# On knots, categories and mass gaps

Kea
Sorry this is all a bit condensed, but people seem interested in
this mass gap question so here are some more thoughts.

Knots and Topos
---------------

In one of his papers on the Temperley-Lieb algebra, Lickorish says
that linear skein theory is useful rather than profound. But like
all magically useful things, it turns out that there is perhaps
depth here after all and that is in the realisation that skein
theory is about the topos theoretic issue of where the numbers
live, because in the topos approach even numerical invariants must
know their context.

The objects of the Temperley-Lieb category are ordinals
represented by dots in the plane. An arrow is a diagram of half
loops and lines. The algebra is given by formal sums over the
complex numbers up to planar isotopy such that the skein relations
of the Kauffman bracket (for A) hold.

The empty diagram on the right hand side of the second relation
has a one dimensional span = C. For a given value of A, this is
where the numerical invariant resides. The number 1 corresponds to
the empty diagram. This follows from the functorality of the
invariant. Similarly, the polynomial ring C[x] arises from the
diagram of a loop in an annulus, where the background 3sphere has
been replaced by a torus.

Loops are labelled with a spin j, representing j strands of the
loop. A ring morphism from C to C[x] arises from the insertion of
a tangle into a torus and the loop j is represented by x^{j} in
the commutative polynomial ring.

The Kauffman bracket becomes a full isotopy invariant under the
addition of a factor (- A)^{- 3 w} where w is the writhe of a link
diagram, which may be used to represent a framing for a knot.

A positive knot is one represesented by a braid diagram with only
positive crossings. For such knots, clearly the writhe is the same
as the total number of crossings. A deep connection between
Feynman diagrams and positive knots has been established by
Broadhurst, Kreimer and now Connes and Marcolli.

For a generalised zeta function (MZV) d is the depth, given by
(n-1) for a representation in B_{n}, and w = s_{1} + s_{2} + ... +
s_{d}.

The genus of a knot is defined to be the smallest integer g such
that the knot is the boundary of an embedded orientable surface in
S^{3}. One has 2 g = w - d.

Ribbon tree diagrams are instances of templates, or branched
surfaces with semiflows, which appear in dynamical systems theory.
The Lorenz attractor template is the pasted ribbon diagram with 2
holes. A positive knot embedded in this template is defined by a
braid word in two generators x_{0} and x_{1}, corresponding to the
two branch cuts of the surface. For example, the (4,3) torus knot
may be represented by the word x_{0}(x_{1}x_{0})^{3}.

There is a template on four letters which is universal in the
sense that all links may be embedded in it. This is very
interesting, because the letters x_{i} become quantum number
indexes appearing below when we consider the problem of solving
coherence conditions.

Now the three dimensionality of weak associativity relies on the
existence of non-trivial 3-arrows in the category in question. In
a braided monoidal category, these are essentially supplied by the
intertwiner maps. However, there is a subtlety to note here. The
tetrahedron treats U(VW) and (UV)W as a single edge. The component
of the weak nerve that distinguishes these objects is a 'parity
square'.

Considering a face as a puncture is what characterises the notion
of a qubit in the application of braids to quantum computers (see
reference below).

Let r = 5 set a value for A. The state space of a quantum computer
built on k qubits is given by the association of C^{2} \otimes k
to a disc with 3k marked points, on which there is naturally an
action of the braid group B_{3k}. Thus the computational power of
the computer translates directly into the number theoretic notion
of depth d introduced above.

If one believes that a higher dimensional analogue of area
operator should describe mass generation, then it is interesting
that such operators are related to sets of qubits, or rather
quantum gates, or intuitively an 'atom of logic'. Maybe the way in
which this idea naturally falls out of topos theoretic
considerations gives it a compelling physical significance.

Let q = A^{4}. Let a, b and c be complex variables such that the
Fermat equation a^{r} + b^{r} = c^{r} holds. The quantum plane
operators satisfy x^{r} = y^{r} = - 1. These ingredients are used
by Kashaev and Faddeev (see ref below) to construct solutions to
the Mac Lane pentagon. Spectral conditions are imposed in order to
set a 'coefficient' for the pentagon to 1. From Joyce's work,
however, it is clear that this parameter may have some physical
significance.

The point is that generalised spin labels are derived from the
requirement of a solution for the coherence conditions. This
replaces their selection by hand in other approaches to quantum
gravity.

It is well recognised that the usual m numbers are related to the
curvature of a puncture on the horizon in the LQG description of
black hole entropy. In a tricategorical context they label edges,
so the single arrow braided monoidal categories cannot hope to
capture the physical origin of mass.

The Parity Cube in Tricategories
--------------------------------

The objects of a tricategory T are labelled p, q etc. For each
pair of objects p and q there is a bicategory T(p,q). The
internalisation of weak associativity is a pseudonatural
transformation 'a'. The parity cube is labelled by bracketed words
on four objects, which may be loosely thought of as the
bicategories of particles. That is, one particle contains the
potential of all its interactions, and this information must be
mathematically realized.

