10 146

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["10 146"];

In[4]:=

PD[K]

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

Out[4]=

X4251 X5,18,6,19 X8394 X2,9,3,10 X11,17,12,16 X7,12,8,13 X15,6,16,7 X17,11,18,10 X13,1,14,20 X19,15,20,14

In[5]:=

GaussCode[K]

Out[5]=

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

In[6]:=

DTCode[K]

Out[6]=

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

(The path below may be different on your system)

In[7]:=

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

In[8]:=

ConwayNotation[K]

Out[8]=

[22,21,21-]

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

In[14]:=

Draw[ap]

Out[14]=

-Graphics-

Further quantum knot invariants for 10_146.

A1 Invariants.

Weight	Invariant
1	−q11 + 2q9−q7 + q5−q3 + 2q−1−q−3 + 2q−5−q−7
2	q32−2q30−2q28 + 6q26−7q22 + 4q20 + 4q18−8q16 + 7q12−3q10−q8 + 5q6 + q4−5q2−1 + 7q−2−5q−4−5q−6 + 9q−8−5q−12 + 4q−14 + q−16−2q−18
3	−q63 + 2q61 + 2q59−3q57−6q55 + 13q51 + 5q49−13q47−14q45 + 8q43 + 23q41−28q37−13q35 + 25q33 + 26q31−18q29−32q27 + 11q25 + 33q23−2q21−31q19−4q17 + 25q15 + 7q13−19q11−11q9 + 15q7 + 16q5−4q3−23q−3q−1 + 27q−3 + 14q−5−26q−7−26q−9 + 22q−11 + 32q−13−12q−15−32q−17 + q−19 + 26q−21 + 8q−23−17q−25−8q−27 + 6q−29 + 8q−31−q−33−4q−35−q−37 + q−41

A2 Invariants.

Weight	Invariant
1,0	−q16 + q14 + q12−q10 + q8−q6 + q2 + 2q−2−q−4 + q−6 + q−8−q−10
1,1	q44−4q42 + 10q40−22q38 + 38q36−54q34 + 72q32−86q30 + 85q28−70q26 + 42q24−4q22−41q20 + 86q18−122q16 + 146q14−153q12 + 146q10−126q8 + 96q6−52q4 + 12q2 + 34−62q−2 + 80q−4−92q−6 + 80q−8−64q−10 + 43q−12−22q−14 + 14q−16−2q−24−2q−26 + q−28
2,0	q42−q40−2q38 + 3q34 + 3q32−4q30−2q28 + 2q26 + 2q24−3q22−4q20 + 3q18 + 2q16−q14 + 4q10 + q8 + q6 + 2q4−2q2−1 + q−4−5q−6−2q−8 + 6q−10 + 3q−12−3q−14 + 4q−18 + 2q−20−2q−22−2q−24

A3 Invariants.

Weight	Invariant
0,1,0	q34−2q32 + 2q28−4q26 + 4q24 + 2q22−5q20 + 4q18 + q16−5q14 + q12 + 2q10−2q8 + q4 + 3q2−q−2 + 7q−4−3q−6−2q−8 + 6q−10−3q−12−3q−14 + 3q−16−q−18−q−20 + q−22
1,0,0	−q21 + q19 + q15−q13 + q11−q9 + q3 + q + 2q−3−q−5 + q−7 + q−11−q−13

A4 Invariants.

Weight	Invariant
0,1,0,0	q44−q42−2q40 + 2q38 + q36−3q34 + 5q30−q28−5q26 + 2q24 + 6q22−3q20−4q18 + 6q16−6q12 + 2q8−4q6−q4 + 7q2 + 3−q−2 + 5q−4 + 8q−6−3q−8−3q−10 + 3q−12−5q−16−2q−18 + 2q−20 + q−22−q−24 + q−28
1,0,0,0	−q26 + q24 + q18−q16 + q14−q12 + q4 + q2 + 1 + 2q−4−q−6 + q−8 + q−14−q−16

B2 Invariants.

Weight	Invariant
0,1	−q34 + 2q32−4q30 + 6q28−6q26 + 6q24−6q22 + 5q20−2q18−q16 + 5q14−7q12 + 10q10−12q8 + 12q6−11q4 + 9q2−6 + 3q−2 + q−4−3q−6 + 6q−8−6q−10 + 7q−12−5q−14 + 5q−16−3q−18 + q−20−q−22
1,0	q56−2q52−2q50 + 2q48 + 4q46−2q44−5q42 + 7q38 + 4q36−6q34−6q32 + 3q30 + 6q28−6q24−2q22 + 4q20 + 3q18−3q16−3q14 + 3q12 + 4q10−q8−5q6 + 6q2 + 2−5q−2−3q−4 + 6q−6 + 5q−8−3q−10−6q−12 + 2q−14 + 7q−16 + 2q−18−5q−20−4q−22 + q−24 + 4q−26−2q−30−q−32 + q−36

D4 Invariants.

