10 164

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["10 164"];

In[4]:=

PD[K]

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

Out[4]=

X6271 X14,7,15,8 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 X20,13,1,14

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]=

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

In[14]:=

Draw[ap]

Out[14]=

-Graphics-

Further quantum knot invariants for 10_164.

A1 Invariants.

Weight	Invariant
1	−q11 + 2q9−2q7 + 2q5−q3 + 2q−1−q−3 + 3q−5−2q−7
2	q32−2q30−q28 + 7q26−4q24−10q22 + 11q20 + 4q18−15q16 + 7q14 + 11q12−11q10−q8 + 9q6−2q4−8q2 + 3 + 10q−2−11q−4−4q−6 + 17q−8−6q−10−9q−12 + 12q−14−7q−18 + q−20 + q−22
3	−q63 + 2q61 + q59−4q57−4q55 + 7q53 + 15q51−8q49−29q47−5q45 + 40q43 + 30q41−40q39−57q37 + 23q35 + 77q33 + 6q31−85q29−33q27 + 75q25 + 54q23−58q21−65q19 + 40q17 + 64q15−19q13−56q11 + q9 + 52q7 + 16q5−42q3−38q + 33q−1 + 58q−3−17q−5−78q−7−4q−9 + 88q−11 + 28q−13−80q−15−51q−17 + 61q−19 + 63q−21−30q−23−59q−25 + 8q−27 + 38q−29 + 11q−31−22q−33−13q−35 + 6q−37 + 5q−39 + 2q−41−2q−43

A2 Invariants.

Weight	Invariant
1,0	−q16 + q14 + q12−2q10 + q8−q6 + 2q2 + 3q−2−q−4 + q−6 + q−8−2q−10
1,1	q44−4q42 + 10q40−22q38 + 46q36−80q34 + 122q32−172q30 + 214q28−236q26 + 224q24−176q22 + 94q20 + 18q18−148q16 + 264q14−353q12 + 418q10−434q8 + 418q6−352q4 + 256q2−144 + 16q−2 + 92q−4−182q−6 + 242q−8−246q−10 + 227q−12−176q−14 + 128q−16−80q−18 + 35q−20−14q−22 + 2q−24−2q−28 + 2q−30
2,0	q42−q40−2q38 + q36 + 5q34 + 2q32−8q30−5q28 + 4q26 + 4q24−6q22−3q20 + 8q18 + 6q16−3q14−2q12 + 5q10−3q8−2q6−2q2−1 + 4q−2 + 5q−4−6q−6 + 9q−10 + 4q−12−9q−14−q−16 + 7q−18 + q−20−5q−22−4q−24 + 2q−26 + q−28

A3 Invariants.

Weight	Invariant
0,1,0	q34−2q32 + 3q28−6q26 + 4q24 + 6q22−10q20 + 6q18 + 4q16−12q14 + 2q12 + 3q10−5q8 + 4q4 + 5q2 + 1−q−2 + 11q−4−3q−6−7q−8 + 11q−10−5q−12−8q−14 + 7q−16−3q−18−3q−20 + 3q−22
1,0,0	−q21 + q19 + q15−2q13 + q11−2q9 + 2q3 + 2q + q−1 + 3q−3−q−5 + q−7−q−9 + q−11−2q−13

A4 Invariants.

Weight	Invariant
0,1,0,0	q44−q42−2q40 + 2q38 + 2q36−4q34−2q32 + 7q30 + 2q28−8q26 + q24 + 11q22−4q20−11q18 + 5q16 + 3q14−12q12−6q10 + 6q8−2q6−4q4 + 14q2 + 13−q−2 + 7q−4 + 15q−6−5q−8−11q−10 + 2q−12 + 2q−14−10q−16−6q−18 + 5q−20 + 2q−22−4q−24 + 3q−28
1,0,0,0	−q26 + q24 + q18−2q16 + q14−2q12−q10 + 2q4 + 2q2 + 3 + q−2 + 3q−4−q−6 + q−8−q−10−q−12 + q−14−2q−16

B2 Invariants.

