K11a196

From Knot Atlas

Jump to: navigation, search

K11a195

K11a197

Contents

Image:K11a196.gif
(Knotscape image) 		See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a196's page at Knotilus!

Visit K11a196's page at the original Knot Atlas!

[edit K11a196 Quick Notes]


[edit K11a196 Further Notes and Views]


[edit] Knot presentations

Planar diagram presentation X4251 X12,3,13,4 X14,6,15,5 X20,7,21,8 X22,10,1,9 X18,11,19,12 X2,13,3,14 X8,16,9,15 X10,17,11,18 X6,19,7,20 X16,21,17,22
Gauss code 1, -7, 2, -1, 3, -10, 4, -8, 5, -9, 6, -2, 7, -3, 8, -11, 9, -6, 10, -4, 11, -5
Dowker-Thistlethwaite code 4 12 14 20 22 18 2 8 10 6 16
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:BraidPart3.gifImage:BraidPart0.gifImage:BraidPart3.gifImage:BraidPart0.gifImage:BraidPart3.gifImage:BraidPart3.gifImage:BraidPart4.gifImage:BraidPart3.gifImage:BraidPart0.gifImage:BraidPart3.gifImage:BraidPart0.gif
Image:BraidPart0.gifImage:BraidPart4.gifImage:BraidPart4.gifImage:BraidPart1.gifImage:BraidPart4.gifImage:BraidPart1.gifImage:BraidPart4.gifImage:BraidPart4.gifImage:BraidPart0.gifImage:BraidPart4.gifImage:BraidPart1.gifImage:BraidPart4.gifImage:BraidPart1.gif
Image:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart2.gifImage:BraidPart0.gifImage:BraidPart2.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart2.gifImage:BraidPart0.gifImage:BraidPart2.gif
A Morse Link Presentation Image:K11a196_ML.gif

[edit] Three dimensional invariants

Symmetry type Reversible
Unknotting number {1,2}
3-genus 4
Bridge index Missing
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a196/ThurstonBennequinNumber
Hyperbolic Volume 17.2855
A-Polynomial See Data:K11a196/A-polynomial

[edit Notes for K11a196's three dimensional invariants]

[edit] Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus 1
Rasmussen s-Invariant 2

[edit Notes for K11a196's four dimensional invariants]