Weight	Invariant
1,0,0,0	q46−2q44 + 2q42−4q40 + 5q38−5q36 + 5q34−5q32 + 6q30−3q28 + 2q26−q22 + 2q20−6q18 + 6q16−8q14 + 8q12−10q10 + 9q8−7q6 + 10q4−5q2 + 6−q−2 + 3q−4 + 3q−6−3q−8 + 3q−10−5q−12 + 6q−14−5q−16 + 3q−18−5q−20 + 4q−22−2q−24 + q−26−q−28 + q−30

G2 Invariants.

Weight

Invariant

1,0

q80−2q78 + 4q76−7q74 + 5q72−2q70−6q68 + 16q66−19q64 + 20q62−13q60−4q58 + 19q56−29q54 + 29q52−14q50−4q48 + 20q46−22q44 + 16q42−q40−18q38 + 23q36−21q34 + 7q32 + 13q30−31q28 + 37q26−24q24 + 10q22 + 6q20−27q18 + 34q16−31q14 + 22q12−3q10−15q8 + 28q6−23q4 + 12q2 + 4−18q−2 + 20q−4−13q−6−q−8 + 22q−10−29q−12 + 27q−14−11q−16−8q−18 + 22q−20−27q−22 + 19q−24−8q−26 + q−28 + 8q−30−12q−32 + 8q−34−3q−36 + 2q−38−2q−42−q−44 + q−48−q−50 + q−52

.

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["10 146"];

In[4]:=

Alexander[K][t]

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

Out[4]=

2t2−8t + 13−8t−1 + 2t−2

In[5]:=

Conway[K][z]

Out[5]=

2z4 + 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]=

{ 33, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

−q3 + 3q2−4q + 6−6q−1 + 5q−2−4q−3 + 3q−4−q−5

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

−z2a4 + z4a2 + z2a2 + z4 + z2 + 1−z2a−2

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

a2z8 + z8 + 3a3z7 + 4az7 + z7a−1 + 3a4z6 + a2z6−2z6 + a5z5−8a3z5−11az5−2z5a−1−8a4z4−6a2z4 + 3z4a−2 + 5z4−2a5z3 + 5a3z3 + 12az3 + 6z3a−1 + z3a−3 + 3a4z2 + 3a2z2−3z2a−2−3z2−a3z−3az−3za−1−za−3 + 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["10 146"];

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]=

{ 2t2−8t + 13−8t−1 + 2t−2, −q3 + 3q2−4q + 6−6q−1 + 5q−2−4q−3 + 3q−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]=

