K11n156

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K11n155

K11n157

Contents

Image:K11n156.gif
(Knotscape image) 		See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11n156's page at Knotilus!

Visit K11n156's page at the original Knot Atlas!

[edit K11n156 Quick Notes]


[edit K11n156 Further Notes and Views]


[edit] Knot presentations

Planar diagram presentation X6271 X3,11,4,10 X12,6,13,5 X14,7,15,8 X16,10,17,9 X11,19,12,18 X22,13,1,14 X20,16,21,15 X17,4,18,5 X2,19,3,20 X8,21,9,22
Gauss code 1, -10, -2, 9, 3, -1, 4, -11, 5, 2, -6, -3, 7, -4, 8, -5, -9, 6, 10, -8, 11, -7
Dowker-Thistlethwaite code 6 -10 12 14 16 -18 22 20 -4 2 8
A Braid Representative
Image:BraidPart1.gifImage:BraidPart0.gifImage:BraidPart1.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart3.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gif
Image:BraidPart2.gifImage:BraidPart3.gifImage:BraidPart2.gifImage:BraidPart1.gifImage:BraidPart0.gifImage:BraidPart1.gifImage:BraidPart4.gifImage:BraidPart1.gifImage:BraidPart0.gifImage:BraidPart1.gif
Image:BraidPart0.gifImage:BraidPart4.gifImage:BraidPart0.gifImage:BraidPart2.gifImage:BraidPart3.gifImage:BraidPart2.gifImage:BraidPart3.gifImage:BraidPart2.gifImage:BraidPart3.gifImage:BraidPart2.gif
Image:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart4.gifImage:BraidPart0.gifImage:BraidPart4.gifImage:BraidPart0.gifImage:BraidPart4.gifImage:BraidPart0.gif
A Morse Link Presentation Image:K11n156_ML.gif

[edit] Three dimensional invariants

Symmetry type Reversible
Unknotting number 1
3-genus 3
Bridge index Missing
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11n156/ThurstonBennequinNumber
Hyperbolic Volume 15.3972
A-Polynomial See Data:K11n156/A-polynomial

[edit Notes for K11n156's three dimensional invariants]

[edit] Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus 3
Rasmussen s-Invariant 0

[edit Notes for K11n156's four dimensional invariants]

[edit] Polynomial invariants

Alexander polynomial −t3 + 7t2−18t + 25−18t−1 + 7t−2−t−3
Conway polynomial −z6 + z4 + z2 + 1
2nd Alexander ideal (db, data sources) {1}
Determinant and Signature { 77, 0 }
Jones polynomial q6−4q5 + 7q4−10q3 + 13q2−13q + 12−9q−1 + 6q−2−2q−3
HOMFLY-PT polynomial (db, data sources) −z6a−2−3z4a−2 + z4a−4 + 3z4−2a2z2−4z2a−2 + z2a−4 + 6z2−a2−a−2 + 3
Kauffman polynomial (db, data sources) 3z9a−1 + 3z9a−3 + 12z8a−2 + 6z8a−4 + 6z8 + 4az7−z7a−1−z7a−3 + 4z7a−5 + a2z6−35z6a−2−18z6a−4 + z6a−6−15z6−3az5−6z5a−1−14z5a−3−11z5a−5 + 6a2z4 + 33z4a−2 + 14z4a−4−2z4a−6 + 23z4 + 3a3z3 + 3az3 + 5z3a−1 + 11z3a−3 + 6z3a−5−6a2z2−12z2a−2−3z2a−4−15z2−a3z−2az−2za−1−za−3 + a2 + a−2 + 3
The A2 invariant −2q10 + q8 + 2q6−2q4 + 3q2−1 + q−2 + 2q−4−q−6 + 3q−8−3q−10 + q−14−2q−16 + q−18
The G2 invariant Data:K11n156/QuantumInvariant/G2/1,0
Further Quantum Invariants
K11n156 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["K11n156"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= −t3 + 7t2−18t + 25−18t−1 + 7t−2−t−3
In[5]:= Conway[K][z]
Out[5]= −z6 + 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]= { 77, 0 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= q6−4q5 + 7q4−10q3 + 13q2−13q + 12−9q−1 + 6q−2−2q−3
In[9]:= HOMFLYPT[K][a, z]
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
Out[9]= −z6a−2−3z4a−2 + z4a−4 + 3z4−2a2z2−4z2a−2 + z2a−4 + 6z2−a2−a−2 + 3
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= 3z9a−1 + 3z9a−3 + 12z8a−2 + 6z8a−4 + 6z8 + 4az7−z7a−1−z7a−3 + 4z7a−5 + a2z6−35z6a−2−18z6a−4 + z6a−6−15z6−3az5−6z5a−1−14z5a−3−11z5a−5 + 6a2z4 + 33z4a−2 + 14z4a−4−2z4a−6 + 23z4 + 3a3z3 + 3az3 + 5z3a−1 + 11z3a−3 + 6z3a−5−6a2z2−12z2a−2−3z2a−4−15z2−a3z−2az−2za−1−za−3 + a2 + a−2 + 3

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

Same Alexander/Conway Polynomial: {10_71, K11n179,}

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["K11n156"];
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]= { −t3 + 7t2−18t + 25−18t−1 + 7t−2−t−3, q6−4q5 + 7q4−10q3 + 13q2−13q + 12−9q−1 + 6q−2−2q−3 }
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]= {10_71, K11n179,}
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, 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 K11n156. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
 -3-2-10123456χ
13         11
11        3 -3
9       41 3
7      63  -3
5     74   3
3    66    0
1   67     -1
-1  47      3
-3 25       -3
-5 4        4
-72         -2
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i = −1 i = 1
r = −3 {\mathbb Z}^{2}
r = −2 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r = −1 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r = 0 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{6}
r = 1 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r = 2 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r = 3 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r = 4 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
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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K11n155

K11n157

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