K11n148

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K11n147

K11n149

Contents

Image:K11n148.gif
(Knotscape image) 		See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11n148's page at Knotilus!

Visit K11n148's page at the original Knot Atlas!

[edit K11n148 Quick Notes]


[edit K11n148 Further Notes and Views]


[edit] Knot presentations

Planar diagram presentation X4251 X14,4,15,3 X5,19,6,18 X7,20,8,21 X9,1,10,22 X11,6,12,7 X2,14,3,13 X15,9,16,8 X17,10,18,11 X19,13,20,12 X21,16,22,17
Gauss code 1, -7, 2, -1, -3, 6, -4, 8, -5, 9, -6, 10, 7, -2, -8, 11, -9, 3, -10, 4, -11, 5
Dowker-Thistlethwaite code 4 14 -18 -20 -22 -6 2 -8 -10 -12 -16
A Braid Representative
Image:BraidPart1.gifImage:BraidPart1.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart1.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gif
Image:BraidPart2.gifImage:BraidPart2.gifImage:BraidPart1.gifImage:BraidPart0.gifImage:BraidPart1.gifImage:BraidPart2.gifImage:BraidPart1.gifImage:BraidPart0.gifImage:BraidPart1.gifImage:BraidPart0.gif
Image:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart2.gifImage:BraidPart3.gifImage:BraidPart2.gifImage:BraidPart3.gifImage:BraidPart2.gifImage:BraidPart3.gifImage:BraidPart2.gifImage:BraidPart3.gif
Image:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart4.gifImage:BraidPart0.gifImage:BraidPart4.gifImage:BraidPart0.gifImage:BraidPart4.gifImage:BraidPart0.gifImage:BraidPart4.gif
A Morse Link Presentation Image:K11n148_ML.gif

[edit] Three dimensional invariants

Symmetry type Reversible
Unknotting number {1,2,3}
3-genus 4
Bridge index Missing
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11n148/ThurstonBennequinNumber
Hyperbolic Volume 15.4617
A-Polynomial See Data:K11n148/A-polynomial

[edit Notes for K11n148's three dimensional invariants]

[edit] Four dimensional invariants

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

[edit Notes for K11n148's four dimensional invariants]

[edit] Polynomial invariants

Alexander polynomial −t4 + 5t3−10t2 + 14t−15 + 14t−1−10t−2 + 5t−3−t−4
Conway polynomial −z8−3z6 + 3z2 + 1
2nd Alexander ideal (db, data sources) \left\{5,t^2+2 t+1\right\}
Determinant and Signature { 75, 2 }
Jones polynomial −2q6 + 5q5−9q4 + 12q3−12q2 + 13q−10 + 7q−1−4q−2 + q−3
HOMFLY-PT polynomial (db, data sources) −z8a−2−5z6a−2 + z6a−4 + z6−7z4a−2 + 4z4a−4 + 3z4−z2a−2 + 4z2a−4−z2a−6 + z2 + 3a−2−a−6−1
Kauffman polynomial (db, data sources) 3z9a−1 + 3z9a−3 + 12z8a−2 + 6z8a−4 + 6z8 + 4az7−z7a−1−z7a−3 + 4z7a−5 + a2z6−36z6a−2−16z6a−4 + z6a−6−18z6−11az5−15z5a−1−10z5a−3−6z5a−5−2a2z4 + 33z4a−2 + 21z4a−4 + 4z4a−6 + 14z4 + 6az3 + 12z3a−1 + 12z3a−3 + 9z3a−5 + 3z3a−7−8z2a−2−8z2a−4−4z2a−6−4z2−za−1−2za−3−4za−5−3za−7−3a−2 + a−6−1
The A2 invariant q8−2q6 + q4−2q2 + 2q−2−q−4 + 6q−6−q−8 + 3q−10−q−12−2q−14 + q−16−2q−18 + q−20−q−22
The G2 invariant Data:K11n148/QuantumInvariant/G2/1,0
Further Quantum Invariants
K11n148 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["K11n148"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= −t4 + 5t3−10t2 + 14t−15 + 14t−1−10t−2 + 5t−3−t−4
In[5]:= Conway[K][z]
Out[5]= −z8−3z6 + 3z2 + 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]= \left\{5,t^2+2 t+1\right\}
In[7]:= {KnotDet[K], KnotSignature[K]}
Out[7]= { 75, 2 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= −2q6 + 5q5−9q4 + 12q3−12q2 + 13q−10 + 7q−1−4q−2 + q−3
In[9]:= HOMFLYPT[K][a, z]
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
Out[9]= −z8a−2−5z6a−2 + z6a−4 + z6−7z4a−2 + 4z4a−4 + 3z4−z2a−2 + 4z2a−4−z2a−6 + z2 + 3a−2−a−6−1
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 + a2z6−36z6a−2−16z6a−4 + z6a−6−18z6−11az5−15z5a−1−10z5a−3−6z5a−5−2a2z4 + 33z4a−2 + 21z4a−4 + 4z4a−6 + 14z4 + 6az3 + 12z3a−1 + 12z3a−3 + 9z3a−5 + 3z3a−7−8z2a−2−8z2a−4−4z2a−6−4z2−za−1−2za−3−4za−5−3za−7−3a−2 + a−6−1

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

Same Alexander/Conway Polynomial: {K11a223,}

Same Jones Polynomial (up to mirroring, q\leftrightarrow q^{-1}): {K11n168,}

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["K11n148"];
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 + 5t3−10t2 + 14t−15 + 14t−1−10t−2 + 5t−3−t−4, −2q6 + 5q5−9q4 + 12q3−12q2 + 13q−10 + 7q−1−4q−2 + q−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]= {K11a223,}
In[6]:= DeleteCases[ Select[ AllKnots[], (J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) & ], K ]
KnotTheory::loading: Loading precomputed data in Jones4Knots11`.
Out[6]= {K11n168,}

[edit] Vassiliev invariants

V2 and V3: (3, 4)

[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 K11n148. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
 -4-3-2-1012345χ
13         2-2
11        3 3
9       62 -4
7      63  3
5     66   0
3    76    1
1   47     3
-1  36      -3
-3 14       3
-5 3        -3
-71         1
Integral Khovanov Homology

(db, data source)

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

[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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K11n147

K11n149

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