K11a189

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K11a188

K11a190

Contents

Image:K11a189.gif
(Knotscape image) 		See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a189's page at Knotilus!

Visit K11a189's page at the original Knot Atlas!

[edit K11a189 Quick Notes]


[edit K11a189 Further Notes and Views]


[edit] Knot presentations

Planar diagram presentation X4251 X12,3,13,4 X14,6,15,5 X16,7,17,8 X22,10,1,9 X18,11,19,12 X2,13,3,14 X20,16,21,15 X10,17,11,18 X6,19,7,20 X8,22,9,21
Gauss code 1, -7, 2, -1, 3, -10, 4, -11, 5, -9, 6, -2, 7, -3, 8, -4, 9, -6, 10, -8, 11, -5
Dowker-Thistlethwaite code 4 12 14 16 22 18 2 20 10 6 8
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:BraidPart3.gifImage:BraidPart0.gifImage:BraidPart3.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart3.gifImage:BraidPart3.gifImage:BraidPart4.gifImage:BraidPart3.gifImage:BraidPart0.gifImage:BraidPart3.gifImage:BraidPart0.gif
Image:BraidPart0.gifImage:BraidPart4.gifImage:BraidPart1.gifImage:BraidPart4.gifImage:BraidPart1.gifImage:BraidPart1.gifImage:BraidPart4.gifImage:BraidPart4.gifImage:BraidPart0.gifImage:BraidPart4.gifImage:BraidPart1.gifImage:BraidPart4.gifImage:BraidPart1.gif
Image:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart2.gifImage:BraidPart0.gifImage:BraidPart2.gifImage:BraidPart2.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart2.gifImage:BraidPart0.gifImage:BraidPart2.gif
A Morse Link Presentation Image:K11a189_ML.gif

[edit] Three dimensional invariants

Symmetry type Chiral
Unknotting number 1
3-genus 4
Bridge index Missing
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a189/ThurstonBennequinNumber
Hyperbolic Volume 17.3742
A-Polynomial See Data:K11a189/A-polynomial

[edit Notes for K11a189's three dimensional invariants]

[edit] Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus 4
Rasmussen s-Invariant 0

[edit Notes for K11a189's four dimensional invariants]

