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September 27, 2007

Detecting Higher Order Necklaces

Posted by Urs Schreiber

You know a conference is a good one when there’s no time to report from it. Categories in Geometries and Physics is of this kind.

But before I forget it, I want to record a couple of things worth mentioning and remembering.

I had very long and interesting discussions with Nils Baas. Let me share the following question he poses:

Suppose you have a lot of silver rings. You join them to form a necklace. Then, given more such silver rings, you form more such necklaces. Then, from all these necklaces, you build, by similarly joining them, yet another necklace: a necklace of necklaces. A second order necklace. And so on.

Now suppose on a table sits a huge pile of silver rings. How do you decide if they form an order nn necklace?

Shouldn’t there be something like a higher order knot/link invariant which detects knots of knots and links of links?

Further topics today: a comparison with Enrico Vitale’s work on weak categorical cokernels and mapping cones, as well as a speculation on weak Lie nn-algebras triggered by discussion with Pavol Ševera.

Weak cokernels and mapping cones

In The Inner Automorphism 3-Group of a Strict 2-Group David Roberts and I considered the mapping cone of the identity morphism on a strict 2-group G (2)G_{(2)} as something we started calling a tangent category (pdf).

This is constructed by mapping the “fat point” 2={}2 =\{\bullet \stackrel{\simeq}{\to}\circ\} into the one-object 3-groupoid ΣG (2)\Sigma G_{(2)} and restricting the morphisms between such maps to fix the left endpoint of the fat point.

As John Baez amplified in Obstructions for nn-Bundle Lifts, forming the mapping cone of a morphism should correspond to forming its weak cokernel.

Today Enrico Vitale gave a talk on his work

P. Carrasco A. R. Garzó́n and E. M. Vitale
On categorical crossed modules
Theory and Applications of Categories, Vol. 16, 2006, No. 22, pp 585-618

in which he discusses the weak cokernel of morphisms of 2-groups. And, indeed, up to different notation, it comes down to exactly this construction.

Namely, in Obstructions, Tangent Categories and Lie NN-tegration I had stated the obvious generalization of the mapping cone construction for non-identity morphisms:

Recall

Mapping cones and tangent categories

Recall from the discussion at The Inner Automorphism 3-Group of a Strict 2-Group that the mapping cone of the identity 2-functor CCC \to C, in the world of strict 2-groupoids, amounts to forming what I called the “tangent category” TCT C, which is obtained by mapping the interval {}\{\bullet \stackrel{\simeq}{\to} \circ\} into CC and admitting only those homotopies between such maps which fix the left endpoint.

Now, you won’t be surprised to learn that the mapping cone of an arbitrary injection t:C 0C 1t : C_0 \to C_1 turns out to correspond to maps of the interval into C 1C_1, whose homotopies are resticted to fix the left endpoint and to have a right endpoint transverse a 2-path in the image of tt. But I mention it nevertheless. Since I think it is true and useful.

Indeed, a short inspection shows that this construction is precisely the same as that which Enrico Vitale describes at the beginning of p.11 from a different point of view and using different notation.

But in order to see how his definition amounts to the above one you simply suspend his categorical group to a one-object 2-groupoid and draw the relevant pictures.

Beware, by the way, that Enrico Vitale and his coauthors are also talking about an inner automorphism 2-group. But the concept they mean is the one orthogonal to the one I was talking about: namely the cokernel of the inclusion GAdAut(G)G \stackrel{\mathrm{Ad}}{\to} \mathrm{Aut}(G) of inner automorphisms into arbitrary autmorphisms. That gives the outer automorphisms!

Compare their example iv) on p. 12.


Weak Lie 2-algebras

A while ago I wrote:

I just had a revelation.

Yesterday I had a followup revelation. (Need to start using a threaded revelation viewer eventually.)

Recall that I conjectured that

general weak Lie nn-algebroids correspond to NPQ manifolds

Had an extensive discussion of this with Pavol Ševera. We agreed that one ought to be looking for some kind of obvious and/or natural thing in between A A_\infty and L L_\infty. You know, whatever truly weak Lie nn-algebras are, they must have a description in terms of something we already sort of know.

A A_\infty comes from tensor co-algebra with co-differential. L L_\infty from (graded) symmetric tensor co-algebra with codifferential.

“Clearly” the strict skew symmetry of the Lie nn-algebra bracket nn-functor translates into the graded commutativity of the coalgebra.

What we are looking for is a situation where two things graded commute only up to an additve correction (since composing higher coherence morphisms in this game is just addition).

