K11a147

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K11a146

K11a148

Contents

Image:K11a147.gif
(Knotscape image) 		See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a147's page at Knotilus!

Visit K11a147's page at the original Knot Atlas!

[edit K11a147 Quick Notes]


[edit K11a147 Further Notes and Views]


[edit] Knot presentations

Planar diagram presentation X4251 X10,4,11,3 X16,6,17,5 X20,8,21,7 X2,10,3,9 X22,11,1,12 X18,13,19,14 X8,16,9,15 X12,17,13,18 X6,20,7,19 X14,21,15,22
Gauss code 1, -5, 2, -1, 3, -10, 4, -8, 5, -2, 6, -9, 7, -11, 8, -3, 9, -7, 10, -4, 11, -6
Dowker-Thistlethwaite code 4 10 16 20 2 22 18 8 12 6 14
A Braid Representative
Image:BraidPart1.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart3.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gif
Image:BraidPart2.gifImage:BraidPart1.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart1.gifImage:BraidPart1.gifImage:BraidPart0.gifImage:BraidPart1.gifImage:BraidPart4.gifImage:BraidPart1.gifImage:BraidPart1.gifImage:BraidPart0.gifImage:BraidPart1.gif
Image:BraidPart0.gifImage:BraidPart2.gifImage:BraidPart3.gifImage:BraidPart3.gifImage:BraidPart2.gifImage:BraidPart2.gifImage:BraidPart3.gifImage:BraidPart2.gifImage:BraidPart0.gifImage:BraidPart2.gifImage:BraidPart2.gifImage:BraidPart3.gifImage:BraidPart2.gif
Image:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart4.gifImage:BraidPart4.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart4.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart4.gifImage:BraidPart0.gif
A Morse Link Presentation Image:K11a147_ML.gif

[edit] Three dimensional invariants

Symmetry type Chiral
Unknotting number {1,2}
3-genus 4
Bridge index Missing
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a147/ThurstonBennequinNumber
Hyperbolic Volume 17.2353
A-Polynomial See Data:K11a147/A-polynomial

[edit Notes for K11a147's three dimensional invariants]

[edit] Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus 4
Rasmussen s-Invariant -2

[edit Notes for K11a147's four dimensional invariants]