Weight	Invariant
0,1	−q34 + 2q32−4q30 + 7q28−10q26 + 12q24−12q22 + 10q20−8q18 + 4q16 + 2q14−8q12 + 15q10−19q8 + 22q6−22q4 + 21q2−17 + 11q−2−3q−4−q−6 + 7q−8−9q−10 + 13q−12−12q−14 + 11q−16−9q−18 + 5q−20−3q−22
1,0	q56−2q52−2q50 + 2q48 + 5q46−q44−8q42−3q40 + 10q38 + 9q36−7q34−12q32 + q30 + 13q28 + 4q26−12q24−8q22 + 6q20 + 8q18−4q16−8q14 + 2q12 + 9q10 + q8−8q6−2q4 + 9q2 + 7−6q−2−8q−4 + 7q−6 + 11q−8−2q−10−12q−12−q−14 + 12q−16 + 8q−18−9q−20−11q−22 + 2q−24 + 9q−26 + q−28−6q−30−4q−32 + q−34 + 3q−36

D4 Invariants.

Weight	Invariant
1,0,0,0	q46−2q44 + 2q42−4q40 + 6q38−8q36 + 9q34−9q32 + 11q30−8q28 + 7q26−5q24 + 3q22−q20−8q18 + 6q16−12q14 + 12q12−18q10 + 17q8−14q6 + 21q4−11q2 + 16−6q−2 + 11q−4 + q−6−2q−8 + 2q−10−7q−12 + 8q−14−11q−16 + 7q−18−12q−20 + 9q−22−6q−24 + 4q−26−4q−28 + 3q−30

G2 Invariants.

Weight

Invariant

1,0

q80−2q78 + 4q76−7q74 + 6q72−5q70−2q68 + 15q66−26q64 + 36q62−32q60 + 11q58 + 17q56−50q54 + 70q52−61q50 + 29q48 + 17q46−54q44 + 69q42−52q40 + 12q38 + 30q36−61q34 + 50q32−15q30−32q28 + 69q26−75q24 + 52q22−7q20−40q18 + 74q16−93q14 + 78q12−34q10−12q8 + 56q6−77q4 + 75q2−36−6q−2 + 40q−4−58q−6 + 45q−8 + q−10−43q−12 + 69q−14−56q−16 + 21q−18 + 28q−20−66q−22 + 72q−24−55q−26 + 22q−28 + 13q−30−40q−32 + 45q−34−33q−36 + 16q−38 + q−40−10q−42 + 7q−44−9q−46 + 5q−48−2q−50 + q−52 + q−54

.

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["10 164"];

In[4]:=

Alexander[K][t]

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

Out[4]=

3t2−11t + 17−11t−1 + 3t−2

In[5]:=

Conway[K][z]

Out[5]=

3z4 + 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]=

{ 45, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

−2q3 + 5q2−6q + 8−8q−1 + 7q−2−5q−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−a2 + 2z4 + 4z2 + 3−2z2a−2−a−2

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

2a2z8 + 2z8 + 4a3z7 + 7az7 + 3z7a−1 + 3a4z6−a2z6 + z6a−2−3z6 + a5z5−10a3z5−17az5−6z5a−1−7a4z4−3a2z4 + 4z4a−2 + 8z4−2a5z3 + 7a3z3 + 16az3 + 10z3a−1 + 3z3a−3 + 3a4z2−6z2a−2−9z2−2a3z−5az−5za−1−2za−3 + a2 + a−2 + 3

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["10 164"];

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]=

{ 3t2−11t + 17−11t−1 + 3t−2, −2q3 + 5q2−6q + 8−8q−1 + 7q−2−5q−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]=

