10 124

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10_123

10_125

Contents

Image:10 124.gif
(KnotPlot image)

See the full Rolfsen Knot Table.

Visit 10 124's page at the Knot Server (KnotPlot driven, includes 3D interactive images!)

Visit 10_124's page at Knotilus!

Visit 10 124's page at the original Knot Atlas!

[edit 10 124 Quick Notes]

10_124 is also known as the torus knot T(5,3) or the pretzel knot P(5,3,-2). It is one of two knots which are both torus knots and pretzel knots, the other being 8_19 = T(4,3) = P(3,3,-2).

[edit 10 124 Further Notes and Views]

If one takes the symmetric diagram for 10_123 and makes it doubly alternating one gets a diagram for 10_124. That's the torus knot view. There is then a nice representation of the quandle of 10_124 into the dodecahedral quandle Q30. See [1].

10_124 is not k-colourable for any k. See The Determinant and the Signature.

[edit] Knot presentations

Planar diagram presentation X4251 X8493 X9,17,10,16 X5,15,6,14 X15,7,16,6 X11,19,12,18 X13,1,14,20 X17,11,18,10 X19,13,20,12 X2837
Gauss code 1, -10, 2, -1, -4, 5, 10, -2, -3, 8, -6, 9, -7, 4, -5, 3, -8, 6, -9, 7
Dowker-Thistlethwaite code 4 8 -14 2 -16 -18 -20 -6 -10 -12
Conway Notation [5,3,2-]


Minimum Braid Representative A Morse Link Presentation An Arc Presentation
Image:BraidPart1.gifImage:BraidPart1.gifImage:BraidPart1.gifImage:BraidPart1.gifImage:BraidPart1.gifImage:BraidPart0.gifImage:BraidPart1.gifImage:BraidPart1.gifImage:BraidPart1.gifImage:BraidPart0.gif
Image:BraidPart2.gifImage:BraidPart2.gifImage:BraidPart2.gifImage:BraidPart2.gifImage:BraidPart2.gifImage:BraidPart1.gifImage:BraidPart2.gifImage:BraidPart2.gifImage:BraidPart2.gifImage:BraidPart1.gif
Image:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart2.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart2.gif

Length is 10, width is 3,

Braid index is 3

Image:10 124_ML.gif Image:10 124_AP.gif
[{4, 12}, {3, 5}, {1, 4}, {6, 11}, {5, 10}, {2, 6}, {12, 3}, {11, 9}, {10, 8}, {9, 7}, {8, 2}, {7, 1}]

[edit Notes on presentations of 10 124]


Computer Talk
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["10 124"];
In[4]:= PD[K]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= X4251 X8493 X9,17,10,16 X5,15,6,14 X15,7,16,6 X11,19,12,18 X13,1,14,20 X17,11,18,10 X19,13,20,12 X2837
In[5]:= GaussCode[K]
Out[5]= 1, -10, 2, -1, -4, 5, 10, -2, -3, 8, -6, 9, -7, 4, -5, 3, -8, 6, -9, 7
In[6]:= DTCode[K]
Out[6]= 4 8 -14 2 -16 -18 -20 -6 -10 -12

(The path below may be different on your system)

In[7]:= AppendTo[$Path, "C:/bin/LinKnot/"];
In[8]:= ConwayNotation[K]
Out[8]= [5,3,2-]
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(3,{1,1,1,1,1,2,1,1,1,2})
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, 10, 3 }
In[11]:= Show[BraidPlot[br]]
Image:BraidPart1.gifImage:BraidPart1.gifImage:BraidPart1.gifImage:BraidPart1.gifImage:BraidPart1.gifImage:BraidPart0.gifImage:BraidPart1.gifImage:BraidPart1.gifImage:BraidPart1.gifImage:BraidPart0.gif
Image:BraidPart2.gifImage:BraidPart2.gifImage:BraidPart2.gifImage:BraidPart2.gifImage:BraidPart2.gifImage:BraidPart1.gifImage:BraidPart2.gifImage:BraidPart2.gifImage:BraidPart2.gifImage:BraidPart1.gif
Image:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart2.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart2.gif
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."
Image:10 124_ML.gif
Out[12]= -Graphics-
In[13]:= ap = ArcPresentation[K]
Out[13]= ArcPresentation[{4, 12}, {3, 5}, {1, 4}, {6, 11}, {5, 10}, {2, 6}, {12, 3}, {11, 9}, {10, 8}, {9, 7}, {8, 2}, {7, 1}]
In[14]:= Draw[ap]
Image:10 124_AP.gif
Out[14]= -Graphics-

