8_19 is the first nonhomologically thin knot in the Rolfsen table. (That is, it's the first knot whose Khovanov homology has 'offdiagonal' elements.)


Symmetrical form ; (3,4) torus knot


Truelover's knot with sticked free ends

Equal to the previous, from knotilus



Knot presentations
Planar diagram presentation

X_{4251} X_{8493} X_{9,15,10,14} X_{5,13,6,12} X_{13,7,14,6} X_{11,1,12,16} X_{15,11,16,10} X_{2837}

Gauss code

1, 8, 2, 1, 4, 5, 8, 2, 3, 7, 6, 4, 5, 3, 7, 6

DowkerThistlethwaite code

4 8 12 2 14 16 6 10

Conway Notation

[3,3,2]

Minimum Braid Representative

A Morse Link Presentation

An Arc Presentation

Length is 8, width is 3,
Braid index is 3


[{4, 10}, {3, 5}, {1, 4}, {6, 9}, {5, 8}, {2, 6}, {10, 3}, {9, 7}, {8, 2}, {7, 1}]

[edit Notes on presentations of 8 19]

A graph which shows knot 8_19.

A part of a knot and a part of a graph.

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`

In[3]:=

K = Knot["8 19"];


KnotTheory::loading: Loading precomputed data in PD4Knots`.

Out[4]=

X_{4251} X_{8493} X_{9,15,10,14} X_{5,13,6,12} X_{13,7,14,6} X_{11,1,12,16} X_{15,11,16,10} X_{2837}

Out[5]=

1, 8, 2, 1, 4, 5, 8, 2, 3, 7, 6, 4, 5, 3, 7, 6

Out[6]=

4 8 12 2 14 16 6 10

(The path below may be different on your system)
In[7]:=

AppendTo[$Path, "C:/bin/LinKnot/"];

In[8]:=

ConwayNotation[K]


KnotTheory::credits: The minimum braids representing the knots with up to 10 crossings were provided by Thomas Gittings. See arXiv:math.GT/0401051.

Out[9]=


In[10]:=

{First[br], Crossings[br], BraidIndex[K]}


KnotTheory::loading: Loading precomputed data in IndianaData`.

In[11]:=

Show[BraidPlot[br]]

In[12]:=

Show[DrawMorseLink[K]]


KnotTheory::credits: "MorseLink was added to KnotTheory` by Siddarth Sankaran at the University of Toronto in the summer of 2005."


KnotTheory::credits: "DrawMorseLink was written by Siddarth Sankaran at the University of Toronto in the summer of 2005."

In[13]:=

ap = ArcPresentation[K]

Out[13]=

ArcPresentation[{4, 10}, {3, 5}, {1, 4}, {6, 9}, {5, 8}, {2, 6}, {10, 3}, {9, 7}, {8, 2}, {7, 1}]

Four dimensional invariants
Polynomial invariants
Further Quantum Invariants
Further quantum knot invariants for 8_19.
A1 Invariants.
Weight

Invariant

1


2


3


4


5


6


A2 Invariants.
Weight

Invariant

1,0


1,1


2,0


A3 Invariants.
Weight

Invariant

0,1,0


1,0,0


1,0,1


A4 Invariants.
Weight

Invariant

0,1,0,0


1,0,0,0


B2 Invariants.
Weight

Invariant

0,1


1,0


D4 Invariants.
Weight

Invariant

1,0,0,0


G2 Invariants.
Weight

Invariant

1,0


.
</div></div>
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`

In[3]:=

K = Knot["8 19"];


KnotTheory::loading: Loading precomputed data in PD4Knots`.

Out[4]=


Out[5]=


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]=


In[7]:=

{KnotDet[K], KnotSignature[K]}


KnotTheory::loading: Loading precomputed data in Jones4Knots`.

Out[8]=


In[9]:=

HOMFLYPT[K][a, z]


KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.

Out[9]=


In[10]:=

Kauffman[K][a, z]


KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.

Out[10]=


"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]:=

AppendTo[$Path, "C:/drorbn/projects/KAtlas/"];
<< KnotTheory`

In[3]:=

K = Knot["8 19"];

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]=

{ , }

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.

In[6]:=

DeleteCases[
Select[
AllKnots[],
(J === Jones[#][q]  (J /. q > 1/q) === Jones[#][q]) &
],
K
]


KnotTheory::loading: Loading precomputed data in Jones4Knots11`.

V_{2,1} through V_{6,9}:

V_{2,1}

V_{3,1}

V_{4,1}

V_{4,2}

V_{4,3}

V_{5,1}

V_{5,2}

V_{5,3}

V_{5,4}

V_{6,1}

V_{6,2}

V_{6,3}

V_{6,4}

V_{6,5}

V_{6,6}

V_{6,7}

V_{6,8}

V_{6,9}




















V_{2,1} through V_{6,9} were provided by Petr DuninBarkowski <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 V_{2} and V_{3}.
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.



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 

The Coloured Jones Polynomials