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Cell["Some HOMFLY-PT One Parameter Knot Theory Computations", "Title",
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Cell["\<\
By Jana Archibald, Dror Bar - Natan, Karene Chu and Thomas Fiedler\
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invariants coming out of Fiedler's \"one-parameter\" approach to knot theory \
as described in ",
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 ", except this time the underlying knot invariant is the HOMFLY-PT \
polynomial instead of the Jones polynomial.\n\nIn short, the idea is to \
associate in some canonical way a \"knot movie\", or more precisely, a \
\"one-parameter family of knot diagrams\" to a given knot diagram. Within a \
generic knot movie some \"frames\" will be singular. In the said article \
Fiedler constructs a state sum invariant of \"knot diagrams with a triple \
crossing\" in much of the same way as knot invariants are constructed using \
state sums involving just standard knot diagrams. When those \
\"triple-crossing state sums\" satisfy certain complicated conditions, they \
yield invariants of knot movies, and when evaluated on the movies canonicaly \
associated with knot diagrams, they yield knot invariants.\n\nThis time \
around we rely on unpublished work by Fiedler, in which the relevant \"state \
sum\" is replaced by a HOMFLY-PT-inspired invariant.\n\nThis construction \
seems new and little is known about it. While the invariants it produces are \
always finite type, they are otherwise potentially new.\n\nIn this notebook \
we compute two of the simplest knot invariants introduced by Fiedler. The \
results are in some sense disappointing - the first of the two invariants \
quite clearly comes out to by a multiple of the HOMFLY-PT polynomial, and \
while we could not compute enough of the second to fully identify it, we \
computed enough to tell that it does not separate ",
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 " from its inverse. So no jackpot here, but a new and unexplained \
construction of the HOMFLY-PT polynomial is surely an interesting thing. \
Where does it come from? What exactly is the universe of all such \
constructions? Are there news in any of the others?\n\nThe main purpose of \
this notebook is to preserve our efforts of June 16-22, 2008. It is not \
easily readable and not meant for readership much beyond its own authors. \
Don't bother complaining.\n\nAlong the way we had to compute the HOMFLY-PT \
polynomial for rather large knots, ones fro which the existing code in the \
package ",
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sparse-matrix code to compute the HOMFLY-PT polynomial (the program \"Homf\", \
below). This code is not yet well tested and is not yet a part of  \
KnotTheory`, but we hope to incorporate it into KnotTheory` soon.\n\nThe .nb \
(original) version of this notebook is at ",
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Cell["\<\
(We conjecture that the uncabled invariant coincides with the HOMFLY - PT \
polynomial up to normalization. See \"The First Invariant\" below.)\
\>", "Text",
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The second invariant: Y2 for the Conway and Kinoshita-Terasaka Knots\
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