10 163

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["10 163"];

In[4]:=

PD[K]

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

Out[4]=

X6271 X8394 X13,19,14,18 X11,1,12,20 X19,13,20,12 X2,16,3,15 X4,17,5,18 X10,6,11,5 X14,7,15,8 X16,10,17,9

In[5]:=

GaussCode[K]

Out[5]=

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

In[6]:=

DTCode[K]

Out[6]=

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

(The path below may be different on your system)

In[7]:=

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

In[8]:=

ConwayNotation[K]

Out[8]=

[8*-30]

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,−1,3,2,−1,2,2,3})

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

In[14]:=

Draw[ap]

Out[14]=

-Graphics-

Further quantum knot invariants for 10_163.

A1 Invariants.

Weight	Invariant
1	−q5 + 3q3−2q + 2q−1−q−3 + 2q−7−2q−9 + 3q−11−2q−13
2	q16−3q14−2q12 + 11q10−3q8−15q6 + 14q4 + 8q2−20 + 6q−2 + 15q−4−12q−6−3q−8 + 12q−10−11q−14 + 2q−16 + 14q−18−14q−20−9q−22 + 22q−24−6q−26−14q−28 + 13q−30 + 2q−32−7q−34 + q−36 + q−38
3	−q33 + 3q31 + 2q29−7q27−10q25 + 7q23 + 31q21−47q17−30q15 + 53q13 + 70q11−39q9−103q7 + 3q5 + 120q3 + 44q−115q−1−80q−3 + 92q−5 + 101q−7−58q−9−107q−11 + 31q−13 + 100q−15−4q−17−85q−19−21q−21 + 70q−23 + 45q−25−52q−27−75q−29 + 30q−31 + 101q−33 + 2q−35−119q−37−42q−39 + 122q−41 + 77q−43−100q−45−101q−47 + 60q−49 + 104q−51−18q−53−85q−55−8q−57 + 48q−59 + 23q−61−21q−63−17q−65 + 5q−67 + 5q−69 + 2q−71−2q−73

A2 Invariants.

Weight	Invariant
1,0	−q6 + 2q4 + 1 + 2q−2−3q−4 + q−6−2q−8 + 2q−10 + 2q−12 + 2q−16−2q−18−q−20
1,1	q20−6q18 + 20q16−50q14 + 95q12−160q10 + 244q8−318q6 + 372q4−384q2 + 336−224q−2 + 62q−4 + 132q−6−332q−8 + 510q−10−629q−12 + 698q−14−694q−16 + 628q−18−496q−20 + 318q−22−126q−24−70q−26 + 232q−28−346q−30 + 396q−32−382q−34 + 326q−36−242q−38 + 158q−40−92q−42 + 41q−44−10q−46−2q−52 + 2q−54
2,0	q18−2q16−3q14 + 4q12 + 4q10−2q8−5q6 + 4q4 + 8q2−7−6q−2 + 6q−4−3q−8 + 2q−10 + 9q−12 + 2q−14 + 4q−18−4q−20−9q−22 + 4q−26−9q−28 + 10q−32 + 5q−34−3q−36−4q−38 + 5q−40−2q−42−5q−44−q−46 + q−48 + 2q−50

A3 Invariants.

Weight	Invariant
0,1,0	q14−3q12 + q10 + 5q8−9q6 + 7q4 + 7q2−13 + 10q−2 + 5q−4−13q−6 + 4q−8 + 5q−10−5q−12−q−14 + 3q−16 + 5q−18−2q−20−4q−22 + 14q−24−6q−26−9q−28 + 14q−30−8q−32−8q−34 + 9q−36−3q−38−3q−40 + 3q−42
1,0,0	−q7 + 2q5−q3 + 3q + 2q−3−2q−5−q−7−q−9−q−11 + 2q−13 + q−15 + 4q−17−q−19 + 2q−21−2q−23−q−25−q−27

A4 Invariants.

Weight	Invariant
0,1,0,0	q16−2q14−q12 + 4q10−3q8−3q6 + 10q4 + 4q2−8 + 3q−2 + 10q−4−7q−6−13q−8 + 8q−10 + 9q−12−12q−14 + 18q−18−7q−20−9q−22 + 12q−24 + q−26−12q−28 + 6q−30 + 13q−32−4q−34−6q−36 + 10q−38 + 3q−40−14q−42−4q−44 + 5q−46−4q−48−4q−50 + 3q−52 + 2q−54 + q−56
1,0,0,0	−q8 + 2q6−q4 + 2q2 + 2 + 2q−4−2q−6−3q−10−q−14 + 2q−16 + q−18 + 3q−20 + 3q−22−q−24 + 2q−26−2q−28−q−30−q−32−q−34

B2 Invariants.

Weight	Invariant
0,1	−q14 + 3q12−7q10 + 11q8−13q6 + 17q4−15q2 + 13−8q−2 + 3q−4 + 5q−6−14q−8 + 21q−10−27q−12 + 29q−14−29q−16 + 25q−18−18q−20 + 12q−22−2q−24−4q−26 + 11q−28−14q−30 + 16q−32−16q−34 + 13q−36−9q−38 + 5q−40−3q−42
1,0	q24−3q20−3q18 + 4q16 + 8q14−2q12−11q10−4q8 + 15q6 + 11q4−11q2−15 + 4q−2 + 17q−4 + 4q−6−15q−8−8q−10 + 10q−12 + 9q−14−6q−16−9q−18 + 4q−20 + 11q−22−12q−26−2q−28 + 12q−30 + 6q−32−10q−34−10q−36 + 10q−38 + 14q−40−5q−42−17q−44 + 16q−48 + 7q−50−12q−52−13q−54 + 4q−56 + 11q−58 + q−60−6q−62−4q−64 + q−66 + 3q−68

D4 Invariants.

