K11a126

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K11a125

K11a127

Contents

Image:K11a126.gif
(Knotscape image) 		See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a126's page at Knotilus!

Visit K11a126's page at the original Knot Atlas!

[edit K11a126 Quick Notes]


[edit K11a126 Further Notes and Views]


[edit] Knot presentations

Planar diagram presentation X4251 X10,4,11,3 X14,5,15,6 X20,8,21,7 X2,10,3,9 X16,11,17,12 X22,13,1,14 X8,15,9,16 X12,17,13,18 X6,20,7,19 X18,22,19,21
Gauss code 1, -5, 2, -1, 3, -10, 4, -8, 5, -2, 6, -9, 7, -3, 8, -6, 9, -11, 10, -4, 11, -7
Dowker-Thistlethwaite code 4 10 14 20 2 16 22 8 12 6 18
A Braid Representative
Image:BraidPart1.gifImage:BraidPart1.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart1.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gif
Image:BraidPart2.gifImage:BraidPart2.gifImage:BraidPart3.gifImage:BraidPart3.gifImage:BraidPart0.gifImage:BraidPart3.gifImage:BraidPart2.gifImage:BraidPart3.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart3.gif
Image:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart4.gifImage:BraidPart4.gifImage:BraidPart1.gifImage:BraidPart4.gifImage:BraidPart0.gifImage:BraidPart4.gifImage:BraidPart1.gifImage:BraidPart1.gifImage:BraidPart4.gif
Image:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart2.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart2.gifImage:BraidPart2.gifImage:BraidPart0.gif
A Morse Link Presentation Image:K11a126_ML.gif

[edit] Three dimensional invariants

Symmetry type Reversible
Unknotting number {1,2,3}
3-genus 4
Bridge index Missing
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a126/ThurstonBennequinNumber
Hyperbolic Volume 17.2092
A-Polynomial See Data:K11a126/A-polynomial

[edit Notes for K11a126'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 K11a126's four dimensional invariants]

[edit] Polynomial invariants

Alexander polynomial t4−5t3 + 16t2−31t + 39−31t−1 + 16t−2−5t−3 + t−4
Conway polynomial z8 + 3z6 + 6z4 + 4z2 + 1
2nd Alexander ideal (db, data sources) {1}
Determinant and Signature { 145, 0 }
Jones polynomial q6−5q5 + 10q4−16q3 + 21q2−23q + 24−19q−1 + 14q−2−8q−3 + 3q−4−q−5
HOMFLY-PT polynomial (db, data sources) z8−a2z6−2z6a−2 + 6z6−4a2z4−7z4a−2 + z4a−4 + 16z4−7a2z2−9z2a−2 + z2a−4 + 19z2−4a2−4a−2 + 9
Kauffman polynomial (db, data sources) 2z10a−2 + 2z10 + 6az9 + 13z9a−1 + 7z9a−3 + 8a2z8 + 17z8a−2 + 9z8a−4 + 16z8 + 6a3z7−16z7a−1−5z7a−3 + 5z7a−5 + 3a4z6−13a2z6−50z6a−2−20z6a−4 + z6a−6−45z6 + a5z5−9a3z5−16az5−13z5a−1−17z5a−3−10z5a−5−4a4z4 + 15a2z4 + 41z4a−2 + 11z4a−4−z4a−6 + 48z4−2a5z3 + 7a3z3 + 23az3 + 25z3a−1 + 15z3a−3 + 4z3a−5 + a4z2−11a2z2−17z2a−2−2z2a−4−27z2 + a5z−3a3z−10az−10za−1−3za−3 + za−5 + 4a2 + 4a−2 + 9
The A2 invariant −q14 + q12−4q10 + q8 + q6−3q4 + 7q2−1 + 5q−2 + q−4−2q−6 + 3q−8−5q−10 + q−12−2q−16 + q−18
The G2 invariant q80−2q78 + 5q76−8q74 + 10q72−10q70 + 4q68 + 10q66−29q64 + 52q62−73q60 + 75q58−57q56 + 5q54 + 81q52−176q50 + 264q48−301q46 + 243q44−90q42−162q40 + 435q38−642q36 + 679q34−495q32 + 97q30 + 391q28−790q26 + 940q24−756q22 + 286q20 + 280q18−717q16 + 822q14−537q12 + 9q10 + 548q8−837q6 + 717q4−214q2−464 + 1042q−2−1252q−4 + 998q−6−336q−8−472q−10 + 1152q−12−1438q−14 + 1242q−16−639q−18−146q−20 + 810q−22−1127q−24 + 1002q−26−494q−28−155q−30 + 664q−32−825q−34 + 565q−36−24q−38−560q−40 + 909q−42−872q−44 + 456q−46 + 157q−48−722q−50 + 1015q−52−936q−54 + 549q−56−32q−58−426q−60 + 665q−62−654q−64 + 456q−66−172q−68−78q−70 + 221q−72−254q−74 + 198q−76−107q−78 + 32q−80 + 21q−82−39q−84 + 35q−86−24q−88 + 11q−90−4q−92 + q−94
Further Quantum Invariants
K11a126 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["K11a126"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= t4−5t3 + 16t2−31t + 39−31t−1 + 16t−2−5t−3 + t−4
In[5]:= Conway[K][z]
Out[5]= z8 + 3z6 + 6z4 + 4z2 + 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]= { 145, 0 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= q6−5q5 + 10q4−16q3 + 21q2−23q + 24−19q−1 + 14q−2−8q−3 + 3q−4−q−5
In[9]:= HOMFLYPT[K][a, z]
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
Out[9]= z8−a2z6−2z6a−2 + 6z6−4a2z4−7z4a−2 + z4a−4 + 16z4−7a2z2−9z2a−2 + z2a−4 + 19z2−4a2−4a−2 + 9
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= 2z10a−2 + 2z10 + 6az9 + 13z9a−1 + 7z9a−3 + 8a2z8 + 17z8a−2 + 9z8a−4 + 16z8 + 6a3z7−16z7a−1−5z7a−3 + 5z7a−5 + 3a4z6−13a2z6−50z6a−2−20z6a−4 + z6a−6−45z6 + a5z5−9a3z5−16az5−13z5a−1−17z5a−3−10z5a−5−4a4z4 + 15a2z4 + 41z4a−2 + 11z4a−4−z4a−6 + 48z4−2a5z3 + 7a3z3 + 23az3 + 25z3a−1 + 15z3a−3 + 4z3a−5 + a4z2−11a2z2−17z2a−2−2z2a−4−27z2 + a5z−3a3z−10az−10za−1−3za−3 + za−5 + 4a2 + 4a−2 + 9

[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["K11a126"];
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]= { t4−5t3 + 16t2−31t + 39−31t−1 + 16t−2−5t−3 + t−4, q6−5q5 + 10q4−16q3 + 21q2−23q + 24−19q−1 + 14q−2−8q−3 + 3q−4−q−5 }
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: (4, 0)

[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 K11a126. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
 -5-4-3-2-10123456χ
13           11
11          4 -4
9         61 5
7        104  -6
5       116   5
3      1210    -2
1     1211     1
-1    813      5
-3   611       -5
-5  28        6
-7 16         -5
-9 2          2
-111           -1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i = −1 i = 1
r = −5 {\mathbb Z}
r = −4 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r = −3 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r = −2 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r = −1 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r = 0 {\mathbb Z}^{13}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{12}
r = 1 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{12} {\mathbb Z}^{12}
r = 2 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{11}
r = 3 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r = 4 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r = 5 {\mathbb Z}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r = 6 {\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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K11a125

K11a127

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