10 115

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["10 115"];

In[4]:=

PD[K]

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

Out[4]=

X6271 X14,6,15,5 X20,15,1,16 X16,7,17,8 X8,19,9,20 X18,11,19,12 X10,4,11,3 X4,10,5,9 X12,17,13,18 X2,14,3,13

In[5]:=

GaussCode[K]

Out[5]=

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

In[6]:=

DTCode[K]

Out[6]=

6 10 14 16 4 18 2 20 12 8

(The path below may be different on your system)

In[7]:=

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

In[8]:=

ConwayNotation[K]

Out[8]=

[8*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,−2,1,3,2,2,−4,−3,2,−3,−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, 11}, {2, 9}, {8, 10}, {9, 12}, {11, 4}, {5, 3}, {4, 7}, {6, 8}, {7, 13}, {12, 6}, {1, 5}, {13, 2}, {10, 1}]

In[14]:=

Draw[ap]

Out[14]=

-Graphics-

Further quantum knot invariants for 10_115.

A1 Invariants.

Weight	Invariant
1	−q11 + 3q9−5q7 + 5q5−3q3 + 2q + 2q−1−3q−3 + 5q−5−5q−7 + 3q−9−q−11
2	q32−3q30 + q28 + 11q26−18q24−9q22 + 44q20−24q18−43q16 + 65q14−64q10 + 42q8 + 27q6−45q4−2q2 + 37−2q−2−45q−4 + 27q−6 + 42q−8−64q−10 + 65q−14−43q−16−24q−18 + 44q−20−9q−22−18q−24 + 11q−26 + q−28−3q−30 + q−32
3	−q63 + 3q61−q59−7q57 + 2q55 + 21q53 + 4q51−58q49−25q47 + 108q45 + 93q43−151q41−226q39 + 156q37 + 389q35−73q33−540q31−100q29 + 629q27 + 303q25−602q23−495q21 + 480q19 + 618q17−298q15−650q13 + 107q11 + 599q9 + 81q7−504q5−239q3 + 379q + 379q−1−239q−3−504q−5 + 81q−7 + 599q−9 + 107q−11−650q−13−298q−15 + 618q−17 + 480q−19−495q−21−602q−23 + 303q−25 + 629q−27−100q−29−540q−31−73q−33 + 389q−35 + 156q−37−226q−39−151q−41 + 93q−43 + 108q−45−25q−47−58q−49 + 4q−51 + 21q−53 + 2q−55−7q−57−q−59 + 3q−61−q−63

A2 Invariants.

Weight	Invariant
1,0	−q16 + q14 + 2q12−4q10 + 2q8−q6−2q4 + 5q2−1 + 5q−2−2q−4−q−6 + 2q−8−4q−10 + 2q−12 + q−14−q−16
2,0	q42−q40−3q38 + 2q36 + 8q34−17q30−5q28 + 24q26 + 4q24−27q22−4q20 + 33q18 + 10q16−38q14−q12 + 26q10−9q8−18q6 + 9q4 + 12q2−6 + 12q−2 + 9q−4−18q−6−9q−8 + 26q−10−q−12−38q−14 + 10q−16 + 33q−18−4q−20−27q−22 + 4q−24 + 24q−26−5q−28−17q−30 + 8q−34 + 2q−36−3q−38−q−40 + q−42

A3 Invariants.

Weight	Invariant
0,1,0	q34−3q32 + q30 + 8q28−15q26 + 3q24 + 26q22−35q20 + 4q18 + 40q16−49q14 + 2q12 + 39q10−35q8−5q6 + 26q4−q2−8−q−2 + 26q−4−5q−6−35q−8 + 39q−10 + 2q−12−49q−14 + 40q−16 + 4q−18−35q−20 + 26q−22 + 3q−24−15q−26 + 8q−28 + q−30−3q−32 + q−34
1,0,0	−q21 + q19 + 2q15−4q13 + 3q11−4q9 + q7−2q5 + 4q3 + 2q + 2q−1 + 4q−3−2q−5 + q−7−4q−9 + 3q−11−4q−13 + 2q−15 + q−19−q−21

B2 Invariants.

Weight	Invariant
0,1	−q34 + 3q32−7q30 + 14q28−25q26 + 37q24−48q22 + 57q20−60q18 + 54q16−41q14 + 20q12 + 7q10−37q8 + 67q6−90q4 + 109q2−114 + 109q−2−90q−4 + 67q−6−37q−8 + 7q−10 + 20q−12−41q−14 + 54q−16−60q−18 + 57q−20−48q−22 + 37q−24−25q−26 + 14q−28−7q−30 + 3q−32−q−34
1,0	q56−3q52−3q50 + 4q48 + 11q46 + q44−20q42−16q40 + 20q38 + 37q36−3q34−51q32−25q30 + 47q28 + 50q26−25q24−64q22−5q20 + 59q18 + 26q16−44q14−37q12 + 28q10 + 40q8−13q6−40q4 + 6q2 + 43 + 6q−2−40q−4−13q−6 + 40q−8 + 28q−10−37q−12−44q−14 + 26q−16 + 59q−18−5q−20−64q−22−25q−24 + 50q−26 + 47q−28−25q−30−51q−32−3q−34 + 37q−36 + 20q−38−16q−40−20q−42 + q−44 + 11q−46 + 4q−48−3q−50−3q−52 + q−56

G2 Invariants.