[edit] Polynomial invariants

Alexander polynomial −t4 + 6t3−17t2 + 32t−39 + 32t−1−17t−2 + 6t−3−t−4
Conway polynomial −z8−2z6−z4 + 2z2 + 1
2nd Alexander ideal (db, data sources) {1}
Determinant and Signature { 151, 2 }
Jones polynomial −q8 + 4q7−10q6 + 16q5−21q4 + 25q3−24q2 + 21q−15 + 9q−1−4q−2 + q−3
HOMFLY-PT polynomial (db, data sources) −z8a−2−5z6a−2 + 2z6a−4 + z6−10z4a−2 + 7z4a−4−z4a−6 + 3z4−8z2a−2 + 9z2a−4−2z2a−6 + 3z2−2a−2 + 4a−4−2a−6 + 1
Kauffman polynomial (db, data sources) 2z10a−2 + 2z10a−4 + 6z9a−1 + 13z9a−3 + 7z9a−5 + 15z8a−2 + 19z8a−4 + 11z8a−6 + 7z8 + 4az7−4z7a−1−13z7a−3 + 4z7a−5 + 9z7a−7 + a2z6−41z6a−2−45z6a−4−16z6a−6 + 4z6a−8−15z6−9az5−12z5a−1−15z5a−3−27z5a−5−14z5a−7 + z5a−9−2a2z4 + 32z4a−2 + 34z4a−4 + 10z4a−6−4z4a−8 + 10z4 + 6az3 + 12z3a−1 + 19z3a−3 + 24z3a−5 + 10z3a−7−z3a−9 + a2z2−12z2a−2−14z2a−4−5z2a−6 + z2a−8−3z2−az−3za−1−5za−3−7za−5−4za−7 + 2a−2 + 4a−4 + 2a−6 + 1
The A2 invariant q8−2q6 + 3q4−2q2−1 + 4q−2−5q−4 + 5q−6−2q−8 + 2q−10 + 3q−12−3q−14 + 4q−16−3q−18−q−20 + q−22−q−24
The G2 invariant q46−3q44 + 8q42−16q40 + 22q38−25q36 + 14q34 + 18q32−66q30 + 128q28−176q26 + 175q24−101q22−63q20 + 289q18−494q16 + 595q14−503q12 + 187q10 + 276q8−746q6 + 1033q4−982q2 + 583 + 52q−2−689q−4 + 1072q−6−1041q−8 + 605q−10 + 49q−12−639q−14 + 889q−16−697q−18 + 143q−20 + 532q−22−1009q−24 + 1072q−26−654q−28−117q−30 + 932q−32−1485q−34 + 1542q−36−1054q−38 + 200q−40 + 732q−42−1392q−44 + 1563q−46−1188q−48 + 435q−50 + 387q−52−955q−54 + 1059q−56−686q−58 + 53q−60 + 563q−62−860q−64 + 723q−66−231q−68−416q−70 + 919q−72−1075q−74 + 828q−76−285q−78−335q−80 + 801q−82−973q−84 + 831q−86−473q−88 + 42q−90 + 307q−92−501q−94 + 504q−96−370q−98 + 187q−100−9q−102−105q−104 + 149q−106−143q−108 + 99q−110−49q−112 + 12q−114 + 12q−116−19q−118 + 17q−120−13q−122 + 7q−124−3q−126 + q−128
Further Quantum Invariants
K11a147 Quantum Invariants.
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["K11a147"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= −t4 + 6t3−17t2 + 32t−39 + 32t−1−17t−2 + 6t−3−t−4
In[5]:= Conway[K][z]
Out[5]= −z8−2z6−z4 + 2z2 + 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]= { 151, 2 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= −q8 + 4q7−10q6 + 16q5−21q4 + 25q3−24q2 + 21q−15 + 9q−1−4q−2 + q−3
In[9]:= HOMFLYPT[K][a, z]
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
Out[9]= −z8a−2−5z6a−2 + 2z6a−4 + z6−10z4a−2 + 7z4a−4−z4a−6 + 3z4−8z2a−2 + 9z2a−4−2z2a−6 + 3z2−2a−2 + 4a−4−2a−6 + 1
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= 2z10a−2 + 2z10a−4 + 6z9a−1 + 13z9a−3 + 7z9a−5 + 15z8a−2 + 19z8a−4 + 11z8a−6 + 7z8 + 4az7−4z7a−1−13z7a−3 + 4z7a−5 + 9z7a−7 + a2z6−41z6a−2−45z6a−4−16z6a−6 + 4z6a−8−15z6−9az5−12z5a−1−15z5a−3−27z5a−5−14z5a−7 + z5a−9−2a2z4 + 32z4a−2 + 34z4a−4 + 10z4a−6−4z4a−8 + 10z4 + 6az3 + 12z3a−1 + 19z3a−3 + 24z3a−5 + 10z3a−7−z3a−9 + a2z2−12z2a−2−14z2a−4−5z2a−6 + z2a−8−3z2−az−3za−1−5za−3−7za−5−4za−7 + 2a−2 + 4a−4 + 2a−6 + 1

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

Same Alexander/Conway Polynomial: {}

Same Jones Polynomial (up to mirroring, q\leftrightarrow q^{-1}): {K11a322,}

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["K11a147"];
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 + 6t3−17t2 + 32t−39 + 32t−1−17t−2 + 6t−3−t−4, −q8 + 4q7−10q6 + 16q5−21q4 + 25q3−24q2 + 21q−15 + 9q−1−4q−2 + q−3 }
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]= {K11a322,}

[edit] Vassiliev invariants

V2 and V3: (2, 4)

[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 = 2 is the signature of K11a147. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
 -4-3-2-101234567χ
17           1-1
15          3 3
13         71 -6
11        93  6
9       127   -5
7      139    4
5     1112     1
3    1013      -3
1   612       6
-1  39        -6
-3 16         5
-5 3          -3
-71           1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i = 1 i = 3
r = −4 {\mathbb Z}
r = −3 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r = −2 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r = −1 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r = 0 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{10}
r = 1 {\mathbb Z}^{13}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{11}
r = 2 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{13} {\mathbb Z}^{13}
r = 3 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{12} {\mathbb Z}^{12}
r = 4 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r = 5 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r = 6 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r = 7 {\mathbb Z}_2 {\mathbb Z}

[edit] Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.


[edit] Modifying This Page

Read me first: Modifying Knot Pages.

See/edit the Hoste-Thistlethwaite Knot Page master template (intermediate).

See/edit the Hoste-Thistlethwaite_Splice_Base (expert).

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K11a146

K11a148

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