{10_10,}

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	q10−8q8 + 8q7 + 12q6−29q5 + 11q4 + 35q3−50q2 + 4q + 56−57q−1−7q−2 + 62q−3−46q−4−17q−5 + 52q−6−24q−7−21q−8 + 30q−9−5q−10−14q−11 + 9q−12 + q−13−3q−14 + q−15
3	−2q20 + 4q19 + 3q18 + q17−21q16−5q15 + 36q14 + 28q13−51q12−72q11 + 65q10 + 121q9−53q8−184q7 + 36q6 + 229q5 + 7q4−276q3−38q2 + 290q + 82−301q−1−109q−2 + 286q−3 + 140q−4−265q−5−160q−6 + 229q−7 + 177q−8−182q−9−184q−10 + 124q−11 + 184q−12−70q−13−163q−14 + 16q−15 + 132q−16 + 21q−17−92q−18−38q−19 + 52q−20 + 38q−21−22q−22−28q−23 + 7q−24 + 14q−25−q−26−5q−27−q−28 + 3q−29−q−30
4	q34−6q32−5q31 + 12q30 + 18q29 + 16q28−31q27−80q26−6q25 + 77q24 + 168q23 + 26q22−268q21−221q20 + 4q19 + 480q18 + 404q17−333q16−643q15−456q14 + 664q13 + 1046q12−17q11−939q10−1180q9 + 485q8 + 1568q7 + 529q6−896q5−1771q4 + 98q3 + 1746q2 + 972q−635−2039q−1−246q−2 + 1657q−3 + 1193q−4−327q−5−2034q−6−505q−7 + 1389q−8 + 1269q−9 + 26q−10−1820q−11−744q−12 + 939q−13 + 1218q−14 + 442q−15−1358q−16−898q−17 + 323q−18 + 932q−19 + 775q−20−682q−21−780q−22−219q−23 + 413q−24 + 765q−25−76q−26−378q−27−381q−28−46q−29 + 424q−30 + 149q−31−11q−32−204q−33−170q−34 + 105q−35 + 78q−36 + 80q−37−33q−38−81q−39 + 5q−40 + 3q−41 + 31q−42 + 5q−43−17q−44 + q−45−3q−46 + 5q−47 + q−48−3q−49 + q−50
5	−2q51 + 4q50 + 2q49 + 3q48−q47−23q46−30q45 + 12q44 + 56q43 + 79q42 + 69q41−96q40−248q39−212q38 + 41q37 + 404q36 + 604q35 + 252q34−553q33−1136q32−874q31 + 354q30 + 1728q29 + 1932q28 + 325q27−2134q26−3245q25−1583q24 + 1988q23 + 4596q22 + 3434q21−1263q20−5652q19−5470q18−186q17 + 6145q16 + 7564q15 + 1987q14−6043q13−9195q12−4024q11 + 5405q10 + 10402q9 + 5826q8−4454q7−10965q6−7379q5 + 3386q4 + 11191q3 + 8414q2−2384q−10991−9169q−1 + 1495q−2 + 10715q−3 + 9557q−4−747q−5−10229q−6−9816q−7 + 10q−8 + 9698q−9 + 9940q−10 + 737q−11−8957q−12−9989q−13−1618q−14 + 8002q−15 + 9897q−16 + 2623q−17−6731q−18−9584q−19−3692q−20 + 5146q−21 + 8900q−22 + 4681q−23−3267q−24−7815q−25−5366q−26 + 1318q−27 + 6246q−28 + 5563q−29 + 507q−30−4376q−31−5178q−32−1847q−33 + 2408q−34 + 4209q−35 + 2578q−36−691q−37−2898q−38−2612q−39−517q−40 + 1544q−41 + 2104q−42 + 1098q−43−434q−44−1349q−45−1144q−46−218q−47 + 622q−48 + 842q−49 + 460q−50−114q−51−474q−52−419q−53−105q−54 + 183q−55 + 250q−56 + 149q−57−17q−58−127q−59−102q−60−17q−61 + 39q−62 + 42q−63 + 28q−64−7q−65−26q−66−7q−67 + 5q−68 + 2q−69 + 3q−70 + 3q−71−5q−72−q−73 + 3q−74−q−75