{K11n18, K11n62,}

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	−2q8 + 3q7 + 3q6−11q5 + 8q4 + 12q3−25q2 + 8q + 24−33q−1 + 4q−2 + 30q−3−29q−4−2q−5 + 28q−6−19q−7−9q−8 + 20q−9−7q−10−9q−11 + 9q−12−3q−14 + q−15
3	q19−q18−q17−3q16 + 4q15 + 8q14−3q13−17q12−5q11 + 33q10 + 15q9−42q8−38q7 + 53q6 + 59q5−52q4−86q3 + 53q2 + 99q−39−116q−1 + 33q−2 + 118q−3−19q−4−117q−5 + 7q−6 + 110q−7 + 7q−8−99q−9−22q−10 + 83q−11 + 36q−12−64q−13−44q−14 + 40q−15 + 50q−16−20q−17−45q−18 + 2q−19 + 35q−20 + 8q−21−22q−22−13q−23 + 13q−24 + 9q−25−4q−26−5q−27 + 3q−29−q−30
4	−q32 + q31 + 3q30−2q28−10q27−5q26 + 16q25 + 18q24 + 13q23−36q22−57q21 + 11q20 + 64q19 + 99q18−30q17−169q16−81q15 + 76q14 + 262q13 + 85q12−265q11−257q10−20q9 + 415q8 + 286q7−271q6−413q5−187q4 + 475q3 + 459q2−204q−472−335q−1 + 450q−2 + 543q−3−126q−4−450q−5−415q−6 + 381q−7 + 544q−8−49q−9−372q−10−448q−11 + 267q−12 + 489q−13 + 47q−14−242q−15−444q−16 + 109q−17 + 367q−18 + 135q−19−64q−20−371q−21−41q−22 + 183q−23 + 149q−24 + 96q−25−217q−26−101q−27 + 12q−28 + 73q−29 + 149q−30−63q−31−58q−32−55q−33−15q−34 + 95q−35 + 8q−36 + q−37−36q−38−36q−39 + 31q−40 + 8q−41 + 14q−42−6q−43−16q−44 + 4q−45 + 5q−47−3q−49 + q−50
5	−2q46 + 5q44 + 5q43−5q41−22q40−17q39 + 15q38 + 45q37 + 49q36 + 12q35−76q34−134q33−71q32 + 90q31 + 239q30 + 211q29−37q28−358q27−443q26−104q25 + 444q24 + 713q23 + 383q22−417q21−1034q20−774q19 + 299q18 + 1271q17 + 1238q16−9q15−1457q14−1706q13−348q12 + 1497q11 + 2121q10 + 774q9−1458q8−2435q7−1150q6 + 1296q5 + 2649q4 + 1503q3−1150q2−2744q−1730 + 942q−1 + 2762q−2 + 1932q−3−798q−4−2730q−5−2016q−6 + 627q−7 + 2647q−8 + 2093q−9−474q−10−2542q−11−2118q−12 + 303q−13 + 2370q−14 + 2138q−15−92q−16−2159q−17−2122q−18−146q−19 + 1864q−20 + 2067q−21 + 410q−22−1501q−23−1946q−24−663q−25 + 1081q−26 + 1737q−27 + 859q−28−633q−29−1428q−30−974q−31 + 215q−32 + 1063q−33 + 946q−34 + 127q−35−658q−36−816q−37−339q−38 + 298q−39 + 586q−40 + 418q−41−20q−42−349q−43−360q−44−142q−45 + 125q−46 + 255q−47 + 190q−48 + 11q−49−125q−50−150q−51−88q−52 + 25q−53 + 101q−54 + 89q−55 + 18q−56−35q−57−60q−58−47q−59 + 9q−60 + 36q−61 + 29q−62 + 6q−63−8q−64−18q−65−14q−66 + 6q−67 + 9q−68 + 3q−69−5q−72 + 3q−74−q−75
6	q68−q67−q66−2q63−3q62 + 10q61 + 8q60 + 5q59 + 2q58−11q57−36q56−51q55 + 2q54 + 52q53 + 90q52 + 114q51 + 60q50−114q49−288q48−262q47−84q46 + 205q45 + 546q44 + 645q43 + 206q42−547q41−1049q40−1041q39−388q38 + 915q37 + 1991q36 + 1786q35 + 248q34−1692q33−2968q32−2693q31−157q30 + 3089q29 + 4578q28 + 3112q27−593q26−4539q25−6351q24−3522q23 + 2249q22 + 6962q21 + 7300q20 + 2819q19−4116q18−9482q17−8029q16−769q15 + 7388q14 + 10770q13 + 7097q12−1757q11−10661q10−11619q9−4382q8 + 6077q7 + 12288q6 + 10342q5 + 1019q4−10185q3−13327q2−7014q + 4337 + 12253q−1 + 11916q−2 + 3013q−3−9143q−4−13633q−5−8352q−6 + 2987q−7 + 11590q−8 + 12374q−9 + 4174q−10−8078q−11−13313q−12−8999q−13 + 1882q−14 + 10654q−15 + 12370q−16 + 5123q−17−6725q−18−12568q−19−9519q−20 + 396q−21 + 9107q−22 + 12003q−23 + 6335q−24−4511q−25−11000q−26−9870q−27−1796q−28 + 6409q−29 + 10771q−30 + 7563q−31−1310q−32−8082q−33−9319q−34−4130q−35 + 2602q−36 + 8045q−37 + 7763q−38 + 1982q−39−3984q−40−7074q−41−5221q−42−1130q−43 + 4071q−44 + 6014q−45 + 3720q−46−115q−47−3501q−48−4123q−49−3012q−50 + 447q−51 + 2891q−52 + 3087q−53 + 1745q−54−378q−55−1682q−56−2487q−57−1152q−58 + 302q−59 + 1223q−60 + 1377q−61 + 814q−62 + 165q−63−955q−64−843q−65−547q−66−48q−67 + 313q−68 + 497q−69 + 567q−70−28q−71−124q−72−285q−73−232q−74−181q−75 + 16q−76 + 262q−77 + 88q−78 + 123q−79−q−80−38q−81−141q−82−96q−83 + 45q−84−2q−85 + 63q−86 + 39q−87 + 38q−88−36q−89−40q−90 + q−91−20q−92 + 9q−93 + 9q−94 + 23q−95−6q−96−9q−97 + 4q−98−7q−99 + 5q−102−3q−104 + q−105