But this can only mean one thing: we need Clifford algebra instead of exterior algebra.

(Sorry for the boldface. Couldn’t help it.)

Wednesday’s conjecture: Lie nn-algebras with arbitrary coherently weak skew symmetry and arbitrary coherent weak Jacobi identity correspond to graded differential Clifford algebras.

You see, take a Lie algebra and the corresponding Chevalley Eilenberg algebra (sg *,d). \wedge^\bullet ( s g^*, d ) \,.

Then replace the exterior algebra (of left invariant differential forms on GG, really), with the Clifford algebra coming from some bilinear form on gg, but such that the \mathbb{Z}-grading is respected.

So we throw in one single degree 2-generator bb and demand that for t at^a and t bt^b any two elements in sg *s g^*, instead of

t at b=t bt a t^a t^b = - t^b t^a

we have

k abt at b=b, k_{ab} t^a t^b = b \,,

where k abk_{ab} are the components of the symmetric bilinear form.

For this to be compatible with the differential, we find that kk has to be invariant. Just as if k abk_{ab} were the components of a symplectic 2-form on the dg-manifold.

Just imagine this turns out to hold water: we’d be able to say

An ordinary Clifford algebra on the vector space VV with bilinear form ,\langle \cdot, \cdot\rangle is an abelian weak Lie 2-algebra with space of objects being VV and space of morphisms being VV \oplus \mathbb{R} where everything is strict, except that the skew symmetry of the bracket functor holds only up to the natural isomorphism [x,y]x,y[y,x]. [x,y] \stackrel{\langle x,y \rangle}{\to} [y,x] \,.

It’d be very nice if this indeed holds water. It would provide a bunch of missing puzzle pieces. Like explaining what Clifford algebra really is, how it categorifies, how the \mathbb{Z}-graded structure introduced by the presence of Lie nn-algebras gives also rise to the 2\mathbb{Z}_2-grading found in supersymmetry, in fact supersymmetry itself would start entering the big picture here.

Pavol’s very thoughtful remark on this last remark of mine:

I knew you nn-category guys are crazy.

When I find the time I should report on Pavol’s cool talk on his work on differentiating Kan simplicial complexes to Lie nn-algebras, as described in

Pavol Ševera
L L_\infty algebras as 1-jets of simplicial manifolds (and a bit beyond)
math/0612349

Posted at September 27, 2007 11:56 PM UTC

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17 Comments & 4 Trackbacks

Re: Detecting Higher Order Necklaces

You’re overusing b in that Clifford equation, right?

Posted by: Allen Knutson on September 28, 2007 12:21 PM | Permalink | Reply to this

Re: Detecting Higher Order Necklaces

You are overusing bb in that Clifford equation, right?

Right. Sorry, this are two stupid conventions I am came to become used to sticking to that are colliding here.

I should have written:

let {t i}\{ t^i \} be a basis for g *g^* in degree one and {b}\{ b \} be the canonical basis of *\mathbb{R}^* in degree 2. Let k ijk_{i j} be the components of a symmetric bilinear form kk on gg in the corresponding basis.

Then weaken the strict graded commutativity of the t it^i by setting

k ijt it j=b. k_{i j} t^i t^j = b \,.

Now demand bb to be closed and check what this implies for kk.

Posted by: Urs Schreiber on September 28, 2007 3:13 PM | Permalink | Reply to this

Re: Detecting Higher Order Necklaces

Another great talk took place today: Konrad Waldorf (one of Urs’s colleagues at Hamburg) gave a beautiful exposition of some of the work that he and Urs have been doing on parallel transport. Even for a diff geom ignoramus like me, it was a delight.

If I’d been less caught up in the mathematics I would have taken photos of the blackboards and posted them on the web. But I know that Konrad also made beamer slides (which he didn’t use in the end because the lectures were outdoors!) Urs, do you think he could be persuaded to make them available online?

Posted by: Tom Leinster on September 28, 2007 6:05 PM | Permalink | Reply to this

Re: Detecting Higher Order Necklaces

I cannot provide links at the moment. But search the n-Cafe for “Waldorf on n-transport” for the slides on a previous talk on much the same topic. I’ll ask Konrad if he puts the new slides on his website. Probably he’ll do so anyway.

Posted by: Urs Schreiber on September 29, 2007 12:16 AM | Permalink | Reply to this

Re: Detecting Higher Order Necklaces

It turns out that Konrad had put the slides on his web page before I even posted that last comment: here they are.