[edit] Three dimensional invariants

Symmetry type Reversible
Unknotting number 4
3-genus 4
Bridge index 3
Super bridge index Missing
Nakanishi index 1
Maximal Thurston-Bennequin number [7][-15]
Hyperbolic Volume Not hyperbolic
A-Polynomial See Data:10 124/A-polynomial

[edit Notes for 10 124's three dimensional invariants]

[edit] Four dimensional invariants

Smooth 4 genus 4
Topological 4 genus 4
Concordance genus 4
Rasmussen s-Invariant -8

[edit Notes for 10 124's four dimensional invariants]

[edit] Polynomial invariants

Alexander polynomial t4−t3 + t−1 + t−1−t−3 + t−4
Conway polynomial z8 + 7z6 + 14z4 + 8z2 + 1
2nd Alexander ideal (db, data sources) {1}
Determinant and Signature { 1, 8 }
Jones polynomial −q10 + q6 + q4
HOMFLY-PT polynomial (db, data sources) z8a−8 + 8z6a−8−z6a−10 + 21z4a−8−7z4a−10 + 21z2a−8−14z2a−10 + z2a−12 + 7a−8−8a−10 + 2a−12
Kauffman polynomial (db, data sources) z8a−8 + z8a−10 + z7a−9 + z7a−11−8z6a−8−8z6a−10−7z5a−9−7z5a−11 + 21z4a−8 + 21z4a−10 + 14z3a−9 + 14z3a−11−21z2a−8−22z2a−10−z2a−12−8za−9−8za−11 + 7a−8 + 8a−10 + 2a−12
The A2 invariant q−14 + q−16 + 2q−18 + 2q−20 + 2q−22 + q−24−2q−28−2q−30−2q−32−q−34 + q−40
The G2 invariant q−70 + q−72 + q−74 + q−76 + q−78 + 2q−80 + 2q−82 + q−84 + 2q−86 + 2q−88 + 2q−90 + 2q−92 + 2q−94 + q−96 + 2q−98 + q−100 + q−104 + q−106−q−112−q−114−q−118−2q−120−2q−122−q−124−q−126−2q−128−2q−130−2q−132−q−134−q−136−2q−138−2q−140−q−142−q−146−q−148 + q−160 + q−162 + q−168 + q−170 + q−180
Further Quantum Invariants
Further quantum knot invariants for 10_124.

A1 Invariants.

Weight Invariant
1 q−7 + q−9 + q−11 + q−13−q−19−q−21
2 q−14 + q−16 + q−18 + q−20 + q−22 + q−24 + q−26−q−36−q−38−q−40−q−42−q−44 + q−60
3 q−21 + q−23 + q−25 + q−27 + q−29 + q−31 + q−33 + q−35 + q−37 + q−39−q−53−q−55−q−57−q−59−q−61−q−63−q−65−q−67 + q−97 + q−99 + q−101 + q−103−q−109−q−111
5 q−35 + q−37 + q−39 + q−41 + q−43 + q−45 + q−47 + q−49 + q−51 + q−53 + q−55 + q−57 + q−59 + q−61 + q−63 + q−65−q−87−q−89−q−91−q−93−q−95−q−97−q−99−q−101−q−103−q−105−q−107−q−109−q−111−q−113 + q−171 + q−173 + q−175 + q−177 + q−179 + q−181 + q−183 + q−185 + q−187 + q−189−q−203−q−205−q−207−q−209−q−211−q−213−q−215−q−217 + q−247 + q−249 + q−251 + q−253−q−259−q−261
6 q−42 + q−44 + q−46 + q−48 + q−50 + q−52 + q−54 + q−56 + q−58 + q−60 + q−62 + q−64 + q−66 + q−68 + q−70 + q−72 + q−74 + q−76 + q−78−q−104−q−106−q−108−q−110−q−112−q−114−q−116−q−118−q−120−q−122−q−124−q−126−q−128−q−130−q−132−q−134−q−136 + q−208 + q−210 + q−212 + q−214 + q−216 + q−218 + q−220 + q−222 + q−224 + q−226 + q−228 + q−230 + q−232−q−250−q−252−q−254−q−256−q−258−q−260−q−262−q−264−q−266−q−268−q−270 + q−314 + q−316 + q−318 + q−320 + q−322 + q−324 + q−326−q−336−q−338−q−340−q−342−q−344 + q−360

A2 Invariants.