Weight	Invariant
1,0,0,0	q18−3q16 + 4q14−6q12 + 9q10−11q8 + 13q6−12q4 + 15q2−9 + 8q−2−4q−4 + 3q−6 + 2q−8−11q−10 + 12q−12−16q−14 + 18q−16−23q−18 + 22q−20−20q−22 + 24q−24−16q−26 + 15q−28−8q−30 + 10q−32 + q−34−5q−36 + 5q−38−11q−40 + 12q−42−14q−44 + 9q−46−14q−48 + 11q−50−6q−52 + 4q−54−4q−56 + 3q−58

G2 Invariants.

Weight

Invariant

1,0

q32−3q30 + 7q28−13q26 + 13q24−8q22−7q20 + 31q18−49q16 + 61q14−48q12 + 5q10 + 46q8−91q6 + 109q4−79q2 + 21 + 50q−2−92q−4 + 96q−6−53q−8−13q−10 + 68q−12−90q−14 + 59q−16 + 6q−18−74q−20 + 115q−22−108q−24 + 61q−26 + 6q−28−81q−30 + 123q−32−134q−34 + 101q−36−34q−38−40q−40 + 99q−42−113q−44 + 90q−46−30q−48−35q−50 + 79q−52−79q−54 + 39q−56 + 32q−58−85q−60 + 106q−62−70q−64 + 3q−66 + 62q−68−104q−70 + 100q−72−65q−74 + 13q−76 + 32q−78−59q−80 + 55q−82−36q−84 + 14q−86 + 5q−88−14q−90 + 9q−92−8q−94 + 5q−96−2q−98 + q−100 + q−102

.

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["10 163"];

In[4]:=

Alexander[K][t]

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

Out[4]=

t3−5t2 + 12t−15 + 12t−1−5t−2 + t−3

In[5]:=

Conway[K][z]

Out[5]=

z6 + z4 + 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]=

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 51, 2 }

In[8]:=

Jones[K][q]

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

Out[8]=

−2q6 + 5q5−7q4 + 9q3−9q2 + 8q−6 + 4q−1−q−2

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

z6a−2 + 3z4a−2−z4a−4−z4 + 2z2a−2−z2−a−2 + 2a−4−a−6 + 1

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

2z8a−2 + 2z8a−4 + 5z7a−1 + 8z7a−3 + 3z7a−5 + 3z6a−2 + z6a−6 + 4z6 + az5−10z5a−1−15z5a−3−4z5a−5−11z4a−2 + z4a−4 + 4z4a−6−8z4−az3 + 3z3a−1 + 8z3a−3 + 7z3a−5 + 3z3a−7 + 2z2a−2−4z2a−4−4z2a−6 + 2z2−za−3−3za−5−2za−7 + a−2 + 2a−4 + a−6 + 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["10 163"];

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]=

{ t3−5t2 + 12t−15 + 12t−1−5t−2 + t−3, −2q6 + 5q5−7q4 + 9q3−9q2 + 8q−6 + 4q−1−q−2 }

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]=

{9_28, 9_29, K11n87,}

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	q18−8q16 + 10q15 + 11q14−35q13 + 18q12 + 39q11−66q10 + 13q9 + 67q8−78q7 + 78q5−66q4−15q3 + 69q2−39q−24 + 43q−1−11q−2−18q−3 + 14q−4 + q−5−4q−6 + q−7
3	−2q35 + 4q34 + 3q33−24q31 + 47q29 + 25q28−80q27−77q26 + 114q25 + 147q24−124q23−238q22 + 115q21 + 324q20−79q19−402q18 + 38q17 + 445q16 + 20q15−473q14−67q13 + 468q12 + 117q11−448q10−158q9 + 404q8 + 198q7−344q6−227q5 + 266q4 + 247q3−185q2−236q + 94 + 212q−1−26q−2−160q−3−23q−4 + 106q−5 + 38q−6−51q−7−40q−8 + 23q−9 + 21q−10−4q−11−9q−12−q−13 + 4q−14−q−15
4	q58−6q56−5q55 + 14q54 + 20q53 + 13q52−45q51−95q50 + 17q49 + 131q48 + 199q47−42q46−420q45−244q44 + 193q43 + 746q42 + 386q41−778q40−975q39−252q38 + 1364q37 + 1404q36−650q35−1793q34−1302q33 + 1511q32 + 2518q31 + 29q30−2147q29−2414q28 + 1131q27 + 3155q26 + 806q25−1975q24−3103q23 + 574q22 + 3243q21 + 1350q20−1534q19−3322q18 + 35q17 + 2942q16 + 1691q15−925q14−3196q13−538q12 + 2306q11 + 1887q10−126q9−2693q8−1091q7 + 1316q6 + 1764q5 + 703q4−1745q3−1298q2 + 225q + 1149 + 1109q−1−634q−2−923q−3−423q−4 + 330q−5 + 851q−6 + 55q−7−296q−8−400q−9−128q−10 + 328q−11 + 148q−12 + 38q−13−131q−14−130q−15 + 49q−16 + 36q−17 + 47q−18−8q−19−33q−20 + q−21−q−22 + 9q−23 + q−24−4q−25 + q−26