Dror BarNatan:
Talks:
Istanbul0606:
Algebraic Knot Theory Summary
an unproven thesis and an untested principle, appealing but mired with
unknowns and known obstractions
(Read: Risks & Opportunities Abound)
 The Thesis. Every 3D property of knots and link is definable in terms of Knotted
Trivalent Graphs.
 3D Property.

Having braid index 6.

Having a projection with 23 crossings.

Having an alternating projection.

Bounding a disk in B^{4}.

Being algebraic.
 Having unknotting number 3.
 Bounding a Seifert Surface of genus < 7.
 Being a boundary link.
 Being a ribbon knot.
 Being fibred.
 Having a vanishing Alexander polynomial.
 Definable. Within some given
collections of objects, in terms of a finite formula utilizing some
operations provided in advance. Examples:
 In algebra.
 In strict knot and link theory.
 Knotted Trivalent Graphs.
 The Principle. Seek a TGmorphism from
KTG to algebra.
 Aside. KTG is finitely presented  to find a TGmorphism
you need to make two guesses and carry out three checks. And when
you're done, you've rediscovered Drinfel'd's theory of associators.
 A Prime Candidate. The theory of finite
type invariants, as studied in Stonhenge.
 A Sad Truth. Z is practically surjective at
bounded degrees for several definable classes of knots (e.g. Ng's arXiv:qalg/9502017).
There are also some bad news about "bounded loop counts"
(GaroufalidisRozansky arXiv:math.GT/0003187).
 Put Your Name Here.
 Do the Alexander test case  recover FoxMilnor (for ribbon knots,
A(t)=f(t)f(1/t)).
 What's in the envelope of the Alexander polynomial, if we're talking
about knots or graphs?
 Control the Alexander associator (see Lieberum's arXiv:math.QA/0204346).
 Pull something out of Jones' envelope.
 Find the envelopes of other interesting finite type invariants.
 Study other internal quotients of the TGalgebra of chord diagrams.
There are plenty, even plenty with polynomial behaviour!
 Formalize and classify such internal quotients.
 Tame one associator and show it to your friends. (Or be brave and
master the theory of multiple ζnumbers).
 Complete the Stonhenge story for KTGs.
 Venture outside of Stonehenge! Surgery quotients?
 Are fibred knots definable?
 Are there other interesting definable classes of knots?
 What's "a TGalgebra"? (See D. Thurston's arXiv:math.GT/0311458).
Make precise the sufficiency of the pentagon and the hexagons for KTG.