K11n116 is not [math]\displaystyle{ k }[/math]-colourable for any [math]\displaystyle{ k }[/math]. See The Determinant and the Signature.
Knot presentations
| Planar diagram presentation
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X4251 X10,3,11,4 X5,14,6,15 X7,19,8,18 X9,17,10,16 X2,11,3,12 X20,13,21,14 X15,22,16,1 X17,9,18,8 X12,19,13,20 X21,7,22,6
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| Gauss code
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1, -6, 2, -1, -3, 11, -4, 9, -5, -2, 6, -10, 7, 3, -8, 5, -9, 4, 10, -7, -11, 8
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| Dowker-Thistlethwaite code
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4 10 -14 -18 -16 2 20 -22 -8 12 -6
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Four dimensional invariants
Polynomial invariants
Further 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]:=
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AppendTo[$Path, "C:/drorbn/projects/KAtlas/"];
<< KnotTheory`
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In[3]:=
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K = Knot["K11n116"];
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KnotTheory::loading: Loading precomputed data in PD4Knots`.
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Out[4]=
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[math]\displaystyle{ -t^2+3- t^{-2} }[/math]
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Out[5]=
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[math]\displaystyle{ -z^4-4 z^2+1 }[/math]
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In[6]:=
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Alexander[K, 2][t]
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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.
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Out[6]=
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[math]\displaystyle{ \{1\} }[/math]
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In[7]:=
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{KnotDet[K], KnotSignature[K]}
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KnotTheory::loading: Loading precomputed data in Jones4Knots`.
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Out[8]=
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[math]\displaystyle{ q^3-q^2+q- q^{-3} + q^{-4} - q^{-5} + q^{-6} }[/math]
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In[9]:=
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HOMFLYPT[K][a, z]
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KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
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Out[9]=
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[math]\displaystyle{ a^6-z^2 a^4-a^4+a^2-z^4-4 z^2-2+z^2 a^{-2} +2 a^{-2} }[/math]
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In[10]:=
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Kauffman[K][a, z]
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KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
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Out[10]=
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[math]\displaystyle{ a^5 z^7+z^7 a^{-1} +a^6 z^6+a^4 z^6+z^6 a^{-2} +z^6-5 a^5 z^5-5 z^5 a^{-1} -5 a^6 z^4-5 a^4 z^4-a^2 z^4-5 z^4 a^{-2} -6 z^4+6 a^5 z^3-a^3 z^3-2 a z^3+5 z^3 a^{-1} +6 a^6 z^2+5 a^4 z^2+a^2 z^2+6 z^2 a^{-2} +8 z^2-2 a^5 z+2 a z-a^6-a^4-a^2-2 a^{-2} -2 }[/math]
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"Similar" Knots (within the Atlas)
Same Alexander/Conway Polynomial:
{K11n49,}
Same Jones Polynomial (up to mirroring, [math]\displaystyle{ q\leftrightarrow q^{-1} }[/math]):
{}
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`
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In[3]:=
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K = Knot["K11n116"];
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In[4]:=
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{A = Alexander[K][t], J = Jones[K][q]}
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KnotTheory::loading: Loading precomputed data in PD4Knots`.
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KnotTheory::loading: Loading precomputed data in Jones4Knots`.
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Out[4]=
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{ [math]\displaystyle{ -t^2+3- t^{-2} }[/math], [math]\displaystyle{ q^3-q^2+q- q^{-3} + q^{-4} - q^{-5} + q^{-6} }[/math] }
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In[5]:=
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DeleteCases[Select[AllKnots[], (A === Alexander[#][t]) &], K]
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KnotTheory::loading: Loading precomputed data in DTCode4KnotsTo11`.
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KnotTheory::credits: The GaussCode to PD conversion was written by Siddarth Sankaran at the University of Toronto in the summer of 2005.
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In[6]:=
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DeleteCases[
Select[
AllKnots[],
(J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) &
],
K
]
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KnotTheory::loading: Loading precomputed data in Jones4Knots11`.
