10 135

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["10 135"];

In[4]:=

PD[K]

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

Out[4]=

X1425 X3849 X9,15,10,14 X12,5,13,6 X6,13,7,14 X11,19,12,18 X15,1,16,20 X19,17,20,16 X17,11,18,10 X7283

In[5]:=

GaussCode[K]

Out[5]=

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

In[6]:=

DTCode[K]

Out[6]=

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

(The path below may be different on your system)

In[7]:=

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

In[8]:=

ConwayNotation[K]

Out[8]=

[221,21,2-]

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

In[14]:=

Draw[ap]

Out[14]=

-Graphics-

Further quantum knot invariants for 10_135.

A1 Invariants.

Weight	Invariant
1	−q11 + q9−2q7 + 2q5 + q + 2q−1−q−3 + 2q−5−2q−7
2	q32−q30−q28 + 4q26−2q24−6q22 + 7q20 + q18−11q16 + 5q14 + 6q12−8q10 + q8 + 7q6−q4−3q2 + 3 + 7q−2−7q−4−2q−6 + 11q−8−6q−10−6q−12 + 8q−14−2q−16−5q−18 + 2q−20 + q−22
3	−q63 + q61 + q59−q57−3q55 + 2q53 + 7q51−q49−12q47−3q45 + 17q43 + 13q41−19q39−24q37 + 15q35 + 34q33−5q31−45q29−7q27 + 41q25 + 19q23−38q21−26q19 + 29q17 + 30q15−19q13−26q11 + 8q9 + 27q7 + 4q5−21q3−14q + 18q−1 + 28q−3−12q−5−36q−7 + 4q−9 + 46q−11 + 4q−13−44q−15−17q−17 + 38q−19 + 22q−21−27q−23−26q−25 + 15q−27 + 19q−29−3q−31−14q−33−2q−35 + 8q−37 + 2q−39−2q−43

A2 Invariants.

Weight	Invariant
1,0	−q16−2q10 + q8 + q4 + 3q2 + 1 + 3q−2−q−4−2q−10
1,1	q44−2q42 + 6q40−12q38 + 23q36−38q34 + 58q32−80q30 + 100q28−116q26 + 114q24−102q22 + 64q20−20q18−44q16 + 104q14−158q12 + 202q10−220q8 + 228q6−199q4 + 170q2−108 + 54q−2 + 12q−4−64q−6 + 102q−8−124q−10 + 123q−12−112q−14 + 86q−16−64q−18 + 36q−20−20q−22 + 8q−24−2q−26 + 2q−30
2,0	q42−q38 + 3q34 + 2q32−3q30−3q28 + 2q26−7q22−5q20 + 2q18 + 2q16−4q14 + 5q10 + q8 + 2q6 + 6q4 + 4q2 + 3 + 5q−2 + 4q−4−5q−6−3q−8 + 3q−10−8q−14−2q−16 + 3q−18−q−20−3q−22−q−24 + 3q−26 + q−28

A3 Invariants.

Weight	Invariant
0,1,0	q34−q32 + q30 + 2q28−4q26 + q24 + 2q22−9q20 + 2q18 + 3q16−9q14 + 2q12 + 3q10−3q8 + q6 + 6q4 + 7q2 + 5 + 3q−2 + 9q−4−3q−6−8q−8 + 4q−10−6q−12−8q−14 + 5q−16−q−18−2q−20 + 3q−22
1,0,0	−q21−q17−2q13 + q11−q9 + q7 + q5 + 3q3 + 3q + 2q−1 + 3q−3−q−5−2q−9−2q−13

A4 Invariants.

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

B2 Invariants.

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

D4 Invariants.

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

G2 Invariants.