The Mac Lane pentagon lives on 5 sides of the cube. The new top
face of the cube is the premonoidality deformation (of Joyce),
which closes the horn. This square appears as a piece of data for
a trimorphism, that is a map between tricategories, so that it
naturally appears in 4-dimensional structures.

The relation of this q to the braiding in the quantum group case
means that: if one believes (a) the deformation parameter q is
associated to the existence of a Planck scale somehow, and (b) one
has cohomological mass generation, then one really ought to
conclude that the appearance of a mass gap is associated with
quantumness.

The tetracategorical breaking of the hexagon amounts to the
breaking of topological invariance as described by Pachner moves
for 4D state sums based on fixed triangulations (spin foams). This
view thus explains why four dimensional gravity is not topological
in the usual sense of the word.

Higher dimensional categories are essential in exactly
characterising interactions between larger ensembles of particles,
since the Gray tensor product is dimension raising.

Some useful REFERENCES
------------------------

J. Roberts
Skein theory and Turaev-Viro invariants
Topology 34, 1995 p.771-787

Quantum Dilogarithm
Mod. Phys. Lett. A9,5: 1994 p.427-434

A. Grothendieck
Pursuing Stacks
available at http://www.math.jussieu.fr/~leila/mathtexts.php [Broken]

C. Mazza V. Voevodsky C. Weibel
Notes on Motivic Cohomology
available at http://www.math.uiuc.edu/K-theory/0486

A. Goncharov
Volumes of hyperbolic manifolds and mixed Tate motives
http://arxiv.org/abs/alg-geom/9601021

J. Stachel
Einstein from B to Z
Birkhauser, 2002

Ross Street
Categorical and Combinatorial Aspects of Descent Theory
available at www.maths.mq.edu.au/~street/DescFlds.pdf

L.H. Kauffman M. Saito M.C. Sullivan
Quantum Invariants of Templates
available at www2.math.uic.edu/~kauffman/Papers.html

R.F. Williams
The universal templates of Ghrist
available at www.ma.utexas.edu/~bob/ghrist.ps

M. Freedman M. Larsen Z. Wang
A modular functor which is universal for quantum computation
http://arxiv.org/abs/quant-ph/0001108

Best regards
Kea

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Gold Member
Dearly Missed
I could not understand your def. of the genus:
"The genus of a knot is defined to be the smallest integer g such
that the knot is the boundary of an embedded orientable surface in
S^{3}..."
I had the feeling that something had been left out, in the haste of typing.
Could you expand on this definition---perhaps give an example or two?

BTW an interesting association with your nickname:
http://www.doc.govt.nz/Conservation/001~Plants-and-Animals/001~Native-Animals/Kea.asp

[edited to muffle the tone of excitement]

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Kea
genus

Hello Marcus

This might be useful:
http://mathworld.wolfram.com/SeifertSurface.html

I should also add the reference: J. S. Birman R.F. Williams
Knotted periodic orbits in dynamical systems I: Lorenz's equations
Topology 22,1 (1983) 47-82

Sorry - very busy today. Regards, Kea.

Before their eyes in sudden view appear
The secrets of the hoary deep, a dark
Illimitable Ocean without bound,
Without dimension, where length, breadth, and height,

And time and place are lost; where eldest Night
And Chaos, Ancestors of Nature, hold
Eternal Anarchy, amidst the noise
Of endless wars, and by confusion stand.

Staff Emeritus
Gold Member
Dearly Missed
marcus said:
I could not understand your def. of the genus:
"The genus of a knot is defined to be the smallest integer g such
that the knot is the boundary of an embedded orientable surface in
S^{3}..."
I had the feeling that something had been left out, in the haste of typing.
Could you expand on this definition---perhaps give an example or two?

[edited to muffle the tone of excitement]

Marcus, you might want to look at these lecture notes: http://www.scholar.google.com/scholar?q=Knot+seifert&hl=en&lr=&start=30&sa=N. They're in postscript, and the Seifert discussion (how to construct the manifold that the knot bounds) starts on page 8, with a diagram for the trefoil knot.

Gold Member
Dearly Missed
I still think there was something left out of that definition of the knot genus.

marcus said:
I could not understand your def. of the genus:
"The genus of a knot is defined to be the smallest integer g such
that the knot is the boundary of an embedded orientable surface in
S^{3}."

I think the omitted phrase was and g is the genus of the surface

So then the completed definition of the knot genus would read:

"The genus of a knot is defined to be the smallest integer g such
that the knot is the boundary of an embedded orientable surface in
S^{3} and g is the genus of the surface."

