K11a106

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K11a105

K11a107

Contents

Image:K11a106.gif
(Knotscape image) 		See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a106's page at Knotilus!

Visit K11a106's page at the original Knot Atlas!

[edit K11a106 Quick Notes]


[edit K11a106 Further Notes and Views]


[edit] Knot presentations

Planar diagram presentation X4251 X10,4,11,3 X14,5,15,6 X16,7,17,8 X2,10,3,9 X20,12,21,11 X18,13,19,14 X6,15,7,16 X8,17,9,18 X22,20,1,19 X12,22,13,21
Gauss code 1, -5, 2, -1, 3, -8, 4, -9, 5, -2, 6, -11, 7, -3, 8, -4, 9, -7, 10, -6, 11, -10
Dowker-Thistlethwaite code 4 10 14 16 2 20 18 6 8 22 12
A Braid Representative
Image:BraidPart1.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart3.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gif
Image:BraidPart2.gifImage:BraidPart1.gifImage:BraidPart1.gifImage:BraidPart0.gifImage:BraidPart1.gifImage:BraidPart4.gifImage:BraidPart1.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart1.gifImage:BraidPart1.gif
Image:BraidPart0.gifImage:BraidPart2.gifImage:BraidPart2.gifImage:BraidPart3.gifImage:BraidPart2.gifImage:BraidPart0.gifImage:BraidPart2.gifImage:BraidPart3.gifImage:BraidPart3.gifImage:BraidPart3.gifImage:BraidPart3.gifImage:BraidPart2.gifImage:BraidPart2.gif
Image:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart4.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart4.gifImage:BraidPart4.gifImage:BraidPart4.gifImage:BraidPart4.gifImage:BraidPart0.gifImage:BraidPart0.gif
A Morse Link Presentation Image:K11a106_ML.gif

[edit] Three dimensional invariants

Symmetry type Reversible
Unknotting number 1
3-genus 4
Bridge index Missing
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a106/ThurstonBennequinNumber
Hyperbolic Volume 13.4001
A-Polynomial See Data:K11a106/A-polynomial

[edit Notes for K11a106'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 K11a106's four dimensional invariants]

[edit] Polynomial invariants

Alexander polynomial t4−5t3 + 12t2−18t + 21−18t−1 + 12t−2−5t−3 + t−4
Conway polynomial z8 + 3z6 + 2z4 + z2 + 1
2nd Alexander ideal (db, data sources) {1}
Determinant and Signature { 93, 0 }
Jones polynomial q6−3q5 + 5q4−9q3 + 13q2−14q + 15−13q−1 + 10q−2−6q−3 + 3q−4−q−5
HOMFLY-PT polynomial (db, data sources) z8−a2z6−2z6a−2 + 6z6−4a2z4−9z4a−2 + z4a−4 + 14z4−5a2z2−12z2a−2 + 3z2a−4 + 15z2−2a2−4a−2 + a−4 + 6
Kauffman polynomial (db, data sources) z10a−2 + z10 + 3az9 + 6z9a−1 + 3z9a−3 + 4a2z8 + 6z8a−2 + 4z8a−4 + 6z8 + 4a3z7−2az7−13z7a−1−4z7a−3 + 3z7a−5 + 3a4z6−4a2z6−27z6a−2−12z6a−4 + z6a−6−21z6 + a5z5−5a3z5 + 7z5a−1−9z5a−3−10z5a−5−6a4z4 + a2z4 + 39z4a−2 + 10z4a−4−3z4a−6 + 33z4−2a5z3−a3z3 + az3 + 9z3a−1 + 17z3a−3 + 8z3a−5 + 3a4z2−4a2z2−22z2a−2−5z2a−4 + z2a−6−23z2 + a5z + a3z−2az−6za−1−6za−3−2za−5 + 2a2 + 4a−2 + a−4 + 6
The A2 invariant −q14 + q12−2q10 + q8 + q6−q4 + 4q2−2 + 3q−2 + 2q−8−3q−10−q−14 + q−18
The G2 invariant q80−2q78 + 5q76−8q74 + 8q72−6q70−2q68 + 16q66−29q64 + 41q62−42q60 + 27q58−2q56−37q54 + 74q52−101q50 + 102q48−79q46 + 25q44 + 47q42−117q40 + 170q38−174q36 + 125q34−38q32−72q30 + 154q28−185q26 + 155q24−59q22−46q20 + 126q18−135q16 + 69q14 + 37q12−137q10 + 180q8−143q6 + 34q4 + 117q2−236 + 292q−2−242q−4 + 106q−6 + 61q−8−211q−10 + 291q−12−270q−14 + 173q−16−22q−18−114q−20 + 200q−22−191q−24 + 99q−26 + 17q−28−118q−30 + 151q−32−107q−34 + q−36 + 114q−38−186q−40 + 195q−42−128q−44−5q−46 + 121q−48−198q−50 + 207q−52−152q−54 + 59q−56 + 37q−58−105q−60 + 135q−62−120q−64 + 77q−66−23q−68−18q−70 + 42q−72−48q−74 + 41q−76−25q−78 + 11q−80 + 2q−82−8q−84 + 7q−86−7q−88 + 4q−90−2q−92 + q−94
Further Quantum Invariants
K11a106 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["K11a106"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= t4−5t3 + 12t2−18t + 21−18t−1 + 12t−2−5t−3 + t−4
In[5]:= Conway[K][z]
Out[5]= z8 + 3z6 + 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]= { 93, 0 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= q6−3q5 + 5q4−9q3 + 13q2−14q + 15−13q−1 + 10q−2−6q−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 + 6z6−4a2z4−9z4a−2 + z4a−4 + 14z4−5a2z2−12z2a−2 + 3z2a−4 + 15z2−2a2−4a−2 + a−4 + 6
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= z10a−2 + z10 + 3az9 + 6z9a−1 + 3z9a−3 + 4a2z8 + 6z8a−2 + 4z8a−4 + 6z8 + 4a3z7−2az7−13z7a−1−4z7a−3 + 3z7a−5 + 3a4z6−4a2z6−27z6a−2−12z6a−4 + z6a−6−21z6 + a5z5−5a3z5 + 7z5a−1−9z5a−3−10z5a−5−6a4z4 + a2z4 + 39z4a−2 + 10z4a−4−3z4a−6 + 33z4−2a5z3−a3z3 + az3 + 9z3a−1 + 17z3a−3 + 8z3a−5 + 3a4z2−4a2z2−22z2a−2−5z2a−4 + z2a−6−23z2 + a5z + a3z−2az−6za−1−6za−3−2za−5 + 2a2 + 4a−2 + a−4 + 6

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

Same Alexander/Conway Polynomial: {K11a194, K11a346,}

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["K11a106"];
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 + 12t2−18t + 21−18t−1 + 12t−2−5t−3 + t−4, q6−3q5 + 5q4−9q3 + 13q2−14q + 15−13q−1 + 10q−2−6q−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]= {K11a194, K11a346,}
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: (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 K11a106. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
 -5-4-3-2-10123456χ
13           11
11          2 -2
9         31 2
7        62  -4
5       73   4
3      76    -1
1     87     1
-1    68      2
-3   47       -3
-5  26        4
-7 14         -3
-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}^{4}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r = −2 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r = −1 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r = 0 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{8}
r = 1 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r = 2 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r = 3 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r = 4 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r = 5 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
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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K11a105

K11a107

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