Edit T(4,3) Further Notes and Views
Knot presentations
Planar diagram presentation

X_{5,11,6,10} X_{16,12,1,11} X_{1726} X_{12,8,13,7} X_{13,3,14,2} X_{8493} X_{9,15,10,14} X_{4,16,5,15}

Gauss code

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

DowkerThistlethwaite code

6 8 10 12 14 16 2 4

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["T(4,3)"];


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:
{8_19,}
Same Jones Polynomial (up to mirroring, ):
{8_19,}
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["T(4,3)"];

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}

Data:T(4,3)/V 2,1

Data:T(4,3)/V 3,1

Data:T(4,3)/V 4,1

Data:T(4,3)/V 4,2

Data:T(4,3)/V 4,3

Data:T(4,3)/V 5,1

Data:T(4,3)/V 5,2

Data:T(4,3)/V 5,3

Data:T(4,3)/V 5,4

Data:T(4,3)/V 6,1

Data:T(4,3)/V 6,2

Data:T(4,3)/V 6,3

Data:T(4,3)/V 6,4

Data:T(4,3)/V 6,5

Data:T(4,3)/V 6,6

Data:T(4,3)/V 6,7

Data:T(4,3)/V 6,8

Data:T(4,3)/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 T(4,3). 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 