Weight

Invariant

1,0

q80−q78 + 3q76−4q74 + 3q72−2q70−3q68 + 9q66−15q64 + 17q62−15q60 + 3q58 + 9q56−25q54 + 35q52−32q50 + 17q48 + 3q46−25q44 + 35q42−31q40 + 14q38 + 7q36−24q34 + 26q32−14q30−9q28 + 30q26−37q24 + 31q22−10q20−18q18 + 42q16−50q14 + 46q12−24q10−q8 + 30q6−41q4 + 45q2−27 + 8q−2 + 18q−4−27q−6 + 25q−8−6q−10−12q−12 + 30q−14−29q−16 + 14q−18 + 7q−20−29q−22 + 39q−24−37q−26 + 19q−28−20q−32 + 26q−34−26q−36 + 16q−38−5q−40−4q−42 + 5q−44−9q−46 + 6q−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["10 135"];

In[4]:=

Alexander[K][t]

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

Out[4]=

3t2−9t + 13−9t−1 + 3t−2

In[5]:=

Conway[K][z]

Out[5]=

3z4 + 3z2 + 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]=

{ 37, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

−2q3 + 4q2−5q + 7−6q−1 + 6q−2−4q−3 + 2q−4−q−5

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

−z2a4−a4 + z4a2 + z2a2 + 2z4 + 5z2 + 4−2z2a−2−2a−2

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

a2z8 + z8 + 2a3z7 + 4az7 + 2z7a−1 + 2a4z6 + a2z6 + z6a−2 + a5z5−3a3z5−8az5−4z5a−1−5a4z4−4a2z4 + 2z4a−2 + 3z4−3a5z3−a3z3 + 8az3 + 9z3a−1 + 3z3a−3 + 3a4z2 + a2z2−4z2a−2−6z2 + 2a5z + a3z−4az−6za−1−3za−3−a4 + 2a−2 + 4

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["10 135"];

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]=

{ 3t2−9t + 13−9t−1 + 3t−2, −2q3 + 4q2−5q + 7−6q−1 + 6q−2−4q−3 + 2q−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_34,}

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 + q9−7q8 + 4q7 + 11q6−21q5 + 4q4 + 28q3−34q2−q + 42−38q−1−7q−2 + 44q−3−30q−4−13q−5 + 35q−6−16q−7−14q−8 + 19q−9−4q−10−8q−11 + 6q−12−2q−14 + q−15
3	−2q20 + 2q19 + 2q18 + 6q17−12q16−10q15 + 13q14 + 28q13−16q12−51q11 + 12q10 + 77q9−106q7−15q6 + 125q5 + 42q4−148q3−55q2 + 149q + 82−158q−1−87q−2 + 142q−3 + 107q−4−135q−5−106q−6 + 108q−7 + 114q−8−86q−9−107q−10 + 53q−11 + 102q−12−29q−13−85q−14 + 5q−15 + 64q−16 + 11q−17−46q−18−14q−19 + 25q−20 + 16q−21−14q−22−10q−23 + 5q−24 + 7q−25−3q−26−2q−27 + 2q−29−q−30
4	q34 + q33−3q32−6q31 + 2q30 + 5q29 + 18q28 + 4q27−37q26−21q25−8q24 + 77q23 + 70q22−70q21−97q20−112q19 + 140q18 + 241q17−19q16−188q15−343q14 + 119q13 + 453q12 + 152q11−200q10−622q9−11q8 + 598q7 + 361q6−115q5−827q4−173q3 + 631q2 + 508q + 12−909q−1−296q−2 + 577q−3 + 566q−4 + 133q−5−875q−6−371q−7 + 450q−8 + 552q−9 + 254q−10−740q−11−410q−12 + 257q−13 + 466q−14 + 362q−15−508q−16−389q−17 + 36q−18 + 298q−19 + 401q−20−237q−21−277q−22−113q−23 + 97q−24 + 318q−25−37q−26−114q−27−130q−28−35q−29 + 168q−30 + 28q−31−3q−32−66q−33−58q−34 + 56q−35 + 15q−36 + 22q−37−16q−38−30q−39 + 14q−40 + 10q−42−q−43−9q−44 + 4q−45−q−46 + 2q−47−2q−49 + q−50