[edit] Polynomial invariants

Alexander polynomial −t4 + 6t3−17t2 + 31t−37 + 31t−1−17t−2 + 6t−3−t−4
Conway polynomial −z8−2z6−z4 + z2 + 1
2nd Alexander ideal (db, data sources) {7,t + 1}
Determinant and Signature { 147, -2 }
Jones polynomial q3−4q2 + 9q−15 + 21q−1−23q−2 + 24q−3−21q−4 + 15q−5−9q−6 + 4q−7−q−8
HOMFLY-PT polynomial (db, data sources) −a2z8 + 2a4z6−5a2z6 + z6−a6z4 + 7a4z4−10a2z4 + 3z4−2a6z2 + 8a4z2−8a2z2 + 3z2−a6 + 2a4−a2 + 1
Kauffman polynomial (db, data sources) 2a4z10 + 2a2z10 + 7a5z9 + 13a3z9 + 6az9 + 10a6z8 + 18a4z8 + 15a2z8 + 7z8 + 8a7z7−16a3z7−4az7 + 4z7a−1 + 4a8z6−15a6z6−47a4z6−44a2z6 + z6a−2−15z6 + a9z5−12a7z5−18a5z5−9a3z5−13az5−9z5a−1−5a8z4 + 10a6z4 + 41a4z4 + 38a2z4−2z4a−2 + 10z4−a9z3 + 7a7z3 + 18a5z3 + 16a3z3 + 12az3 + 6z3a−1 + a8z2−4a6z2−15a4z2−15a2z2 + z2a−2−4z2−2a7z−4a5z−4a3z−3az−za−1 + a6 + 2a4 + a2 + 1
The A2 invariant −q24 + q22−2q18 + 4q16−4q14 + q12 + q10−2q8 + 6q6−4q4 + 4q2−1−2q−2 + 3q−4−2q−6 + q−8
The G2 invariant q128−3q126 + 7q124−13q122 + 16q120−16q118 + 7q116 + 16q114−45q112 + 83q110−114q108 + 116q106−81q104−10q102 + 143q100−285q98 + 395q96−412q94 + 288q92−19q90−340q88 + 676q86−847q84 + 752q82−393q80−154q78 + 682q76−983q74 + 934q72−512q70−90q68 + 619q66−839q64 + 636q62−113q60−501q58 + 913q56−917q54 + 476q52 + 253q50−968q48 + 1385q46−1320q44 + 767q42 + 61q40−878q38 + 1394q36−1426q34 + 992q32−239q30−517q28 + 1002q26−1043q24 + 647q22−12q20−573q18 + 842q16−688q14 + 185q12 + 451q10−920q8 + 1023q6−718q4 + 123q2 + 493−913q−2 + 997q−4−744q−6 + 304q−8 + 158q−10−480q−12 + 587q−14−495q−16 + 292q−18−70q−20−97q−22 + 174q−24−178q−26 + 130q−28−66q−30 + 18q−32 + 14q−34−25q−36 + 22q−38−16q−40 + 8q−42−3q−44 + q−46
Further Quantum Invariants
K11a196 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["K11a196"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= −t4 + 6t3−17t2 + 31t−37 + 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]= {7,t + 1}
In[7]:= {KnotDet[K], KnotSignature[K]}
Out[7]= { 147, -2 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= q3−4q2 + 9q−15 + 21q−1−23q−2 + 24q−3−21q−4 + 15q−5−9q−6 + 4q−7−q−8
In[9]:= HOMFLYPT[K][a, z]
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
Out[9]= −a2z8 + 2a4z6−5a2z6 + z6−a6z4 + 7a4z4−10a2z4 + 3z4−2a6z2 + 8a4z2−8a2z2 + 3z2−a6 + 2a4−a2 + 1
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= 2a4z10 + 2a2z10 + 7a5z9 + 13a3z9 + 6az9 + 10a6z8 + 18a4z8 + 15a2z8 + 7z8 + 8a7z7−16a3z7−4az7 + 4z7a−1 + 4a8z6−15a6z6−47a4z6−44a2z6 + z6a−2−15z6 + a9z5−12a7z5−18a5z5−9a3z5−13az5−9z5a−1−5a8z4 + 10a6z4 + 41a4z4 + 38a2z4−2z4a−2 + 10z4−a9z3 + 7a7z3 + 18a5z3 + 16a3z3 + 12az3 + 6z3a−1 + a8z2−4a6z2−15a4z2−15a2z2 + z2a−2−4z2−2a7z−4a5z−4a3z−3az−za−1 + a6 + 2a4 + a2 + 1

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

Same Alexander/Conway Polynomial: {K11a216, K11a286,}

Same Jones Polynomial (up to mirroring, q\leftrightarrow q^{-1}): {K11a216,}

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["K11a196"];
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−37 + 31t−1−17t−2 + 6t−3−t−4, q3−4q2 + 9q−15 + 21q−1−23q−2 + 24q−3−21q−4 + 15q−5−9q−6 + 4q−7−q−8 }
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]= {K11a216, K11a286,}
In[6]:= DeleteCases[ Select[ AllKnots[], (J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) & ], K ]
KnotTheory::loading: Loading precomputed data in Jones4Knots11`.
Out[6]= {K11a216,}

[edit] Vassiliev invariants

V2 and V3: (1, -2)

[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 K11a196. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
 -7-6-5-4-3-2-101234χ
7           11
5          3 -3
3         61 5
1        93  -6
-1       126   6
-3      1210    -2
-5     1211     1
-7    912      3
-9   612       -6
-11  39        6
-13 16         -5
-15 3          3
-171           -1
Integral Khovanov Homology

(db, data source)

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

Back to the top.

K11a195

K11a197

Retrieved from "http://katlas.org/wiki/K11a196"
Personal tools