K11a26

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K11a25

K11a27

Contents

Image:K11a26.gif
(Knotscape image) 		See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a26's page at Knotilus!

Visit K11a26's page at the original Knot Atlas!

[edit K11a26 Quick Notes]


[edit K11a26 Further Notes and Views]


[edit] Knot presentations

Planar diagram presentation X4251 X8493 X12,5,13,6 X2837 X20,9,21,10 X18,12,19,11 X6,13,7,14 X10,16,11,15 X22,17,1,18 X14,20,15,19 X16,21,17,22
Gauss code 1, -4, 2, -1, 3, -7, 4, -2, 5, -8, 6, -3, 7, -10, 8, -11, 9, -6, 10, -5, 11, -9
Dowker-Thistlethwaite code 4 8 12 2 20 18 6 10 22 14 16
A Braid Representative
Image:BraidPart1.gifImage:BraidPart1.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart1.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gif
Image:BraidPart2.gifImage:BraidPart2.gifImage:BraidPart3.gifImage:BraidPart0.gifImage:BraidPart3.gifImage:BraidPart3.gifImage:BraidPart2.gifImage:BraidPart3.gifImage:BraidPart3.gifImage:BraidPart0.gif
Image:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart4.gifImage:BraidPart1.gifImage:BraidPart4.gifImage:BraidPart4.gifImage:BraidPart1.gifImage:BraidPart4.gifImage:BraidPart4.gifImage:BraidPart1.gif
Image:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart2.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart2.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart2.gif
A Morse Link Presentation Image:K11a26_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:K11a26/ThurstonBennequinNumber
Hyperbolic Volume 17.9006
A-Polynomial See Data:K11a26/A-polynomial

[edit Notes for K11a26'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 K11a26's four dimensional invariants]

