 + 164q9−41q8−174q7 + 212q6 + 7q5−239q4 + 207q3 + 59q2−246q + 152 + 85q−1−183q−2 + 74q−3 + 66q−4−88q−5 + 23q−6 + 25q−7−25q−8 + 7q−9 + 4q−10−4q−11 + q−12
3	q36−5q35 + 3q34 + 14q33−42q31−29q30 + 97q29 + 92q28−125q27−241q26 + 121q25 + 429q24−9q23−638q22−208q21 + 788q20 + 534q19−856q18−897q17 + 799q16 + 1254q15−631q14−1578q13 + 413q12 + 1801q11−125q10−1978q9−139q8 + 2034q7 + 443q6−2048q5−682q4 + 1920q3 + 929q2−1730q−1066 + 1404q−1 + 1145q−2−1050q−3−1080q−4 + 677q−5 + 910q−6−360q−7−683q−8 + 152q−9 + 438q−10−39q−11−243q−12 + 6q−13 + 111q−14 + q−15−50q−16 + 10q−17 + 15q−18−7q−19−5q−20 + 3q−21 + 4q−22−4q−23 + q−24
4	q60−5q59 + 3q58 + 14q57−6q56−9q55−56q54 + 8q53 + 126q52 + 73q51 + 14q50−370q49−285q48 + 281q47 + 586q46 + 741q45−701q44−1468q43−673q42 + 880q41 + 2997q40 + 747q39−2369q38−3644q37−1547q36 + 4943q35 + 4858q34 + 171q33−6159q32−7414q31 + 2969q30 + 8585q29 + 6684q28−4473q27−13248q26−3251q25 + 8247q24 + 13620q23 + 1429q22−15524q21−10334q20 + 4000q19 + 17807q18 + 8401q17−14277q16−15584q15−1524q14 + 19134q13 + 14190q12−11277q11−18735q10−6782q9 + 18435q8 + 18492q7−7057q6−19850q5−11705q4 + 15253q3 + 20794q2−1299q−17650−15351q−1 + 8955q−2 + 19206q−3 + 4533q−4−11417q−5−15187q−6 + 1703q−7 + 13052q−8 + 7028q−9−3926q−10−10460q−11−2413q−12 + 5657q−13 + 5148q−14 + 477q−15−4606q−16−2335q−17 + 1225q−18 + 1994q−19 + 1092q−20−1214q−21−882q−22 + 41q−23 + 342q−24 + 451q−25−215q−26−141q−27−5q−28−13q−29 + 97q−30−43q−31−2q−32 + 12q−33−17q−34 + 13q−35−9q−36 + 3q−37 + 4q−38−4q−39 + q−40