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| V2,1 through V6,9:
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| V2,1
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V3,1
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V4,1
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V4,2
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V4,3
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V5,1
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V5,2
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V5,3
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V5,4
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V6,1
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V6,2
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V6,3
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V6,4
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V6,5
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V6,6
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V6,7
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V6,8
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V6,9
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| [math]\displaystyle{ -16 }[/math]
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[math]\displaystyle{ 24 }[/math]
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[math]\displaystyle{ 128 }[/math]
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[math]\displaystyle{ \frac{472}{3} }[/math]
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[math]\displaystyle{ \frac{224}{3} }[/math]
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[math]\displaystyle{ -384 }[/math]
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[math]\displaystyle{ -592 }[/math]
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[math]\displaystyle{ -128 }[/math]
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[math]\displaystyle{ -136 }[/math]
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[math]\displaystyle{ -\frac{2048}{3} }[/math]
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[math]\displaystyle{ 288 }[/math]
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[math]\displaystyle{ -\frac{7552}{3} }[/math]
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[math]\displaystyle{ -\frac{3584}{3} }[/math]
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[math]\displaystyle{ -\frac{22022}{15} }[/math]
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[math]\displaystyle{ \frac{8128}{15} }[/math]
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[math]\displaystyle{ -\frac{87728}{45} }[/math]
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[math]\displaystyle{ \frac{3158}{9} }[/math]
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[math]\displaystyle{ -\frac{7622}{15} }[/math]
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V2,1 through V6,9 were provided by Petr Dunin-Barkowski <barkovs@itep.ru>, Andrey Smirnov <asmirnov@itep.ru>, and Alexei Sleptsov <sleptsov@itep.ru> and uploaded on October 2010 by User:Drorbn. Note that they are normalized differently than V2 and V3.
| The coefficients of the monomials [math]\displaystyle{ t^rq^j }[/math] are shown, along with their alternating sums [math]\displaystyle{ \chi }[/math] (fixed [math]\displaystyle{ j }[/math], alternation over [math]\displaystyle{ r }[/math]). The squares with yellow highlighting are those on the "critical diagonals", where [math]\displaystyle{ j-2r=s+1 }[/math] or [math]\displaystyle{ j-2r=s-1 }[/math], where [math]\displaystyle{ s= }[/math]0 is the signature of K11n116. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.
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-6 | -5 | -4 | -3 | -2 | -1 | 0 | 1 | 2 | 3 | 4 | χ |
| 7 | | | | | | | | | | | 1 | 1 |
| 5 | | | | | | | | | | | | 0 |
| 3 | | | | | | | | | 1 | 1 | | 0 |
| 1 | | | | | | | 2 | 1 | | | | 1 |
| -1 | | | | | | | 1 | 1 | | | | 0 |
| -3 | | | | | 1 | 2 | 1 | | | | | 0 |
| -5 | | | | 1 | | | | | | | | -1 |
| -7 | | | | 1 | 1 | | | | | | | 0 |
| -9 | | 1 | 1 | | | | | | | | | 0 |
| -11 | | | | | | | | | | | | 0 |
| -13 | 1 | | | | | | | | | | | 1 |
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| Integral Khovanov Homology
(db, data source)
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| [math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math]
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[math]\displaystyle{ i=-3 }[/math]
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[math]\displaystyle{ i=-1 }[/math]
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[math]\displaystyle{ i=1 }[/math]
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| [math]\displaystyle{ r=-6 }[/math]
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[math]\displaystyle{ {\mathbb Z} }[/math]
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| [math]\displaystyle{ r=-5 }[/math]
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[math]\displaystyle{ {\mathbb Z}_2 }[/math]
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[math]\displaystyle{ {\mathbb Z} }[/math]
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| [math]\displaystyle{ r=-4 }[/math]
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[math]\displaystyle{ {\mathbb Z} }[/math]
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| [math]\displaystyle{ r=-3 }[/math]
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[math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math]
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[math]\displaystyle{ {\mathbb Z} }[/math]
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| [math]\displaystyle{ r=-2 }[/math]
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[math]\displaystyle{ {\mathbb Z} }[/math]
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[math]\displaystyle{ {\mathbb Z}_2 }[/math]
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[math]\displaystyle{ {\mathbb Z} }[/math]
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| [math]\displaystyle{ r=-1 }[/math]
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[math]\displaystyle{ {\mathbb Z}_2 }[/math]
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[math]\displaystyle{ {\mathbb Z}^{2} }[/math]
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| [math]\displaystyle{ r=0 }[/math]
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[math]\displaystyle{ {\mathbb Z} }[/math]
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[math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math]
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[math]\displaystyle{ {\mathbb Z}^{2} }[/math]
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| [math]\displaystyle{ r=1 }[/math]
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[math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math]
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[math]\displaystyle{ {\mathbb Z} }[/math]
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| [math]\displaystyle{ r=2 }[/math]
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[math]\displaystyle{ {\mathbb Z}_2 }[/math]
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[math]\displaystyle{ {\mathbb Z} }[/math]
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| [math]\displaystyle{ r=3 }[/math]
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[math]\displaystyle{ {\mathbb Z} }[/math]
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| [math]\displaystyle{ r=4 }[/math]
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[math]\displaystyle{ {\mathbb Z}_2 }[/math]
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[math]\displaystyle{ {\mathbb Z} }[/math]
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