Weight Invariant
1,0 q−14 + q−16 + 2q−18 + 2q−20 + 2q−22 + q−24−2q−28−2q−30−2q−32−q−34 + q−40
1,1 q−28 + 2q−30 + 4q−32 + 6q−34 + 7q−36 + 8q−38 + 6q−40 + 4q−42 + q−44−2q−46−6q−48−8q−50−9q−52−8q−54−6q−56−4q−58−2q−60 + 2q−64 + 2q−66 + 4q−68 + 4q−70 + 4q−72 + 2q−74 + q−76−2q−78−2q−80−2q−82−q−84 + 2q−90
2,0 q−28 + q−30 + 2q−32 + 2q−34 + 3q−36 + 3q−38 + 4q−40 + 3q−42 + 3q−44 + 2q−46 + 2q−48−q−52−3q−54−4q−56−5q−58−5q−60−5q−62−4q−64−3q−66−2q−68−q−70 + 2q−74 + 2q−76 + 4q−78 + 4q−80 + 4q−82 + 2q−84 + q−86−2q−88−2q−90−2q−92−q−94 + q−100

A3 Invariants.

Weight Invariant
0,1,0 q−28 + q−30 + 3q−32 + 4q−34 + 6q−36 + 6q−38 + 7q−40 + 3q−42 + q−44−4q−46−7q−48−9q−50−9q−52−6q−54−3q−56 + q−58 + 3q−60 + 4q−62 + 2q−64 + 2q−66
1,0,0 q−21 + q−23 + 2q−25 + 3q−27 + 3q−29 + 3q−31 + 2q−33−2q−37−3q−39−4q−41−3q−43−2q−45 + q−49 + q−51 + q−53
1,0,1 q−42 + 2q−44 + 5q−46 + 9q−48 + 14q−50 + 19q−52 + 23q−54 + 22q−56 + 19q−58 + 10q−60−q−62−14q−64−26q−66−34q−68−38q−70−35q−72−26q−74−13q−76−q−78 + 11q−80 + 18q−82 + 21q−84 + 19q−86 + 14q−88 + 8q−90 + 4q−92−3q−96−3q−98−4q−100−3q−102−3q−104−q−106−q−108 + 2q−120

A4 Invariants.

Weight Invariant
0,1,0,0 q−42 + q−44 + 3q−46 + 5q−48 + 8q−50 + 10q−52 + 14q−54 + 13q−56 + 13q−58 + 8q−60 + 2q−62−7q−64−14q−66−21q−68−23q−70−21q−72−16q−74−8q−76−q−78 + 7q−80 + 9q−82 + 11q−84 + 9q−86 + 7q−88 + 3q−90 + 2q−92−q−96−q−98−q−100−q−102−q−104
1,0,0,0 q−28 + q−30 + 2q−32 + 3q−34 + 4q−36 + 4q−38 + 4q−40 + 2q−42 + q−44−2q−46−4q−48−5q−50−5q−52−4q−54−2q−56 + q−60 + 2q−62 + q−64 + q−66

B2 Invariants.

Weight Invariant
0,1 q−28 + q−30 + q−32 + 2q−34 + 2q−36 + 2q−38 + q−40 + q−42 + q−44−q−48−q−50−q−52−2q−54−q−56−q−58−q−60
1,0 q−42 + q−46 + q−48 + 2q−50 + q−52 + 3q−54 + 2q−56 + 3q−58 + 2q−60 + 3q−62 + 2q−64 + 2q−66−q−72−2q−74−3q−76−3q−78−3q−80−4q−82−3q−84−3q−86−2q−88−2q−90 + q−96 + q−98 + 2q−100 + q−102 + q−104 + q−106 + q−108

D4 Invariants.

Weight Invariant
1,0,0,0 q−42 + q−44 + 2q−46 + 4q−48 + 5q−50 + 7q−52 + 8q−54 + 8q−56 + 7q−58 + 5q−60 + q−62−3q−64−7q−66−10q−68−11q−70−11q−72−9q−74−6q−76−2q−78 + q−80 + 3q−82 + 4q−84 + 4q−86 + 3q−88 + 2q−90 + q−92

G2 Invariants.

Weight Invariant
1,0 q−70 + q−72 + q−74 + q−76 + q−78 + 2q−80 + 2q−82 + q−84 + 2q−86 + 2q−88 + 2q−90 + 2q−92 + 2q−94 + q−96 + 2q−98 + q−100 + q−104 + q−106−q−112−q−114−q−118−2q−120−2q−122−q−124−q−126−2q−128−2q−130−2q−132−q−134−q−136−2q−138−2q−140−q−142−q−146−q−148 + q−160 + q−162 + q−168 + q−170 + q−180

.