[edit] Polynomial invariants

Alexander polynomial t4−6t3 + 18t2−33t + 41−33t−1 + 18t−2−6t−3 + t−4
Conway polynomial z8 + 2z6 + 2z4 + z2 + 1
2nd Alexander ideal (db, data sources) {1}
Determinant and Signature { 157, 0 }
Jones polynomial q6−4q5 + 9q4−16q3 + 22q2−25q + 26−22q−1 + 17q−2−10q−3 + 4q−4−q−5
HOMFLY-PT polynomial (db, data sources) z8−a2z6−2z6a−2 + 5z6−3a2z4−7z4a−2 + z4a−4 + 11z4−4a2z2−9z2a−2 + 2z2a−4 + 12z2−2a2−4a−2 + a−4 + 6
Kauffman polynomial (db, data sources) 2z10a−2 + 2z10 + 8az9 + 14z9a−1 + 6z9a−3 + 12a2z8 + 18z8a−2 + 7z8a−4 + 23z8 + 9a3z7 + 2az7−12z7a−1−z7a−3 + 4z7a−5 + 4a4z6−19a2z6−50z6a−2−14z6a−4 + z6a−6−58z6 + a5z5−13a3z5−26az5−24z5a−1−21z5a−3−9z5a−5−4a4z4 + 15a2z4 + 42z4a−2 + 9z4a−4−2z4a−6 + 50z4−a5z3 + 9a3z3 + 25az3 + 30z3a−1 + 22z3a−3 + 7z3a−5 + a4z2−8a2z2−18z2a−2−3z2a−4 + z2a−6−23z2−3a3z−8az−10za−1−7za−3−2za−5 + 2a2 + 4a−2 + a−4 + 6
The A2 invariant −q14 + 2q12−4q10 + 2q8 + q6−3q4 + 7q2−3 + 5q−2−q−4−2q−6 + 3q−8−5q−10 + 2q−12−q−16 + q−18
The G2 invariant q80−3q78 + 7q76−13q74 + 17q72−19q70 + 12q68 + 11q66−46q64 + 94q62−139q60 + 154q58−123q56 + 20q54 + 160q52−369q50 + 543q48−584q46 + 408q44−30q42−482q40 + 949q38−1167q36 + 1001q34−454q32−314q30 + 1007q28−1337q26 + 1155q24−520q22−297q20 + 930q18−1100q16 + 723q14 + 43q12−830q10 + 1288q8−1170q6 + 494q4 + 492q2−1389 + 1846q−2−1655q−4 + 877q−6 + 226q−8−1251q−10 + 1840q−12−1780q−14 + 1119q−16−121q−18−823q−20 + 1338q−22−1252q−24 + 639q−26 + 208q−28−899q−30 + 1117q−32−782q−34 + 40q−36 + 763q−38−1275q−40 + 1280q−42−790q−44 + 9q−46 + 733q−48−1173q−50 + 1190q−52−823q−54 + 270q−56 + 253q−58−591q−60 + 671q−62−539q−64 + 300q−66−52q−68−120q−70 + 194q−72−188q−74 + 134q−76−66q−78 + 16q−80 + 15q−82−26q−84 + 22q−86−16q−88 + 8q−90−3q−92 + q−94
Further Quantum Invariants
K11a26 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["K11a26"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= t4−6t3 + 18t2−33t + 41−33t−1 + 18t−2−6t−3 + t−4
In[5]:= Conway[K][z]
Out[5]= z8 + 2z6 + 2z4 + 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]= { 157, 0 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= q6−4q5 + 9q4−16q3 + 22q2−25q + 26−22q−1 + 17q−2−10q−3 + 4q−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 + 5z6−3a2z4−7z4a−2 + z4a−4 + 11z4−4a2z2−9z2a−2 + 2z2a−4 + 12z2−2a2−4a−2 + a−4 + 6
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= 2z10a−2 + 2z10 + 8az9 + 14z9a−1 + 6z9a−3 + 12a2z8 + 18z8a−2 + 7z8a−4 + 23z8 + 9a3z7 + 2az7−12z7a−1−z7a−3 + 4z7a−5 + 4a4z6−19a2z6−50z6a−2−14z6a−4 + z6a−6−58z6 + a5z5−13a3z5−26az5−24z5a−1−21z5a−3−9z5a−5−4a4z4 + 15a2z4 + 42z4a−2 + 9z4a−4−2z4a−6 + 50z4−a5z3 + 9a3z3 + 25az3 + 30z3a−1 + 22z3a−3 + 7z3a−5 + a4z2−8a2z2−18z2a−2−3z2a−4 + z2a−6−23z2−3a3z−8az−10za−1−7za−3−2za−5 + 2a2 + 4a−2 + a−4 + 6

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

Same Alexander/Conway Polynomial: {K11a24, K11a315,}

Same Jones Polynomial (up to mirroring, q\leftrightarrow q^{-1}): {K11a24, K11a315,}

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["K11a26"];
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 + 18t2−33t + 41−33t−1 + 18t−2−6t−3 + t−4, q6−4q5 + 9q4−16q3 + 22q2−25q + 26−22q−1 + 17q−2−10q−3 + 4q−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]= {K11a24, K11a315,}
In[6]:= DeleteCases[ Select[ AllKnots[], (J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) & ], K ]
KnotTheory::loading: Loading precomputed data in Jones4Knots11`.
Out[6]= {K11a24, K11a315,}

[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 K11a26. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
 -5-4-3-2-10123456χ
13           11
11          3 -3
9         61 5
7        103  -7
5       126   6
3      1310    -3
1     1312     1
-1    1014      4
-3   712       -5
-5  310        7
-7 17         -6
-9 3          3
-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}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r = −3 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r = −2 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r = −1 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r = 0 {\mathbb Z}^{14}\oplus{\mathbb Z}_2^{12} {\mathbb Z}^{13}
r = 1 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{13} {\mathbb Z}^{13}
r = 2 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{12} {\mathbb Z}^{12}
r = 3 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r = 4 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r = 5 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
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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K11a25

K11a27

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