4 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.org/wiki/KnotTheory.

In[3]:=

K = Knot["4 1"];

In[4]:=

PD[K]

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

Out[4]=

X4251 X8615 X6374 X2738

In[5]:=

GaussCode[K]

Out[5]=

1, -4, 3, -1, 2, -3, 4, -2

In[6]:=

DTCode[K]

Out[6]=

4 6 8 2

(The path below may be different on your system)

In[7]:=

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

In[8]:=

ConwayNotation[K]

Out[8]=

[22]

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(3,{−1,2,−1,2})

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]=

{ 3, 4, 3 }

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, 5}, {6, 4}, {5, 2}, {1, 3}, {2, 6}, {4, 1}]

In[14]:=

Draw[ap]

Out[14]=

-Graphics-

Further quantum knot invariants for 4_1.

A1 Invariants.

Weight	Invariant
1	q5 + q−5
2	q14−q10 + q2 + 1 + q−2−q−10 + q−14
3	q27−q23−q21 + q17 + q11 + q9 + q−9 + q−11 + q−17−q−21−q−23 + q−27
4	q44−q40−q38−q36 + q34 + q32 + q30−q26 + q24 + q22−q18−q16 + q4 + q2 + 1 + q−2 + q−4−q−16−q−18 + q−22 + q−24−q−26 + q−30 + q−32 + q−34−q−36−q−38−q−40 + q−44
5	q65−q61−q59−q57 + q53 + 2q51 + q49−q45−q43 + q39 + q37−q35−2q33−q31 + q27 + q25 + q17 + q15 + q13 + q−13 + q−15 + q−17 + q−25 + q−27−q−31−2q−33−q−35 + q−37 + q−39−q−43−q−45 + q−49 + 2q−51 + q−53−q−57−q−59−q−61 + q−65
6	q90−q86−q84−q82 + 2q76 + 2q74 + q72−q68−2q66−2q64 + q62 + q60 + q58−q54−2q52−2q50 + q48 + 2q46 + 2q44 + q42−q38−q36 + q34 + q32 + q30−q26−q24−q22 + q6 + q4 + q2 + 1 + q−2 + q−4 + q−6−q−22−q−24−q−26 + q−30 + q−32 + q−34−q−36−q−38 + q−42 + 2q−44 + 2q−46 + q−48−2q−50−2q−52−q−54 + q−58 + q−60 + q−62−2q−64−2q−66−q−68 + q−72 + 2q−74 + 2q−76−q−82−q−84−q−86 + q−90

A2 Invariants.

Weight	Invariant
1,0	q8 + q6−1 + q−6 + q−8
1,1	q20 + 2q16−2q10−2q8 + 4q2 + 2 + 4q−2−2q−8−2q−10 + 2q−16 + q−20
2,0	q20 + q18 + q16−q14−q12−q10−q8 + q4 + 2q2 + 2 + 2q−2 + q−4−q−8−q−10−q−12−q−14 + q−16 + q−18 + q−20
3,0	q36 + q34 + q32−2q28−2q26−2q24 + q18 + 2q16 + 3q14 + 3q12 + 2q10 + q8−q4−2q2−2−2q−2−q−4 + q−8 + 2q−10 + 3q−12 + 3q−14 + 2q−16 + q−18−2q−24−2q−26−2q−28 + q−32 + q−34 + q−36

A3 Invariants.

Weight	Invariant
0,1,0	q16 + q12 + q10 + q−10 + q−12 + q−16
1,0,0	q11 + q9 + q7−q−q−1 + q−7 + q−9 + q−11
1,0,1	q26 + 2q22 + 2q20 + q18 + 2q16−2q14−2q12−4q10−4q8 + 2q4 + 6q2 + 7 + 6q−2 + 2q−4−4q−8−4q−10−2q−12−2q−14 + 2q−16 + q−18 + 2q−20 + 2q−22 + q−26

A4 Invariants.

Weight	Invariant
0,1,0,0	q22 + q20 + q18 + q16 + q14−q10−q8 + q2 + 2 + q−2−q−8−q−10 + q−14 + q−16 + q−18 + q−20 + q−22
1,0,0,0	q14 + q12 + q10 + q8−q2−1−q−2 + q−8 + q−10 + q−12 + q−14

B2 Invariants.

Weight	Invariant
0,1	q16 + q12 + q10−2 + q−10 + q−12 + q−16
1,0	q26 + q18 + 1 + q−18 + q−26

B3 Invariants.

Weight	Invariant
1,0,0	q38 + q30 + q26 + q22−q2 + 1−q−2 + q−22 + q−26 + q−30 + q−38

B4 Invariants.

Weight	Invariant
1,0,0,0	q50 + q42 + q38 + q34 + q30 + q26−q6−q2 + 1−q−2−q−6 + q−26 + q−30 + q−34 + q−38 + q−42 + q−50

C3 Invariants.

Weight	Invariant
1,0,0	q22 + q18 + q16 + q14 + q12−q2−2−q−2 + q−12 + q−14 + q−16 + q−18 + q−22

C4 Invariants.