Posted by: Tom Leinster on September 30, 2007 1:02 PM | Permalink | Reply to this

Re: Detecting Higher Order Necklaces

Problem: in general, one must deal with WILD knots and links.

There are uncountably many pathological examples. Wild knots are mostly swept under the Knot Theory rug, but your problem as stated is not likely to be solved by merely a new polynomial invariant, much as that might be nice.

Posted by: Jonathan Vos Post on September 28, 2007 6:41 PM | Permalink | Reply to this

Re: Detecting Higher Order Necklaces

Speaking of wild phenomena in his Sketch of a Programme, Grothendieck has this to say (Section 5: Denunciation of so-called “general” topology, and heuristic reflections toward a so-called “tame” topology):

… Even now, just as in the heroic times when one anxiously witnessed for the first time curves cheerfully filling squares and cubes, when one tries to do topological geometry in the technical context of topological spaces, one is confronted at each step with spurious difficulties related to wild phenomena… Topologists elude the difficulty, without tackling it, moving to contexts which are close to the topological one and less subject to wildness, such as differentiable manifolds, PL spaces (piecewise linear) etc., of which it is clear that none is “good”, i.e. stable under the most obvious topological operations, such as contraction-gluing operations… This is a way of beating about the bush! This situation, like so often already in the history of our science, simply reveals the almost insurmountable inertia of the mind, burdened by a heavy weight of conditioning, which makes it difficult to take a real look at a foundational question… It is certainly this inertia which explains why it took millenia before such childish ideas as that of a zero, of a group, of a topological shape found their place in mathematics. It is this again which explains why the rigid framework of general topology is patiently dragged along by generation after generation of topologists for whom “wildness” is a fatal necessity, rooted in the nature of things.

(Extracted from Leila Schneps’s translation in Geometric Galois Actions 1. Around Grothendieck’s Esquisse d’un Programme. LMS Lecture Note Series 242, pp. 264-265.)

Of course Urs and Nils were contemplating tame links, since they take silver rings as tubular neighborhoods! :-)

Posted by: Todd Trimble on September 28, 2007 9:58 PM | Permalink | Reply to this

Antoine’s Necklace; Re: Detecting Higher Order Necklaces

Okay, forgt for a moment about wild knots and links. Still open are higher-order necklaces where higher-order means infinite, yet countable. The classic example is:

Weisstein, Eric W. “Antoine’s Necklace.” From MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/AntoinesNecklace.html

So how do you tell if you have a necklace which starts with an Antoine’s Necklace and has links branching off from that? How do you tell if your countable pile of tori are two interlocked Antoine’s Necklaces?

The knot-invarient polynomials involved seem to have infinitely many terms, i.e. to be infinite series.

Posted by: Jonathan Vos Post on September 30, 2007 5:12 PM | Permalink | Reply to this

Re: Antoine’s Necklace; Re: Detecting Higher Order Necklaces

I thought Antoine’s necklace would also considered “wild” (technically it’s not a link at all, if it’s homeomorphic to the Cantor set).

Someone like John Armstrong would be more knowledgeable about this than I, but I believe it is common practice to restrict attention to knots and links which are tame, i.e., ambient-isotopic to unions of finitely many line segments in 3\mathbb{R}^3 which form a subspace homeomorphic to a disjoint sum of circles. For example, when one says that “every” link is the closure of a braid, one is clearly referring just to tame links.

Your question might be interesting, but I’d be surprised if Nils and Urs were referring to anything outside of tame links, even if they used the word “necklace”.

Posted by: Todd Trimble on September 30, 2007 6:23 PM | Permalink | Reply to this

Re: Antoine’s Necklace; Re: Detecting Higher Order Necklaces

Yes, we restrict to tame knots and links. Actually, what we do is say that every point on the link has a neighborhood homeomorphic to a ball-arc pair, which fails at a singular point.

I’ve been thinking about this recently, inspired by the post. What we avoid is having an infinite number of “convolutions” (crossings, cups, caps) in an arbitrarily small space.

When we build the category of tangles up by taking a free monoidal category and adding in new morphisms for crossings – braiding – and cups and caps – duals – we talk about building finite words out of finite monoidal products. This finiteness condition is what keeps all of our tangles tame.

How can we bring wild tangles into the categorical picture? Well, if we have a good way of talking about infinite composition of monoidal products (I think we can push the infinities all into one direction and keep a bounded width) then we can capture some of the standard examples of wild knots. So what’s an infinite composition?

