10 110

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["10 110"];

In[4]:=

PD[K]

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

Out[4]=

X1627 X7,20,8,1 X3,11,4,10 X5,16,6,17 X17,8,18,9 X9,14,10,15 X11,3,12,2 X15,4,16,5 X13,19,14,18 X19,13,20,12

In[5]:=

GaussCode[K]

Out[5]=

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

In[6]:=

DTCode[K]

Out[6]=

6 10 16 20 14 2 18 4 8 12

(The path below may be different on your system)

In[7]:=

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

In[8]:=

ConwayNotation[K]

Out[8]=

[2.2.2.20]

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,−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[{5, 3}, {2, 4}, {3, 1}, {6, 15}, {7, 5}, {11, 6}, {14, 8}, {9, 7}, {15, 12}, {8, 10}, {4, 9}, {13, 11}, {12, 2}, {10, 13}, {1, 14}]

In[14]:=

Draw[ap]

Out[14]=

-Graphics-

Further quantum knot invariants for 10_110.

A1 Invariants.

Weight	Invariant
1	q15−2q13 + 4q11−4q9 + 2q7−q5−q3 + 3q−3q−1 + 4q−3−2q−5 + q−7
2	q42−2q40 + q38 + 5q36−11q34 + 3q32 + 19q30−26q28−5q26 + 37q24−23q22−19q20 + 32q18−2q16−22q14 + 8q12 + 19q10−13q8−18q6 + 28q4 + 3q2−35 + 22q−2 + 20q−4−33q−6 + 4q−8 + 23q−10−14q−12−6q−14 + 9q−16−q−18−2q−20 + q−22
3	q81−2q79 + q77 + 2q75−2q73−5q71 + 6q69 + 13q67−18q65−30q63 + 34q61 + 63q59−40q57−122q55 + 31q53 + 189q51 + 8q49−235q47−84q45 + 253q43 + 158q41−218q39−215q37 + 144q35 + 239q33−52q31−227q29−37q27 + 190q25 + 109q23−140q21−169q19 + 90q17 + 211q15−36q13−244q11−16q9 + 257q7 + 86q5−250q3−154q + 213q−1 + 210q−3−142q−5−241q−7 + 54q−9 + 233q−11 + 27q−13−182q−15−81q−17 + 110q−19 + 99q−21−48q−23−77q−25 + 3q−27 + 46q−29 + 12q−31−20q−33−9q−35 + 6q−37 + 4q−39−q−41−2q−43 + q−45

A2 Invariants.

Weight	Invariant
1,0	q22−q18 + 3q16−2q14 + q10−3q8 + 2q6−3q4 + 2q2 + 1−q−2 + 3q−4−q−6 + q−10
2,0	q56−q52 + q50 + 2q48−q46−6q44 + 3q42 + 10q40−9q38−11q36 + 7q34 + 15q32−10q30−16q28 + 14q26 + 10q24−10q22−4q20 + 12q18−2q16−3q14 + 6q12−q10−9q8 + 2q6 + 12q4−12q2−10 + 15q−2 + 9q−4−13q−6−7q−8 + 11q−10 + 7q−12−9q−14−6q−16 + 6q−18 + 5q−20−q−22−2q−24 + q−28

A3 Invariants.

Weight	Invariant
0,1,0	q48−2q46 + 6q42−7q40−4q38 + 18q36−12q34−13q32 + 27q30−13q28−16q26 + 27q24−6q22−11q20 + 12q18 + 3q16−5q14−10q12 + 9q10 + 6q8−23q6 + 9q4 + 18q2−25 + 8q−2 + 17q−4−20q−6 + 8q−8 + 8q−10−9q−12 + 4q−14 + 2q−16−2q−18 + q−20
1,0,0	q29 + q25−q23 + 3q21−3q19 + 2q17−2q15 + q13−2q11−2q5 + 2q3−q + 3q−1−2q−3 + 3q−5−q−7 + q−9 + q−13

B2 Invariants.

Weight	Invariant
0,1	q48−2q46 + 4q44−8q42 + 13q40−18q38 + 26q36−30q34 + 33q32−31q30 + 25q28−14q26−q24 + 16q22−33q20 + 46q18−59q16 + 63q14−62q12 + 55q10−42q8 + 27q6−9q4−6q2 + 19−28q−2 + 33q−4−32q−6 + 30q−8−24q−10 + 17q−12−10q−14 + 6q−16−2q−18 + q−20
1,0	q78−2q74−2q72 + 2q70 + 7q68 + 2q66−10q64−11q62 + 5q60 + 22q58 + 7q56−23q54−23q52 + 12q50 + 33q48 + 3q46−33q44−17q42 + 25q40 + 26q38−13q36−27q34 + 5q32 + 27q30 + 4q28−24q26−8q24 + 19q22 + 12q20−17q18−17q16 + 14q14 + 21q12−11q10−30q8 + q6 + 33q4 + 15q2−28−29q−2 + 16q−4 + 35q−6 + q−8−28q−10−14q−12 + 18q−14 + 18q−16−5q−18−13q−20−2q−22 + 7q−24 + 4q−26−2q−28−2q−30 + q−34

G2 Invariants.