Weight	Invariant
1,0,0,0	q28 + q24 + q22 + q20 + q18 + q16 + q14−q4−q2−2−q−2−q−4 + q−14 + q−16 + q−18 + q−20 + q−22 + q−24 + q−28

D4 Invariants.

Weight	Invariant
0,1,0,0	q38 + q34 + 3q32 + 2q30 + q28 + 4q26 + q24−2q18−5q16−3q14−3q12−4q10 + 3q6 + 5q4 + 6q2 + 8 + 6q−2 + 5q−4 + 3q−6−4q−10−3q−12−3q−14−5q−16−2q−18 + q−24 + 4q−26 + q−28 + 2q−30 + 3q−32 + q−34 + q−38
1,0,0,0	q22 + q18 + q16 + q14 + q12−q2−q−2 + q−12 + q−14 + q−16 + q−18 + q−22

G2 Invariants.

Weight	Invariant
1,0	q38 + q34−q30 + q28 + q26 + q24 + q18 + q16−q10−q4−1−q−4−q−10 + q−16 + q−18 + q−24 + q−26 + q−28−q−30 + q−34 + q−38

.

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["4 1"];

In[4]:=

Alexander[K][t]

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

Out[4]=

−t + 3−t−1

In[5]:=

Conway[K][z]

Out[5]=

1−z2

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]=

{ 5, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

q2−q + 1−q−1 + q−2

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

a2−z2−1 + a−2

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

az3 + z3a−1 + a2z2 + z2a−2 + 2z2−az−za−1−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["4 1"];

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]=

{ −t + 3−t−1, q2−q + 1−q−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]=

{}

In[6]:=

DeleteCases[ Select[ AllKnots[], (J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) & ], K ]

KnotTheory::loading: Loading precomputed data in Jones4Knots11`.

Out[6]=

{K11n19,}

n	Jn
2	q6−q5−q4 + 2q3−q2−q + 3−q−1−q−2 + 2q−3−q−4−q−5 + q−6
3	q12−q11−q10 + 2q8−2q6 + 3q4−3q2 + 3−3q−2 + 3q−4−2q−6 + 2q−8−q−10−q−11 + q−12
4	q20−q19−q18 + 3q15−q14−q13−q12−q11 + 5q10−q9−2q8−2q7−q6 + 6q5−q4−2q3−2q2−q + 7−q−1−2q−2−2q−3−q−4 + 6q−5−q−6−2q−7−2q−8−q−9 + 5q−10−q−11−q−12−q−13−q−14 + 3q−15−q−18−q−19 + q−20
5	q30−q29−q28 + q25 + 2q24−2q22−q21−q20 + q19 + 3q18 + q17−2q16−3q15−2q14 + 2q13 + 4q12 + 2q11−2q10−4q9−2q8 + 2q7 + 5q6 + 2q5−2q4−5q3−2q2 + 2q + 5 + 2q−1−2q−2−5q−3−2q−4 + 2q−5 + 5q−6 + 2q−7−2q−8−4q−9−2q−10 + 2q−11 + 4q−12 + 2q−13−2q−14−3q−15−2q−16 + q−17 + 3q−18 + q−19−q−20−q−21−2q−22 + 2q−24 + q−25−q−28−q−29 + q−30
6	q42−q41−q40 + q37 + 3q35−q34−2q33−q32−q31 + 6q28−q27−2q26−2q25−2q24−q23 + 9q21−2q19−3q18−3q17−2q16 + 11q14−2q12−4q11−4q10−2q9 + 12q7−2q5−4q4−4q3−2q2 + 13−2q−2−4q−3−4q−4−2q−5 + 12q−7−2q−9−4q−10−4q−11−2q−12 + 11q−14−2q−16−3q−17−3q−18−2q−19 + 9q−21−q−23−2q−24−2q−25−2q−26−q−27 + 6q−28−q−31−q−32−2q−33−q−34 + 3q−35 + q−37−q−40−q−41 + q−42
7	q56−q55−q54 + q51 + q49 + 2q48−q47−2q46−q45−2q44 + q43 + q41 + 5q40−2q38−2q37−4q36 + 2q33 + 7q32 + q31−q30−2q29−7q28−2q27 + 2q25 + 9q24 + 2q23−3q21−9q20−3q19 + 3q17 + 10q16 + 3q15−3q13−10q12−3q11 + 3q9 + 11q8 + 3q7−3q5−11q4−3q3 + 3q + 11 + 3q−1−3q−3−11q−4−3q−5 + 3q−7 + 11q−8 + 3q−9−3q−11−10q−12−3q−13 + 3q−15 + 10q−16 + 3q−17−3q−19−9q−20−3q−21 + 2q−23 + 9q−24 + 2q−25−2q−27−7q−28−2q−29−q−30 + q−31 + 7q−32 + 2q−33−4q−36−2q−37−2q−38 + 5q−40 + q−41 + q−43−2q−44−q−45−2q−46−q−47 + 2q−48 + q−49 + q−51−q−54−q−55 + q−56