K11n157

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K11n156

K11n158

Contents

Image:K11n157.gif
(Knotscape image) 		See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11n157's page at Knotilus!

Visit K11n157's page at the original Knot Atlas!

[edit K11n157 Quick Notes]


[edit K11n157 Further Notes and Views]


[edit] Knot presentations

Planar diagram presentation X6271 X3,11,4,10 X5,12,6,13 X14,7,15,8 X9,16,10,17 X11,19,12,18 X22,13,1,14 X20,16,21,15 X17,4,18,5 X19,3,20,2 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:BraidPart1.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gif
Image:BraidPart2.gifImage:BraidPart3.gifImage:BraidPart2.gifImage:BraidPart3.gifImage:BraidPart0.gifImage:BraidPart1.gifImage:BraidPart2.gifImage:BraidPart1.gifImage:BraidPart0.gifImage:BraidPart3.gif
Image:BraidPart0.gifImage:BraidPart4.gifImage:BraidPart0.gifImage:BraidPart4.gifImage:BraidPart3.gifImage:BraidPart2.gifImage:BraidPart3.gifImage:BraidPart2.gifImage:BraidPart3.gifImage:BraidPart4.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:K11n157_ML.gif

[edit] Three dimensional invariants

Symmetry type Reversible
Unknotting number {1,2}
3-genus 3
Bridge index Missing
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11n157/ThurstonBennequinNumber
Hyperbolic Volume 14.7471
A-Polynomial See Data:K11n157/A-polynomial

[edit Notes for K11n157'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 K11n157's four dimensional invariants]

[edit] Polynomial invariants

Alexander polynomial −t3 + 6t2−15t + 21−15t−1 + 6t−2−t−3
Conway polynomial 1−z6
2nd Alexander ideal (db, data sources) \left\{3,t^2+1\right\}
Determinant and Signature { 65, 0 }
Jones polynomial −q3 + 4q2−7q + 10−11q−1 + 11q−2−9q−3 + 7q−4−4q−5 + q−6
HOMFLY-PT polynomial (db, data sources) −a2z6 + a4z4−3a2z4 + 2z4 + a4z2−3a2z2−z2a−2 + 3z2 + 1
Kauffman polynomial (db, data sources) 2a3z9 + 2az9 + 5a4z8 + 8a2z8 + 3z8 + 4a5z7 + 2a3z7−az7 + z7a−1 + a6z6−13a4z6−20a2z6−6z6−11a5z5−16a3z5−2az5 + 3z5a−1−2a6z4 + 5a4z4 + 13a2z4 + 4z4a−2 + 10z4 + 6a5z3 + 9a3z3 + az3−z3a−1 + z3a−3 + a6z2 + a4z2−3a2z2−2z2a−2−5z2 + 1
The A2 invariant q18−2q16 + q14−2q10 + 3q8−q6 + 2q4−1 + 2q−2−2q−4 + 2q−6 + q−8−q−10
The G2 invariant Data:K11n157/QuantumInvariant/G2/1,0
Further Quantum Invariants
K11n157 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["K11n157"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= −t3 + 6t2−15t + 21−15t−1 + 6t−2−t−3
In[5]:= Conway[K][z]
Out[5]= 1−z6
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]= \left\{3,t^2+1\right\}
In[7]:= {KnotDet[K], KnotSignature[K]}
Out[7]= { 65, 0 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= −q3 + 4q2−7q + 10−11q−1 + 11q−2−9q−3 + 7q−4−4q−5 + q−6
In[9]:= HOMFLYPT[K][a, z]
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
Out[9]= −a2z6 + a4z4−3a2z4 + 2z4 + a4z2−3a2z2−z2a−2 + 3z2 + 1
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= 2a3z9 + 2az9 + 5a4z8 + 8a2z8 + 3z8 + 4a5z7 + 2a3z7−az7 + z7a−1 + a6z6−13a4z6−20a2z6−6z6−11a5z5−16a3z5−2az5 + 3z5a−1−2a6z4 + 5a4z4 + 13a2z4 + 4z4a−2 + 10z4 + 6a5z3 + 9a3z3 + az3−z3a−1 + z3a−3 + a6z2 + a4z2−3a2z2−2z2a−2−5z2 + 1

[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["K11n157"];
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 + 6t2−15t + 21−15t−1 + 6t−2−t−3, −q3 + 4q2−7q + 10−11q−1 + 11q−2−9q−3 + 7q−4−4q−5 + q−6 }
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: (0, 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 K11n157. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
 -6-5-4-3-2-10123χ
7         1-1
5        3 3
3       41 -3
1      63  3
-1     65   -1
-3    55    0
-5   46     2
-7  35      -2
-9 14       3
-11 3        -3
-131         1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i = −1 i = 1
r = −6 {\mathbb Z}
r = −5 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r = −4 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r = −3 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r = −2 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r = −1 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r = 0 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{6}
r = 1 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r = 2 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r = 3 {\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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K11n156

K11n158

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