10 145

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["10 145"];

In[4]:=

PD[K]

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

Out[4]=

X4251 X5,12,6,13 X8394 X2,9,3,10 X11,16,12,17 X17,10,18,11 X7,18,8,19 X13,20,14,1 X19,14,20,15 X15,6,16,7

In[5]:=

GaussCode[K]

Out[5]=

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

In[6]:=

DTCode[K]

Out[6]=

4 8 -12 -18 2 -16 -20 -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]=

[22,3,3-]

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

In[14]:=

Draw[ap]

Out[14]=

-Graphics-

Further quantum knot invariants for 10_145.

A1 Invariants.

Weight	Invariant
1	−q21 + q13 + q5 + q3
2	q60−q56−q46 + q44 + q42−q40−q34−q32−q30−q28 + q26 + q24 + 2q22−q18 + 2q16−q12 + q10 + q8 + q6
3	−q117 + q113 + q111−q107 + q97−q95−2q93 + 2q89 + 2q87−q85−q83 + q79 + 2q77−2q73−q71 + q69 + q67−2q65−q63 + q61−q57−q55 + q53−q49−q47−q41−q39 + 2q35 + 2q33−q29 + 2q25 + 2q23−q19−q17 + q15 + q13 + q11 + q9

A2 Invariants.

Weight	Invariant
1,0	−q32−q30 + q24 + q14 + q10 + q8 + q6
1,1	q84 + 2q80−2q74−4q72−2q68 + 2q66 + 2q64 + 2q62 + 6q60−2q58−4q54−3q52−2q50−2q48 + 2q46−2q44 + 2q42−4q40 + 2q38−4q36 + 2q34 + 2q32−2q30 + 4q28 + 4q24 + q20 + 2q18 + 2q16 + 2q14 + q12
2,0	q82 + q80 + q78−q76−q74−q72−q70 + q64 + q60 + q58−q56−q54−q52−2q48−2q46−q44−2q42−q40 + 2q36 + q34 + 2q32 + 2q30 + q28 + q26 + q24 + q22 + 2q16 + q14 + q12

A3 Invariants.

Weight	Invariant
0,1,0	q66 + q62 + q60−2q56−q54−2q52−3q50−q46 + 2q40 + q38 + q34−q30−q28 + q26 + q22 + 3q20 + q18 + 2q16 + q14 + q12
1,0,0	−q43−q41−q39 + q33 + q31−q25 + q19 + q17 + q15 + q13 + q11 + q9

A4 Invariants.

Weight	Invariant
0,1,0,0	q88 + q86 + q84 + q82 + q80−q74−2q72−2q70−2q68−3q66−3q64−2q62−2q60−2q58−q56 + 2q54 + 2q52 + 3q50 + 4q48 + 2q46−q42−2q40−3q38−q36 + q34 + 2q32 + 3q30 + 3q28 + 4q26 + 2q24 + 2q22 + q20 + q18
1,0,0,0	−q54−q52−q50−q48 + q42 + q40 + q38−q32−q30 + q24 + q22 + 2q20 + q18 + q16 + q14 + q12

B2 Invariants.

Weight	Invariant
0,1	−q66−q62−q60 + q54 + q50 + q46−q38−q34−q30 + q28 + q26 + q22 + q20 + q18 + q14 + q12
1,0	q108 + q100−q90−2q88−q82−q80−q72 + q56 + q48−q44 + q42 + q40−q36 + q34 + q32 + q30 + q26 + q24 + q22 + q18

D4 Invariants.

Weight	Invariant
1,0,0,0	q90 + q86 + q84 + q82−q78−q76−3q74−2q72−3q70−2q68−2q66 + q60 + 2q58 + 2q56 + 2q54 + q52 + q50−2q44−q42−2q40 + 2q32 + 2q30 + 3q28 + 2q26 + 2q24 + q22 + q20 + q18

G2 Invariants.

Weight	Invariant
1,0	q156 + q152−q148 + q142−2q138−q130−3q128−q126 + q124−q122−q120−q118−2q116 + 2q114−2q112−q110 + q108 + 2q104 + 2q102 + q98 + q96 + 2q92 + q88 + 2q82−q78−3q76 + q74−q72−3q66 + 2q64 + q62−2q60−q54 + 3q52 + 2q48 + q46 + 3q42 + q38 + q32 + q30

.

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["10 145"];

In[4]:=

Alexander[K][t]

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

Out[4]=

t2 + t−3 + t−1 + t−2

In[5]:=

Conway[K][z]

Out[5]=

z4 + 5z2 + 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]=

{ 3, -2 }

In[8]:=

Jones[K][q]

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

Out[8]=

q−2 + q−7−q−8 + q−9−q−10

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

−a10 + z2a8 + a8−a6 + z4a4 + 4z2a4 + 2a4

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

z7a11−6z5a11 + 10z3a11−5za11 + z8a10−6z6a10 + 10z4a10−6z2a10 + a10 + 2z7a9−12z5a9 + 18z3a9−6za9 + z8a8−6z6a8 + 9z4a8−4z2a8 + a8 + z7a7−6z5a7 + 8z3a7−2za7−2z2a6 + a6−za5 + z4a4−4z2a4 + 2a4

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["10 145"];

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]=

{ t2 + t−3 + t−1 + t−2, q−2 + q−7−q−8 + q−9−q−10 }

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	q−4−q−7 + q−8 + 2q−9−4q−10 + 2q−11 + 4q−12−5q−13 + 2q−14 + 2q−15−5q−16 + 2q−17 + 2q−18−4q−19 + 2q−20 + q−21−2q−22 + 2q−23−q−24−q−25 + 2q−26−q−27−q−28 + q−29
3	q−6−q−10 + q−12 + 2q−13−q−14−2q−15 + 3q−17 + q−18−2q−19−2q−20 + 2q−21 + q−22−q−23−2q−24 + q−25 + q−26−q−28−q−29 + q−30 + q−31−3q−33 + 4q−35−5q−37−q−38 + 6q−39 + 2q−40−6q−41−2q−42 + 5q−43 + 2q−44−3q−45−2q−46 + 3q−47−2q−49 + 2q−51−2q−53 + q−55 + q−56−q−57
4	q−8−q−13 + q−16 + 2q−17−q−18 + 2q−19−4q−20−q−21 + q−22 + 9q−24−3q−25−4q−26−8q−27−2q−28 + 17q−29 + 2q−30−4q−31−14q−32−5q−33 + 20q−34 + 3q−35−2q−36−16q−37−6q−38 + 21q−39 + 3q−40−4q−41−16q−42−5q−43 + 21q−44 + 4q−45−5q−46−14q−47−5q−48 + 18q−49 + 5q−50−3q−51−12q−52−6q−53 + 13q−54 + 7q−55−q−56−10q−57−6q−58 + 8q−59 + 6q−60−q−61−5q−62−3q−63 + 4q−64 + 3q−65−4q−66−2q−67 + q−68 + 6q−69 + 2q−70−8q−71−3q−72 + q−73 + 7q−74 + 3q−75−5q−76−3q−77−2q−78 + 5q−79 + q−80−2q−81−q−82−q−83 + 4q−84−q−85−q−86−q−87−q−88 + 3q−89−q−92−q−93 + q−94