10 105

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["10 105"];

In[4]:=

PD[K]

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

Out[4]=

X4251 X12,4,13,3 X20,8,1,7 X16,5,17,6 X6,15,7,16 X10,17,11,18 X18,9,19,10 X8,14,9,13 X14,20,15,19 X2,12,3,11

In[5]:=

GaussCode[K]

Out[5]=

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

In[6]:=

DTCode[K]

Out[6]=

4 12 16 20 18 2 8 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]=

[21:20:20]

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

In[14]:=

Draw[ap]

Out[14]=

-Graphics-

Further quantum knot invariants for 10_105.

A1 Invariants.

Weight	Invariant
1	q7−2q5 + 4q3−4q + 3q−1−q−3 + 3q−7−4q−9 + 4q−11−3q−13 + q−15
2	q22−2q20−q18 + 9q16−7q14−14q12 + 26q10 + q8−38q6 + 28q4 + 23q2−44 + 10q−2 + 33q−4−26q−6−12q−8 + 24q−10 + 5q−12−26q−14 + 2q−16 + 36q−18−27q−20−24q−22 + 46q−24−11q−26−31q−28 + 28q−30 + 2q−32−15q−34 + 8q−36 + q−38−3q−40 + q−42
3	q45−2q43−q41 + 4q39 + 6q37−10q35−20q33 + 15q31 + 48q29−4q27−88q25−38q23 + 126q21 + 108q19−128q17−202q15 + 86q13 + 289q11−4q9−330q7−106q5 + 325q3 + 209q−278q−1−279q−3 + 199q−5 + 314q−7−110q−9−315q−11 + 27q−13 + 295q−15 + 53q−17−249q−19−135q−21 + 190q−23 + 212q−25−111q−27−283q−29 + 9q−31 + 328q−33 + 103q−35−326q−37−213q−39 + 284q−41 + 279q−43−197q−45−296q−47 + 100q−49 + 262q−51−24q−53−192q−55−15q−57 + 115q−59 + 27q−61−61q−63−20q−65 + 30q−67 + 6q−69−10q−71−3q−73 + 5q−75 + q−77−3q−79 + q−81

A2 Invariants.

Weight	Invariant
1,0	q10−q6 + 3q4−2q2 + 2q−2−3q−4 + 3q−6−2q−8 + 2q−10 + q−12−2q−14 + 3q−16−2q−18−q−20 + q−22
2,0	q28−2q24−q22 + 5q20 + 5q18−7q16−9q14 + 9q12 + 10q10−12q8−14q6 + 13q4 + 19q2−14−11q−2 + 19q−4 + 5q−6−13q−8−3q−10 + 9q−12−5q−14−4q−16 + 11q−18−4q−20−12q−22 + 12q−24 + 15q−26−20q−28−10q−30 + 20q−32 + 7q−34−17q−36−9q−38 + 16q−40 + 4q−42−10q−44−q−46 + 5q−48 + 2q−50−3q−52−q−54 + q−56

A3 Invariants.

Weight	Invariant
0,1,0	q20−2q18 + 2q16 + 4q14−10q12 + 8q10 + 9q8−24q6 + 18q4 + 12q2−32 + 21q−2 + 13q−4−29q−6 + 9q−8 + 14q−10−12q−12−5q−14 + 6q−16 + 12q−18−13q−20−9q−22 + 32q−24−18q−26−18q−28 + 34q−30−15q−32−17q−34 + 23q−36−5q−38−10q−40 + 9q−42−3q−46 + q−48
1,0,0	q13 + q9−q7 + 3q5−3q3 + 2q−2q−1 + 2q−3−2q−5 + q−7 + q−9−q−11 + 2q−13−q−15 + 3q−17−3q−19 + 3q−21−2q−23−q−27 + q−29

B2 Invariants.

Weight	Invariant
0,1	q20−2q18 + 6q16−10q14 + 18q12−26q10 + 33q8−38q6 + 40q4−36q2 + 26−13q−2−5q−4 + 25q−6−45q−8 + 62q−10−72q−12 + 77q−14−72q−16 + 62q−18−45q−20 + 27q−22−6q−24−12q−26 + 26q−28−36q−30 + 39q−32−39q−34 + 33q−36−25q−38 + 18q−40−11q−42 + 6q−44−3q−46 + q−48
1,0	q34−2q30−2q28 + 4q26 + 7q24−2q22−14q20−6q18 + 19q16 + 20q14−14q12−33q10−2q8 + 40q6 + 22q4−32q2−36 + 14q−2 + 42q−4 + 5q−6−36q−8−17q−10 + 25q−12 + 21q−14−17q−16−22q−18 + 11q−20 + 25q−22−6q−24−28q−26 + 31q−30 + 9q−32−30q−34−20q−36 + 28q−38 + 32q−40−17q−42−41q−44 + 40q−48 + 18q−50−27q−52−31q−54 + 8q−56 + 28q−58 + 8q−60−15q−62−14q−64 + 3q−66 + 10q−68 + 3q−70−3q−72−3q−74 + q−78

G2 Invariants.

