Hi Udo,
thanks for your answer. Your suggestion only works for knots and
links for which there are already tables implemented, right? What, if
I want to have a link with more crossings (about 22 or even more).
There are no tables for them, right? Since this is part of a very long
going project (years) I don't mind long calculation time.
Best wishes
Stephan
Quoting Udo und Susanne Krause <su.krause@XXXXXXX.ch>:
Hello Stephan,
you shouldn't, period. The manual help once again
http://katlas.org/wiki/DT_(Dowker-Thistlethwaite)_Codes
says
DTCode also acts as a "type caster", so for example, DTCode[K]
where K is is a named knot or link returns the DT code of K.
So one looks to the link table in the case of links
(http://katlas.org/wiki/The_Thistlethwaite_Link_Table), all links
KnotTheory knows about are listed. No need for iteration on integer
lists (that might take a very very very long time if the list is an
n-tupel with n > 1).
First get the DTCodes[] out of the list
In[13]:= DTCode[Link[8, Alternating, 1]]
During evaluation of In[13]:= KnotTheory::loading: Loading
precomputed data in PD4Links`.
Out[13]= DTCode[{6, 8}, {10, 14, 12, 16, 2, 4}]
In[14]:= DTCode[Link[8, NonAlternating, 1]]
Out[14]= DTCode[{6, -8}, {-10, 14, -12, -16, -2, 4}]
if for example a link with 8 crossings
(http://katlas.org/wiki/The_Thistlethwaite_Link_Table_L8n1-L8n8) has
the name 'L8n1' than this is
Link[8, NonAlternating, 1], and if it has the name 'L8a19' in the
table, then it is Link[8,Alternating, 19]. This can be plotted
without actually realizing the DTCode[]:
In[17]:= DrawPD[DTCode[Link[8, Alternating, 1]], {Gap -> 0.025}]
In[18]:= DrawPD[DTCode[Link[8, NonAlternating, 1]], {Gap -> 0.025}]
In[19]:= DrawPD[DTCode[Link[8, Alternating, 19]], {Gap -> 0.025}]
if you really want to iterate, here is a linear opportunity (may
be you meant that):
In[53]:= Clear[howManyLinks]
howManyLinks[k_Integer, a_Integer] := Block[{o = 1},
If[a == 1,
While[Depth[DTCode[Link[k, Alternating, o]]] == 3, o += 1];
Print["There are ", o - 1, " alternating links with ", k, "
crossings in KnotTheory."], (* else *)
While[Depth[DTCode[Link[k, NonAlternating, o]]] == 3, o += 1];
Print["There are ", o - 1, " non-alternating links with ", k, "
crossings in KnotTheory."]
]
] /; k > 1 && k != 3 && k < 12 && (a == 0 || a == 1)
In[56}:= howManyLinks[11, 0]
There are 459 non-alternating links with 11 crossings in KnotTheory.
In[58]:= howManyLinks[11, 1]
There are 548 alternating links with 11 crossings in KnotTheory.
Mit den besten grüssen
Udo.
On Mon, 06 Aug 2018 18:23:13 +0200, Stephan Rosebrock
<rosebrock@XXXXXXX.de> wrote:
Hi Udo and Peter, it is possible, to draw a knot or a link by
giving its Dowker notation: Show[DrawPD[DTCode[{6, -8}, {-10, 12,
-14, 2, -4}], {Gap -> 0.025}]] Since I am interested to draw knots
and links with many crossings, this is interesting to me. I would
like to write a small mathematica program which takes as input some
dowker notation and a number n. As output I would like to have the
pictures of the next n exisiting knots or links, where the program
makes the dowker notation bigger in lexicographic order. I think I
might be able to do this by myself up to one point which I don't
know: If I come to a dowker notation of a link which does not
exist, I want the program just to forget about it and go on to the
next link. How can I do this? Best wishes Stephan
*********************************************************
Dr. Stephan Rosebrock
Paedagogische Hochschule Karlsruhe
Bismarckstr. 10
76133 Karlsruhe
Deutschland / Germany
e-mail: rosebrock@XXXXXXX.de
Homepage: http://www.rosebrock.ph-karlsruhe.de/
Tel: 0721-925-4275
Fax: 0721-925-4249
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