K11n34 is not colourable for any . See The Determinant and the Signature.


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

Knot presentations
Planar diagram presentation

X_{4251} X_{8493} X_{12,5,13,6} X_{2837} X_{9,17,10,16} X_{11,18,12,19} X_{6,13,7,14} X_{15,20,16,21} X_{17,1,18,22} X_{19,14,20,15} X_{21,10,22,11}

Gauss code

1, 4, 2, 1, 3, 7, 4, 2, 5, 11, 6, 3, 7, 10, 8, 5, 9, 6, 10, 8, 11, 9

DowkerThistlethwaite code

4 8 12 2 16 18 6 20 22 14 10

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]:=

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

In[3]:=

K = Knot["K11n34"];


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

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:
{0_1, K11n42,}
Same Jones Polynomial (up to mirroring, ):
{K11n42,}
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["K11n34"];

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]=

{ 1, }

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 0 is the signature of K11n34. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.



6  5  4  3  2  1  0  1  2  3  4  5  χ 
9             1  1 
7            1   1 
5           1  1   0 
3         1  2  1    0 
1        2  1  1     2 
1       1  3  2      0 
3      2  2  1       1 
5     1  1  1        1 
7    1  2  1         0 
9   1  1           0 
11   1            1 
13  1             1 
