8_19 is the first non-homologically thin knot in the Rolfsen table. (That is, it's the first knot whose Khovanov homology has 'off-diagonal' elements.)
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Symmetrical form ; (3,4) torus knot
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True-lover's knot with sticked free ends
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Equal to the previous, from knotilus
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Knot presentations
Planar diagram presentation
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X4251 X8493 X9,15,10,14 X5,13,6,12 X13,7,14,6 X11,1,12,16 X15,11,16,10 X2837
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Gauss code
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1, -8, 2, -1, -4, 5, 8, -2, -3, 7, -6, 4, -5, 3, -7, 6
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Dowker-Thistlethwaite code
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4 8 -12 2 -14 -16 -6 -10
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Conway Notation
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[3,3,2-]
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Minimum Braid Representative
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A Morse Link Presentation
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An Arc Presentation
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Length is 8, width is 3,
Braid index is 3
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![8 19 AP.gif](/images/c/cf/8_19_AP.gif) [{4, 10}, {3, 5}, {1, 4}, {6, 9}, {5, 8}, {2, 6}, {10, 3}, {9, 7}, {8, 2}, {7, 1}]
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[edit Notes on presentations of 8 19]
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A graph which shows knot 8_19.
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A part of a knot and a part of a graph.
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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]:=
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AppendTo[$Path, "C:/drorbn/projects/KAtlas/"];
<< KnotTheory`
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In[3]:=
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K = Knot["8 19"];
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KnotTheory::loading: Loading precomputed data in PD4Knots`.
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Out[4]=
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X4251 X8493 X9,15,10,14 X5,13,6,12 X13,7,14,6 X11,1,12,16 X15,11,16,10 X2837
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Out[5]=
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1, -8, 2, -1, -4, 5, 8, -2, -3, 7, -6, 4, -5, 3, -7, 6
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Out[6]=
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4 8 -12 2 -14 -16 -6 -10
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(The path below may be different on your system)
In[7]:=
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AppendTo[$Path, "C:/bin/LinKnot/"];
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In[8]:=
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ConwayNotation[K]
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KnotTheory::credits: The minimum braids representing the knots with up to 10 crossings were provided by Thomas Gittings. See arXiv:math.GT/0401051.
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Out[9]=
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In[10]:=
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{First[br], Crossings[br], BraidIndex[K]}
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KnotTheory::loading: Loading precomputed data in IndianaData`.
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In[11]:=
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Show[BraidPlot[br]]
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In[12]:=
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Show[DrawMorseLink[K]]
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KnotTheory::credits: "MorseLink was added to KnotTheory` by Siddarth Sankaran at the University of Toronto in the summer of 2005."
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KnotTheory::credits: "DrawMorseLink was written by Siddarth Sankaran at the University of Toronto in the summer of 2005."
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In[13]:=
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ap = ArcPresentation[K]
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Out[13]=
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ArcPresentation[{4, 10}, {3, 5}, {1, 4}, {6, 9}, {5, 8}, {2, 6}, {10, 3}, {9, 7}, {8, 2}, {7, 1}]
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Four dimensional invariants
Polynomial invariants
Alexander polynomial |
![{\displaystyle t^{3}-t^{2}+1-t^{-2}+t^{-3}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6a179fb23cc53f9856f0c725c8524792084c3f89) |
Conway polynomial |
![{\displaystyle z^{6}+5z^{4}+5z^{2}+1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5219e7066c0aac767912f39f21186b66cc4bd178) |
2nd Alexander ideal (db, data sources) |
![{\displaystyle \{1\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5acdcac635f883f8b4f0a01aa03b16b22f23b124) |
Determinant and Signature |
{ 3, 6 } |
Jones polynomial |
![{\displaystyle -q^{8}+q^{5}+q^{3}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/63044f48892740521f0d983dcb5a5921587949ca) |
HOMFLY-PT polynomial (db, data sources) |
![{\displaystyle z^{6}a^{-6}+6z^{4}a^{-6}-z^{4}a^{-8}+10z^{2}a^{-6}-5z^{2}a^{-8}+5a^{-6}-5a^{-8}+a^{-10}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/298fab1c1417c350de2834136000f08974796466) |
Kauffman polynomial (db, data sources) |
![{\displaystyle z^{6}a^{-6}+z^{6}a^{-8}+z^{5}a^{-7}+z^{5}a^{-9}-6z^{4}a^{-6}-6z^{4}a^{-8}-5z^{3}a^{-7}-5z^{3}a^{-9}+10z^{2}a^{-6}+10z^{2}a^{-8}+5za^{-7}+5za^{-9}-5a^{-6}-5a^{-8}-a^{-10}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7194f0c54b504dff49221783470afd7ef722cbda) |
The A2 invariant |
![{\displaystyle q^{-10}+q^{-12}+2q^{-14}+2q^{-16}+2q^{-18}-q^{-22}-2q^{-24}-2q^{-26}-q^{-28}+q^{-32}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/afea2538f865016f6e6bc326d931fc3e40ea9b4d) |
The G2 invariant |
![{\displaystyle q^{-50}+q^{-52}+q^{-54}+q^{-56}+q^{-58}+q^{-60}+2q^{-62}+2q^{-64}+q^{-66}+q^{-68}+2q^{-70}+2q^{-72}+2q^{-74}+q^{-76}+q^{-80}+2q^{-82}-q^{-94}-2q^{-96}-q^{-98}-q^{-100}-2q^{-102}-2q^{-104}-2q^{-106}-q^{-108}-q^{-110}-2q^{-112}-2q^{-114}-q^{-116}-q^{-122}-q^{-124}+q^{-126}+q^{-128}+q^{-136}+q^{-138}+q^{-144}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fe00cc72ab7e2d5b4cb6e193abe399eb10ec0683) |
Further Quantum Invariants
Further quantum knot invariants for 8_19.
A1 Invariants.
Weight
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Invariant
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1
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2
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3
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4
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5
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6
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A2 Invariants.
Weight
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Invariant
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1,0
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1,1
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2,0
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A3 Invariants.
Weight
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Invariant
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0,1,0
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1,0,0
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1,0,1
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A4 Invariants.
Weight
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Invariant
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0,1,0,0
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1,0,0,0
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B2 Invariants.
Weight
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Invariant
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0,1
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1,0
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D4 Invariants.
Weight
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Invariant
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1,0,0,0
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G2 Invariants.
Weight
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Invariant
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1,0
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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["8 19"];
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KnotTheory::loading: Loading precomputed data in PD4Knots`.
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Out[4]=
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Out[5]=
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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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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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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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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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"Similar" Knots (within the Atlas)
Same Alexander/Conway Polynomial:
{}
Same Jones Polynomial (up to mirroring,
):
{}
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]:=
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AppendTo[$Path, "C:/drorbn/projects/KAtlas/"];
<< KnotTheory`
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In[3]:=
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K = Knot["8 19"];
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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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{ , }
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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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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 are shown, along with their alternating sums (fixed , alternation over ). The squares with yellow highlighting are those on the "critical diagonals", where or , where 6 is the signature of 8 19. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.
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0 | 1 | 2 | 3 | 4 | 5 | χ |
17 | | | | | | 1 | -1 |
15 | | | | | | 1 | -1 |
13 | | | | 1 | 1 | | 0 |
11 | | | | | 1 | | 1 |
9 | | | 1 | | | | 1 |
7 | 1 | | | | | | 1 |
5 | 1 | | | | | | 1 |
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The Coloured Jones Polynomials