Weight

Invariant

1,0

q46−2q44 + 6q42−10q40 + 13q38−13q36 + 4q34 + 18q32−47q30 + 82q28−97q26 + 75q24−8q22−95q20 + 200q18−257q16 + 230q14−110q12−81q10 + 262q8−360q6 + 332q4−170q2−47 + 233q−2−311q−4 + 242q−6−67q−8−131q−10 + 261q−12−259q−14 + 124q−16 + 92q−18−293q−20 + 397q−22−359q−24 + 181q−26 + 73q−28−324q−30 + 472q−32−465q−34 + 308q−36−48q−38−210q−40 + 372q−42−381q−44 + 242q−46−23q−48−174q−50 + 266q−52−210q−54 + 44q−56 + 152q−58−277q−60 + 283q−62−164q−64−29q−66 + 203q−68−305q−70 + 301q−72−199q−74 + 53q−76 + 85q−78−173q−80 + 192q−82−159q−84 + 96q−86−26q−88−29q−90 + 57q−92−66q−94 + 54q−96−33q−98 + 16q−100 + q−102−8q−104 + 10q−106−10q−108 + 6q−110−3q−112 + q−114

.

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["10 105"];

In[4]:=

Alexander[K][t]

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

Out[4]=

t3−8t2 + 22t−29 + 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]=

{ 91, 2 }

In[8]:=

Jones[K][q]

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

Out[8]=

q7−4q6 + 8q5−12q4 + 15q3−15q2 + 14q−11 + 7q−1−3q−2 + q−3

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

z6a−2 + 2z4a−2−2z4a−4−2z4 + a2z2 + 2z2a−2−2z2a−4 + z2a−6−3z2 + a2 + a−2−1

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

2z9a−1 + 2z9a−3 + 11z8a−2 + 7z8a−4 + 4z8 + 3az7 + 6z7a−1 + 13z7a−3 + 10z7a−5 + a2z6−19z6a−2−3z6a−4 + 8z6a−6−7z6−8az5−24z5a−1−33z5a−3−13z5a−5 + 4z5a−7−3a2z4 + 2z4a−2−9z4a−4−8z4a−6 + z4a−8−z4 + 7az3 + 18z3a−1 + 19z3a−3 + 6z3a−5−2z3a−7 + 3a2z2 + 4z2a−2 + 5z2a−4 + 3z2a−6 + 5z2−2az−4za−1−3za−3−za−5−a2−a−2−1

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["10 105"];

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−29 + 22t−1−8t−2 + t−3, q7−4q6 + 8q5−12q4 + 15q3−15q2 + 14q−11 + 7q−1−3q−2 + q−3 }

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]=

{K11n163,}

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	q20−4q19 + 4q18 + 8q17−27q16 + 21q15 + 34q14−86q13 + 41q12 + 91q11−156q10 + 38q9 + 154q8−190q7 + 10q6 + 185q5−171q4−26q3 + 171q2−112q−49 + 117q−1−45q−2−44q−3 + 51q−4−6q−5−19q−6 + 11q−7 + q−8−3q−9 + q−10
3	q39−4q38 + 4q37 + 4q36−7q35−11q34 + 20q33 + 28q32−57q31−52q30 + 108q29 + 116q28−187q27−229q26 + 276q25 + 402q24−349q23−625q22 + 375q21 + 878q20−344q19−1122q18 + 262q17 + 1307q16−119q15−1441q14−30q13 + 1479q12 + 204q11−1463q10−355q9 + 1365q8 + 506q7−1221q6−623q5 + 1023q4 + 711q3−797q2−738q + 545 + 712q−1−310q−2−622q−3 + 114q−4 + 488q−5 + 16q−6−329q−7−89q−8 + 200q−9 + 90q−10−93q−11−71q−12 + 36q−13 + 40q−14−9q−15−19q−16 + 3q−17 + 5q−18 + q−19−3q−20 + q−21
4	q64−4q63 + 4q62 + 4q61−11q60 + 9q59−12q58 + 24q57 + 7q56−76q55 + 40q54 + 10q53 + 129q52−10q51−373q50 + 22q49 + 199q48 + 630q47 + 47q46−1278q45−515q44 + 512q43 + 2147q42 + 863q41−2840q40−2437q39 + 91q38 + 4733q37 + 3454q36−3927q35−5700q34−2233q33 + 6969q32 + 7567q31−3134q30−8620q29−6107q28 + 7270q27 + 11269q26−621q25−9553q24−9721q23 + 5650q22 + 12982q21 + 2218q20−8484q19−11785q18 + 3161q17 + 12649q16 + 4506q15−6213q14−12283q13 + 428q12 + 10833q11 + 6174q10−3204q9−11434q8−2333q7 + 7759q6 + 6975q5 + 220q4−9063q3−4430q2 + 3766q + 6190 + 3049q−1−5330q−2−4744q−3 + 101q−4 + 3740q−5 + 3936q−6−1629q−7−3115q−8−1653q−9 + 1024q−10 + 2760q−11 + 363q−12−1016q−13−1372q−14−357q−15 + 1071q−16 + 543q−17 + 51q−18−489q−19−401q−20 + 188q−21 + 168q−22 + 150q−23−62q−24−131q−25 + 10q−26 + 8q−27 + 42q−28 + 3q−29−22q−30 + 3q−31−3q−32 + 5q−33 + q−34−3q−35 + q−36