K11n128

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K11n127

K11n129

Contents

Image:K11n128.gif
(Knotscape image) 		See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11n128's page at Knotilus!

Visit K11n128's page at the original Knot Atlas!

[edit K11n128 Quick Notes]


[edit K11n128 Further Notes and Views]


[edit] Knot presentations

Planar diagram presentation X4251 X10,3,11,4 X5,19,6,18 X7,15,8,14 X9,16,10,17 X2,11,3,12 X13,7,14,6 X15,20,16,21 X17,1,18,22 X19,12,20,13 X21,9,22,8
Gauss code 1, -6, 2, -1, -3, 7, -4, 11, -5, -2, 6, 10, -7, 4, -8, 5, -9, 3, -10, 8, -11, 9
Dowker-Thistlethwaite code 4 10 -18 -14 -16 2 -6 -20 -22 -12 -8
A Braid Representative
Image:BraidPart3.gifImage:BraidPart0.gifImage:BraidPart3.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gif
Image:BraidPart4.gifImage:BraidPart1.gifImage:BraidPart4.gifImage:BraidPart3.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart3.gifImage:BraidPart0.gifImage:BraidPart3.gifImage:BraidPart0.gif
Image:BraidPart0.gifImage:BraidPart2.gifImage:BraidPart1.gifImage:BraidPart4.gifImage:BraidPart1.gifImage:BraidPart1.gifImage:BraidPart4.gifImage:BraidPart1.gifImage:BraidPart4.gifImage:BraidPart1.gif
Image:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart2.gifImage:BraidPart0.gifImage:BraidPart2.gifImage:BraidPart2.gifImage:BraidPart0.gifImage:BraidPart2.gifImage:BraidPart0.gifImage:BraidPart2.gif
A Morse Link Presentation Image:K11n128_ML.gif

[edit] Three dimensional invariants

Symmetry type Chiral
Unknotting number {1,2}
3-genus 3
Bridge index Missing
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11n128/ThurstonBennequinNumber
Hyperbolic Volume 12.2619
A-Polynomial See Data:K11n128/A-polynomial

[edit Notes for K11n128's three dimensional invariants]

[edit] Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus 3
Rasmussen s-Invariant -2

[edit Notes for K11n128's four dimensional invariants]

[edit] Polynomial invariants

Alexander polynomial t3−5t2 + 10t−11 + 10t−1−5t−2 + t−3
Conway polynomial z6 + z4−z2 + 1
2nd Alexander ideal (db, data sources) {1}
Determinant and Signature { 43, 2 }
Jones polynomial q5−3q4 + 5q3−6q2 + 7q−7 + 6q−1−4q−2 + 3q−3−q−4
HOMFLY-PT polynomial (db, data sources) z6−a2z4−2z4a−2 + 4z4−2a2z2−5z2a−2 + z2a−4 + 5z2−2a−2 + a−4 + 2
Kauffman polynomial (db, data sources) 2az9 + 2z9a−1 + 3a2z8 + 4z8a−2 + 7z8 + a3z7−6az7−4z7a−1 + 3z7a−3−14a2z6−15z6a−2 + z6a−4−30z6−4a3z5−az5−5z5a−1−8z5a−3 + 18a2z4 + 19z4a−2 + z4a−4 + 36z4 + 4a3z3 + 8az3 + 8z3a−1 + 7z3a−3 + 3z3a−5−6a2z2−12z2a−2−2z2a−4 + z2a−6−15z2−a3z−3az−3za−1−2za−3−za−5 + 2a−2 + a−4 + 2
The A2 invariant −q12 + q10 + q6 + 2q4−q2 + 1−q−2 + q−6−q−8 + q−10−q−12 + q−16
The G2 invariant Data:K11n128/QuantumInvariant/G2/1,0
Further Quantum Invariants
K11n128 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["K11n128"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= t3−5t2 + 10t−11 + 10t−1−5t−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]= { 43, 2 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= q5−3q4 + 5q3−6q2 + 7q−7 + 6q−1−4q−2 + 3q−3−q−4
In[9]:= HOMFLYPT[K][a, z]
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
Out[9]= z6−a2z4−2z4a−2 + 4z4−2a2z2−5z2a−2 + z2a−4 + 5z2−2a−2 + a−4 + 2
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= 2az9 + 2z9a−1 + 3a2z8 + 4z8a−2 + 7z8 + a3z7−6az7−4z7a−1 + 3z7a−3−14a2z6−15z6a−2 + z6a−4−30z6−4a3z5−az5−5z5a−1−8z5a−3 + 18a2z4 + 19z4a−2 + z4a−4 + 36z4 + 4a3z3 + 8az3 + 8z3a−1 + 7z3a−3 + 3z3a−5−6a2z2−12z2a−2−2z2a−4 + z2a−6−15z2−a3z−3az−3za−1−2za−3−za−5 + 2a−2 + a−4 + 2

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

Same Alexander/Conway Polynomial: {9_22,}

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["K11n128"];
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]= { t3−5t2 + 10t−11 + 10t−1−5t−2 + t−3, q5−3q4 + 5q3−6q2 + 7q−7 + 6q−1−4q−2 + 3q−3−q−4 }
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]= {9_22,}
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 = 2 is the signature of K11n128. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
 -5-4-3-2-101234χ
11         11
9        2 -2
7       31 2
5      32  -1
3     43   1
1    44    0
-1   23     -1
-3  24      2
-5 12       -1
-7 2        2
-91         -1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i = 1 i = 3
r = −5 {\mathbb Z}
r = −4 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r = −3 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r = −2 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r = −1 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r = 0 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{4}
r = 1 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r = 2 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r = 3 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r = 4 {\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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K11n127

K11n129

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