If the definition says "smallest integer g such that..." and then fails to give any condition on the integer g, then it is meaningless. That was all I had in mind

That is a great quote from Paradise Lost, and right apt to Quantum Gravity.

selfAdjoint, Kea chose a really interesting handle. You should look at the page about the actual new zealand bird called Kea. I posted the link a couple back.

Gold Member
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selfAdjoint, I just wanted to consult a roadmap at this point, namely the
recent Ashtekar survey "Gravity and the Quantum"

There is a key paragraph on page 29 where he indicates the 4 lines of research that he sees addressing the issue of dynamics. the second of thes is Knots.

I might as well list all 4 lines of development and then i will quote what he says about knots.

1. Thiemann's master-constraint (see recent papers by Dittrich and Thiemann)

2. knot approach---may eventually even dispense with underlying diff. manif.

3. spinfoam (recent papers by Freidel, Noui, Perez)

4. Gambini Pullin Porto consistent discretizations (gets rid of hamiltonian constraint altogether!)

these are all basically ways of coping with difficulty with the hamiltonian constraint and the general problem of introducing dynamics in the full theory. in the Loop Quantum Cosmology restriction the hamiltonian constraint becomes a very serviceable difference equation and dynamics are going along quite well and give the right semiclassical limit etc. but in the full theory there is difficulty with dynamics, so these 4 separate efforts.

Now it turns out that Gambini Pullin have been involved in the knot approach too. here is what Ashtekar says on page 29:

---quote---
A second strategy to solve the quantum scalar constraint is due to Gambini, Pullin and their collaborators. It builds on their extensive work on the interplay between quantum gravity and knot theory [27]. The more recent developments use the relatively new invariants of intersecting knots discovered by Vassiliev. This is a novel approach which furthermore has a potential of enhancing the relation between topological field theories and quantum gravity. As our knowledge of invariants of intersecting knots deepens, this approach is likely to provide increasingly significant insights. In particular, it has the potential of leading to a formulation of quantum gravity which does not refer even to a background manifold (see footnote 9).

Footnote 9)
In 2+1 dimensions, although one begins in a completely analogous fashion, in the final picture one can get rid of the background manifold as well. Thus, the fundamental theory can be formulated combinatorially [21,23]. To achieve this in 3+1 dimensions, one needs more complete theory of (intersecting) knots in 3 dimensions.

---end quote---

Reference [27] is
[27] Gambini R and Pullin J 1996 Loops, knots, gauge theories and quantum gravity (Cambridge UP, Cambridge)
=======
Well Kea you sound familiar with some LQG-related areas of research. there may be a lot more going with knots on that Ashtekar didnt indicate, since his survey is quite abbreviated.
Also selfAdjoint do you remember that "Causal Sites" paper? It was by Dan Christensen and Louis Crane.

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Marcus said:
selfAdjoint do you remember that "Causal Sites" paper? It was by Dan Christensen and Louis Crane.

Yes, it's

http://www.arxiv.org/PS_cache/gr-qc/pdf/0410/0410104.pdf [Broken]

These things are coming along too fast to keep track of!

And thick and fast they came at last
And more and more and more.

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Kea
connections

marcus said:
Well Kea you sound familiar with some LQG-related areas of research. there may be a lot more going with knots on that Ashtekar didnt indicate, since his survey is quite abbreviated.
Also selfAdjoint do you remember that "Causal Sites" paper? It was by Dan Christensen and Louis Crane.

I don't think the latter paper addresses the subtleties of
category theory sufficiently well, but then I might be
Street, the right way to look at stacks (categorified sheaves)
is in terms of descent theory as studied by Street and others.
This is all rather technical, and will take me years to
understand. He says to think of stacks as the process of
'localising the combinatorics'.

Also, apparently the category of stacks over a site is a really
nice 2-category with a biadjunction to the pseudofunctor
category into Cat (categorifed presheaves). So its really
like 'higher toposes'.

Gold Member
Dearly Missed
Kea said:

I don't think the latter paper addresses the subtleties of
category theory sufficiently well, but then I might be
Street, the right way to look at stacks (categorified sheaves)
is in terms of descent theory as studied by Street and others.
This is all rather technical, and will take me years to
understand. He says to think of stacks as the process of
'localising the combinatorics'.

Also, apparently the category of stacks over a site is a really
nice 2-category with a biadjunction to the pseudofunctor
category into Cat (categorifed presheaves). So its really
like 'higher toposes'.

Kea since your post here is largely about what Ross Street said (and since I'm unfamiliar with a lot of things you are studying and discussing) I thought I'd introduce Ross Street (Maquarie Univ. NSW Australia) to anyone with about my own level of familiarity who might be curious. Here's his homepage
http://scg.levels.unisa.edu.au/mass/members/street.html [Broken]
And his name crops up a lot in John Baez TWF #209
which was just posted yesterday

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Kea