Computer Talk
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["10 124"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= t4−t3 + t−1 + t−1−t−3 + t−4
In[5]:= Conway[K][z]
Out[5]= z8 + 7z6 + 14z4 + 8z2 + 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]= { 1, 8 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= −q10 + q6 + q4
In[9]:= HOMFLYPT[K][a, z]
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
Out[9]= z8a−8 + 8z6a−8−z6a−10 + 21z4a−8−7z4a−10 + 21z2a−8−14z2a−10 + z2a−12 + 7a−8−8a−10 + 2a−12
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= z8a−8 + z8a−10 + z7a−9 + z7a−11−8z6a−8−8z6a−10−7z5a−9−7z5a−11 + 21z4a−8 + 21z4a−10 + 14z3a−9 + 14z3a−11−21z2a−8−22z2a−10−z2a−12−8za−9−8za−11 + 7a−8 + 8a−10 + 2a−12

[edit] "Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

Same Jones Polynomial (up to mirroring, q\leftrightarrow q^{-1}): {}

Computer Talk
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["10 124"];
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]= { t4−t3 + t−1 + t−1−t−3 + t−4, −q10 + q6 + q4 }
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]= {}

[edit] Vassiliev invariants

V2 and V3: (8, 20)

[edit] Khovanov Homology

The coefficients of the monomials trqj are shown, along with their alternating sums χ (fixed j, alternation over r). The squares with yellow highlighting are those on the "critical diagonals", where j−2r = s + 1 or j−2r = s−1, where s = 8 is the signature of 10 124. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
 01234567χ
21       1-1
19     1  -1
17     11 0
15   11   0
13    1   1
11  1     1
91       1
71       1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i = 5 i = 7 i = 9
r = 0 {\mathbb Z} {\mathbb Z}
r = 1
r = 2 {\mathbb Z}
r = 3 {\mathbb Z}_2 {\mathbb Z}
r = 4 {\mathbb Z} {\mathbb Z}
r = 5 {\mathbb Z} {\mathbb Z}
r = 6 {\mathbb Z}
r = 7 {\mathbb Z}_2 {\mathbb Z}

[edit] The Coloured Jones Polynomials

The Coloured Jones Polynomials (in the n + 1-dimensional representation of sl(2))

 n  Jn
2 q29−q28 + q26−q25 + q23−q22−q21 + q20−q19−q18 + q17−q15 + q14 + q11 + q8
3 −q54 + q52 + q48−q46 + q44−q42 + q40−q38 + q36−q34−q30−q26 + q24−q22 + q20 + q16 + q12
4 q88−q87 + q83−q82−q80 + q78−q77 + q73−q72 + q71 + q68−q67 + q66−q64 + q63−q62 + q61−q59 + q58−q57 + q56−q54 + q53−q52 + q51−q49 + q48−q47 + q46−q44−q42 + q41−q39−q37 + q36−q34 + q31−q29 + q26 + q21 + q16
5 −q128 + q126 + q124−q122 + q118−q116 + q112−q110−q104 + q92 + q86−q82 + q80−q76 + q74−q70 + q68−q64 + q62−q58 + q56−q54−q52 + q50−q48−q46 + q44−q42 + q38−q36 + q32 + q26 + q20
6 q177−q176 + q170−2q169 + q164 + q163−2q162 + q160 + q157 + q156−2q155 + q150 + q149−2q148 + q143 + q142−2q141 + q136 + q135−2q134−q132 + q129 + q128−2q127−q125 + q122 + 2q121−2q120−q118 + q115 + 2q114−q113−q111 + q108 + 2q107−q106−q104 + q101 + q100−q99−q97 + q94 + q93−q92−q90 + q87 + q86−q85−q83 + q80 + q79−q78−q76 + q73 + q72−q71−q69 + q66−q64−q62 + q59−q57−q55 + q52−q50 + q45−q43 + q38 + q31 + q24
7 −q232 + q230 + q228−2q224 + q222 + q220−2q216 + q214 + q212−q210−2q208 + q206 + q204−2q200 + q198 + 2q196−2q192 + q190 + 2q188−q186−2q184 + q182 + 2q180−q178−2q176 + q174 + 2q172−q170−2q168 + q166 + 2q164−q162−2q160 + 2q156−q154−2q152 + 2q148−q146−q144 + 2q140−q138−q136 + q134 + 2q132−q130−q128 + q126 + 2q124−q122−q120 + 2q116−q114−q112 + 2q108−q106−q104 + 2q100−q98−q96 + 2q92−q90−q88 + 2q84−q82−q80 + q76−q74−q72 + q68−q66−q64 + q60−q58 + q52−q50 + q44 + q36 + q28

[edit] Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session, or any of the Computer Talk sections above.

[edit] Modifying This Page

Read me first: Modifying Knot Pages

See/edit the Rolfsen Knot Page master template (intermediate).

See/edit the Rolfsen_Splice_Base (expert).

Back to the top.

10_123

10_125

Retrieved from "http://katlas.org/wiki/10_124"
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