Weight

Invariant

1,0

q114−2q112 + 4q110−6q108 + 6q106−5q104 + 10q100−20q98 + 31q96−39q94 + 35q92−20q90−10q88 + 55q86−94q84 + 121q82−118q80 + 71q78 + 10q76−112q74 + 200q72−226q70 + 178q68−61q66−84q64 + 200q62−231q60 + 167q58−31q56−116q54 + 198q52−176q50 + 53q48 + 118q46−250q44 + 281q42−194q40 + 14q38 + 182q36−325q34 + 358q32−275q30 + 101q28 + 99q26−256q24 + 317q22−265q20 + 124q18 + 44q16−179q14 + 221q12−157q10 + 20q8 + 134q6−223q4 + 206q2−87−82q−2 + 226q−4−279q−6 + 228q−8−96q−10−60q−12 + 176q−14−213q−16 + 180q−18−94q−20 + 3q−22 + 60q−24−87q−26 + 76q−28−46q−30 + 19q−32 + 3q−34−12q−36 + 13q−38−10q−40 + 6q−42−2q−44 + q−46

.

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["10 110"];

In[4]:=

Alexander[K][t]

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

Out[4]=

t3−8t2 + 20t−25 + 20t−1−8t−2 + t−3

In[5]:=

Conway[K][z]

Out[5]=

z6−2z4−3z2 + 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]=

{ 83, -2 }

In[8]:=

Jones[K][q]

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

Out[8]=

q3−3q2 + 7q−10 + 13q−1−14q−2 + 13q−3−11q−4 + 7q−5−3q−6 + q−7

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

z2a6 + a6−2z4a4−3z2a4−a4 + z6a2 + 2z4a2 + z2a2−2z4−3z2 + z2a−2 + a−2

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

2a3z9 + 2az9 + 6a4z8 + 10a2z8 + 4z8 + 8a5z7 + 9a3z7 + 4az7 + 3z7a−1 + 6a6z6−5a4z6−20a2z6 + z6a−2−8z6 + 3a7z5−13a5z5−27a3z5−19az5−8z5a−1 + a8z4−7a6z4−4a4z4 + 8a2z4−3z4a−2 + z4−2a7z3 + 12a5z3 + 21a3z3 + 13az3 + 6z3a−1−a8z2 + 5a6z2 + 6a4z2−a2z2 + 3z2a−2 + 2z2−4a5z−6a3z−3az−za−1−a6−a4−a−2

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["10 110"];

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 + 20t−25 + 20t−1−8t−2 + t−3, q3−3q2 + 7q−10 + 13q−1−14q−2 + 13q−3−11q−4 + 7q−5−3q−6 + q−7 }

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	q10−3q9 + q8 + 11q7−18q6−7q5 + 48q4−37q3−44q2 + 101q−35−101q−1 + 139q−2−10q−3−147q−4 + 144q−5 + 22q−6−158q−7 + 114q−8 + 42q−9−124q−10 + 63q−11 + 38q−12−64q−13 + 21q−14 + 17q−15−19q−16 + 5q−17 + 3q−18−3q−19 + q−20
3	q21−3q20 + q19 + 5q18 + 3q17−18q16−10q15 + 37q14 + 37q13−61q12−90q11 + 66q10 + 184q9−50q8−281q7−35q6 + 393q5 + 156q4−460q3−330q2 + 492q + 508−457q−1−697q−2 + 396q−3 + 844q−4−286q−5−970q−6 + 168q−7 + 1052q−8−39q−9−1091q−10−91q−11 + 1081q−12 + 210q−13−1010q−14−318q−15 + 891q−16 + 385q−17−719q−18−413q−19 + 532q−20 + 382q−21−343q−22−318q−23 + 195q−24 + 231q−25−100q−26−137q−27 + 37q−28 + 78q−29−18q−30−34q−31 + 8q−32 + 14q−33−6q−34−3q−35 + q−36 + 3q−37−3q−38 + q−39
4	q36−3q35 + q34 + 5q33−3q32 + 3q31−21q30 + 2q29 + 39q28 + 9q27 + 14q26−120q25−66q24 + 121q23 + 151q22 + 202q21−315q20−439q19−45q18 + 379q17 + 969q16−109q15−1009q14−990q13−71q12 + 2090q11 + 1161q10−744q9−2354q8−1950q7 + 2308q6 + 2993q5 + 1157q4−2760q3−4608q2 + 807q + 3963 + 4009q−1−1499q−2−6615q−3−1739q−4 + 3449q−5 + 6490q−6 + 756q−7−7352q−8−4194q−9 + 2018q−10 + 8024q−11 + 3037q−12−7124q−13−6052q−14 + 331q−15 + 8656q−16 + 4972q−17−6150q−18−7223q−19−1495q−20 + 8244q−21 + 6420q−22−4265q−23−7298q−24−3320q−25 + 6391q−26 + 6806q−27−1667q−28−5768q−29−4343q−30 + 3453q−31 + 5516q−32 + 479q−33−3098q−34−3771q−35 + 881q−36 + 3111q−37 + 1124q−38−840q−39−2133q−40−202q−41 + 1108q−42 + 657q−43 + 79q−44−770q−45−214q−46 + 237q−47 + 165q−48 + 137q−49−190q−50−52q−51 + 40q−52 + 3q−53 + 49q−54−39q−55−3q−56 + 11q−57−9q−58 + 10q−59−7q−60 + q−61 + 3q−62−3q−63 + q−64