[edit] Polynomial invariants

Alexander polynomial t4−6t3 + 17t2−31t + 39−31t−1 + 17t−2−6t−3 + t−4
Conway polynomial z8 + 2z6 + z4−z2 + 1
2nd Alexander ideal (db, data sources) {1}
Determinant and Signature { 149, 0 }
Jones polynomial −q5 + 4q4−9q3 + 16q2−21q + 24−24q−1 + 21q−2−15q−3 + 9q−4−4q−5 + q−6
HOMFLY-PT polynomial (db, data sources) z8−2a2z6−z6a−2 + 5z6 + a4z4−7a2z4−3z4a−2 + 10z4 + 2a4z2−8a2z2−3z2a−2 + 8z2 + a4−2a2 + 2
Kauffman polynomial (db, data sources) 2a2z10 + 2z10 + 6a3z9 + 13az9 + 7z9a−1 + 7a4z8 + 15a2z8 + 10z8a−2 + 18z8 + 4a5z7−4a3z7−15az7 + z7a−1 + 8z7a−3 + a6z6−15a4z6−42a2z6−13z6a−2 + 4z6a−4−43z6−9a5z5−12a3z5−8az5−17z5a−1−11z5a−3 + z5a−5−2a6z4 + 10a4z4 + 34a2z4 + 6z4a−2−5z4a−4 + 33z4 + 6a5z3 + 11a3z3 + 11az3 + 13z3a−1 + 6z3a−3−z3a−5 + a6z2−3a4z2−13a2z2−z2a−2 + 2z2a−4−12z2−a5z−2a3z−3az−3za−1−za−3 + a4 + 2a2 + 2
The A2 invariant q18−q16 + 2q12−4q10 + 4q8−q6−q4 + 3q2−5 + 5q−2−3q−4 + 2q−6 + 3q−8−3q−10 + 2q−12−q−14
The G2 invariant q94−3q92 + 8q90−16q88 + 22q86−25q84 + 14q82 + 18q80−66q78 + 128q76−176q74 + 175q72−102q70−63q68 + 291q66−498q64 + 599q62−499q60 + 178q58 + 290q56−759q54 + 1036q52−973q50 + 548q48 + 98q46−728q44 + 1080q42−1000q40 + 535q38 + 131q36−693q34 + 892q32−650q30 + 51q28 + 623q26−1060q24 + 1059q22−580q20−201q18 + 996q16−1494q14 + 1489q12−975q10 + 111q8 + 783q6−1390q4 + 1499q2−1079 + 330q−2 + 455q−4−964q−6 + 1000q−8−594q−10−53q−12 + 636q−14−876q−16 + 682q−18−137q−20−501q−22 + 962q−24−1052q−26 + 755q−28−204q−30−393q−32 + 806q−34−921q−36 + 752q−38−387q−40−5q−42 + 304q−44−454q−46 + 439q−48−320q−50 + 157q−52−7q−54−89q−56 + 127q−58−122q−60 + 89q−62−47q−64 + 15q−66 + 8q−68−17q−70 + 16q−72−13q−74 + 7q−76−3q−78 + q−80
Further Quantum Invariants
K11a189 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["K11a189"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= t4−6t3 + 17t2−31t + 39−31t−1 + 17t−2−6t−3 + t−4
In[5]:= Conway[K][z]
Out[5]= z8 + 2z6 + z4−z2 + 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]= { 149, 0 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= −q5 + 4q4−9q3 + 16q2−21q + 24−24q−1 + 21q−2−15q−3 + 9q−4−4q−5 + q−6
In[9]:= HOMFLYPT[K][a, z]
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
Out[9]= z8−2a2z6−z6a−2 + 5z6 + a4z4−7a2z4−3z4a−2 + 10z4 + 2a4z2−8a2z2−3z2a−2 + 8z2 + a4−2a2 + 2
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= 2a2z10 + 2z10 + 6a3z9 + 13az9 + 7z9a−1 + 7a4z8 + 15a2z8 + 10z8a−2 + 18z8 + 4a5z7−4a3z7−15az7 + z7a−1 + 8z7a−3 + a6z6−15a4z6−42a2z6−13z6a−2 + 4z6a−4−43z6−9a5z5−12a3z5−8az5−17z5a−1−11z5a−3 + z5a−5−2a6z4 + 10a4z4 + 34a2z4 + 6z4a−2−5z4a−4 + 33z4 + 6a5z3 + 11a3z3 + 11az3 + 13z3a−1 + 6z3a−3−z3a−5 + a6z2−3a4z2−13a2z2−z2a−2 + 2z2a−4−12z2−a5z−2a3z−3az−3za−1−za−3 + a4 + 2a2 + 2

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

Same Alexander/Conway Polynomial: {}

Same Jones Polynomial (up to mirroring, q\leftrightarrow q^{-1}): {K11a30, K11a272,}

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["K11a189"];
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 + 17t2−31t + 39−31t−1 + 17t−2−6t−3 + t−4, −q5 + 4q4−9q3 + 16q2−21q + 24−24q−1 + 21q−2−15q−3 + 9q−4−4q−5 + q−6 }
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]= {K11a30, K11a272,}

[edit] Vassiliev invariants

V2 and V3: (-1, 1)

[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 = 0 is the signature of K11a189. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
 -6-5-4-3-2-1012345χ
11           1-1
9          3 3
7         61 -5
5        103  7
3       116   -5
1      1310    3
-1     1212     0
-3    912      -3
-5   612       6
-7  39        -6
-9 16         5
-11 3          -3
-131           1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i = −1 i = 1
r = −6 {\mathbb Z}
r = −5 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r = −4 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r = −3 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r = −2 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r = −1 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{12} {\mathbb Z}^{12}
r = 0 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{12} {\mathbb Z}^{13}
r = 1 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{11}
r = 2 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r = 3 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r = 4 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r = 5 {\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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K11a188

K11a190

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