Knot presentations
Planar diagram presentation

X_{1425} X_{11,14,12,15} X_{3,13,4,12} X_{13,3,14,2} X_{5,18,6,19} X_{7,20,8,1} X_{19,6,20,7} X_{9,16,10,17} X_{15,10,16,11} X_{17,8,18,9}

Gauss code

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

DowkerThistlethwaite code

4 12 18 20 16 14 2 10 8 6

Conway Notation

[352]

Minimum Braid Representative

A Morse Link Presentation

An Arc Presentation

Length is 12, width is 5,
Braid index is 5


[{12, 3}, {4, 2}, {3, 11}, {1, 4}, {10, 12}, {11, 9}, {8, 10}, {9, 5}, {2, 6}, {5, 7}, {6, 8}, {7, 1}]

[edit Notes on presentations of 10 20]
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["10 20"];


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

Out[4]=

X_{1425} X_{11,14,12,15} X_{3,13,4,12} X_{13,3,14,2} X_{5,18,6,19} X_{7,20,8,1} X_{19,6,20,7} X_{9,16,10,17} X_{15,10,16,11} X_{17,8,18,9}

Out[5]=

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

Out[6]=

4 12 18 20 16 14 2 10 8 6

(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[{12, 3}, {4, 2}, {3, 11}, {1, 4}, {10, 12}, {11, 9}, {8, 10}, {9, 5}, {2, 6}, {5, 7}, {6, 8}, {7, 1}]

Four dimensional invariants
Polynomial invariants
Further Quantum Invariants
Further quantum knot invariants for 10_20.
A1 Invariants.
Weight

Invariant

1


2


3


4


5


A2 Invariants.
Weight

Invariant

1,0


1,1


2,0


A3 Invariants.
Weight

Invariant

0,1,0


1,0,0


B2 Invariants.
Weight

Invariant

0,1


1,0


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["10 20"];


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:
{10_162, K11n117,}
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["10 20"];

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 2 is the signature of 10 20. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.



8  7  6  5  4  3  2  1  0  1  2  χ 
3            1  1 
1             0 
1          3  1   2 
3         2  1    1 
5        3  2     1 
7       3  2      1 
9      2  3       1 
11     2  3        1 
13    1  2         1 
15   1  2          1 
17   1           1 
19  1            1 

The Coloured Jones Polynomials