Weight

Invariant

1,0

q80−3q78 + 7q76−13q74 + 16q72−17q70 + 8q68 + 17q66−53q64 + 98q62−130q60 + 121q58−62q56−61q54 + 225q52−360q50 + 410q48−311q46 + 62q44 + 258q42−536q40 + 646q38−522q36 + 193q34 + 206q32−514q30 + 589q28−396q26 + 28q24 + 339q22−530q20 + 436q18−110q16−314q14 + 652q12−743q10 + 555q8−133q6−361q4 + 759q2−907 + 759q−2−361q−4−133q−6 + 555q−8−743q−10 + 652q−12−314q−14−110q−16 + 436q−18−530q−20 + 339q−22 + 28q−24−396q−26 + 589q−28−514q−30 + 206q−32 + 193q−34−522q−36 + 646q−38−536q−40 + 258q−42 + 62q−44−311q−46 + 410q−48−360q−50 + 225q−52−61q−54−62q−56 + 121q−58−130q−60 + 98q−62−53q−64 + 17q−66 + 8q−68−17q−70 + 16q−72−13q−74 + 7q−76−3q−78 + q−80

.

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["10 115"];

In[4]:=

Alexander[K][t]

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

Out[4]=

−t3 + 9t2−26t + 37−26t−1 + 9t−2−t−3

In[5]:=

Conway[K][z]

Out[5]=

−z6 + 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]=

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 109, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

−q5 + 4q4−9q3 + 14q2−17q + 19−17q−1 + 14q−2−9q−3 + 4q−4−q−5

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

−z6 + 2a2z4 + 2z4a−2−z4−a4z2 + a2z2 + z2a−2−z2a−4 + z2−a2−a−2 + 3

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

3az9 + 3z9a−1 + 8a2z8 + 8z8a−2 + 16z8 + 8a3z7 + 13az7 + 13z7a−1 + 8z7a−3 + 4a4z6−9a2z6−9z6a−2 + 4z6a−4−26z6 + a5z5−13a3z5−34az5−34z5a−1−13z5a−3 + z5a−5−5a4z4 + a2z4 + z4a−2−5z4a−4 + 12z4−a5z3 + 8a3z3 + 22az3 + 22z3a−1 + 8z3a−3−z3a−5 + 2a4z2−a2z2−z2a−2 + 2z2a−4−6z2−2a3z−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["10 115"];

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 + 9t2−26t + 37−26t−1 + 9t−2−t−3, −q5 + 4q4−9q3 + 14q2−17q + 19−17q−1 + 14q−2−9q−3 + 4q−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]=

{}

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	q15−4q14 + 4q13 + 11q12−33q11 + 13q10 + 64q9−101q8−6q7 + 172q6−166q5−70q4 + 278q3−181q2−142q + 321−142q−1−181q−2 + 278q−3−70q−4−166q−5 + 172q−6−6q−7−101q−8 + 64q−9 + 13q−10−33q−11 + 11q−12 + 4q−13−4q−14 + q−15
3	−q30 + 4q29−4q28−6q27 + 8q26 + 23q25−21q24−68q23 + 41q22 + 156q21−36q20−312q19−34q18 + 538q17 + 197q16−774q15−501q14 + 978q13 + 926q12−1100q11−1406q10 + 1085q9 + 1901q8−962q7−2322q6 + 733q5 + 2658q4−470q3−2840q2 + 148q + 2923 + 148q−1−2840q−2−470q−3 + 2658q−4 + 733q−5−2322q−6−962q−7 + 1901q−8 + 1085q−9−1406q−10−1100q−11 + 926q−12 + 978q−13−501q−14−774q−15 + 197q−16 + 538q−17−34q−18−312q−19−36q−20 + 156q−21 + 41q−22−68q−23−21q−24 + 23q−25 + 8q−26−6q−27−4q−28 + 4q−29−q−30
4	q50−4q49 + 4q48 + 6q47−13q46 + 2q45−15q44 + 40q43 + 49q42−94q41−61q40−89q39 + 246q38 + 385q37−253q36−518q35−744q34 + 642q33 + 1807q32 + 368q31−1400q30−3385q29−217q28 + 4490q27 + 3818q26−590q25−8228q24−5045q23 + 5642q22 + 10246q21 + 5143q20−11875q19−13751q18 + 1667q17 + 15820q16 + 15318q15−10566q14−22033q13−6857q12 + 16840q11 + 25391q10−4814q9−26115q8−15813q7 + 13604q6 + 31686q5 + 2140q4−25795q3−22277q2 + 8365q + 33665 + 8365q−1−22277q−2−25795q−3 + 2140q−4 + 31686q−5 + 13604q−6−15813q−7−26115q−8−4814q−9 + 25391q−10 + 16840q−11−6857q−12−22033q−13−10566q−14 + 15318q−15 + 15820q−16 + 1667q−17−13751q−18−11875q−19 + 5143q−20 + 10246q−21 + 5642q−22−5045q−23−8228q−24−590q−25 + 3818q−26 + 4490q−27−217q−28−3385q−29−1400q−30 + 368q−31 + 1807q−32 + 642q−33−744q−34−518q−35−253q−36 + 385q−37 + 246q−38−89q−39−61q−40−94q−41 + 49q−42 + 40q−43−15q−44 + 2q−45−13q−46 + 6q−47 + 4q−48−4q−49 + q−50