Posted by: John Armstrong on September 30, 2007 9:15 PM | Permalink | Reply to this

Re: Antoine’s Necklace; Re: Detecting Higher Order Necklaces

Hmm… wonder if Kauffman has thought about this stuff.

This isn’t too responsive to your comment, but I remember Jim Stasheff’s teaching me the ‘Milnor slide trick’ for proving that proving that fiber bundles over paracompact base spaces are fibrations: that used an “infinite composite” of bundle isomorphisms (albeit “locally finite”, so there is no problem with well-definedness). Somehow that trick has always amused me…

Posted by: Todd Trimble on September 30, 2007 10:05 PM | Permalink | Reply to this

Uncountable necklaces; Re: Detecting Higher Order Necklaces

Just wanted to back up my earlier vague comment with a specific quotation from

Antoine’s Necklace or How to Keep a Necklace from Falling Apart,
B.L. Brechner, J.C. Mayer - The College Mathematics Journal, 1988
:

“… the fact that there are infinitely (in fact uncountably) many different Antoine’s Necklaces.”

I do admit that your categorification of the Morse theory of this looks lovely even as a goal.

“So what’s an infinite composition?” That presumably has a different answer depending on whether the infinity is countable, uncountable, of large cardinality…

Posted by: Jonathan Vos Post on October 2, 2007 12:15 AM | Permalink | Reply to this

Analogues in in E^n; Re: Uncountable necklaces; Re: Detecting Higher Order Necklaces

Posted by: Jonathan Vos Post on October 2, 2007 12:29 AM | Permalink | Reply to this

Experiment on how knots form; Re: Detecting Higher Order Necklaces

Experimental Physicists and Biologists, including a really clever undergrad, only care about tame knots and tame links.

Physicists Tackle Knotty Puzzle.

Electrical cables, garden hoses and strands of holiday lights seem to get themselves hopelessly tangled with no help at all. Now research initiated by an undergraduate student at the University of California, San Diego has resulted in the first model of how knots form…

Posted by: Jonathan Vos Post on October 2, 2007 6:23 AM | Permalink | Reply to this

Re: Detecting Higher Order Necklaces

A conversation with Nils Baas this lunchtime on a completely different topic reminded me of an example of higher-order structure. The Magic Roundabout should probably be called a 2-roundabout. On the other hand, I think even committed (in either sense) knot theorists would have a hard time classifying the original spaghetti junction.

Posted by: Andrew Stacey on October 9, 2007 12:36 PM | Permalink | Reply to this
Read the post Categorified Clifford Algebra and weak Lie n-Algebras
Weblog: The n-Category Café
Excerpt: On weak Lie n-algebras, differential graded Clifford algebra and Roytenberg's work on weak Lie 2-algebras.
Tracked: October 9, 2007 4:57 PM
Read the post n-Curvature, Part III
Weblog: The n-Category Café
Excerpt: Curvature is the obstruction to flatness. Believe it or not.
Tracked: October 16, 2007 10:51 PM
Read the post On weak Cokernels for 2-Groups
Weblog: The n-Category Café
Excerpt: On weak cokernels of 2-groups.
Tracked: October 17, 2007 11:02 PM
Read the post 2-Vectors in Trondheim
Weblog: The n-Category Café
Excerpt: On line 2-bundles.
Tracked: November 5, 2007 9:55 PM

Wild knots in higher dimensions; Re: Detecting Higher Order Necklaces

Wild knots in higher dimensions as limit sets of Kleinian groups, by Margareta Boege, Gabriela Hinojosa, Alberto Verjovsky, Accepted in Conformal Theory and Dynamical Systems (AMS), 25 Aug 2009.

“Such wild knots are examples of self-similar fractal sets and they are extremely beautiful to contemplate. For instance, one can admire the pictures in the classic book by R. Fricke and F. Klein ([11]) or the pictures of limit sets of classical Kleinian groups in the book Indra’s Pearls: The vision of Felix Klein by D. Mumford, C. Series, D. Wright ([27])…. we give the preliminaries:
the definition of oriented tangles and knots in high dimensions and some basic facts of Kleinian groups….”

COROLLARY 6.3 There exist infinitely many nonequivalent wild n-knots
in R^(n+2).

THEOREM 4.1 There exist infinitely many non-equivalent knots ψ : S^n → S^(n+2) wildly embedded as limit sets of geometrically finite Kleinian groups,
for n = 1, 2, 3, 4, 5.

Posted by: Jonathan Vos Post on August 26, 2009 7:05 PM | Permalink | Reply to this

Re: Detecting Higher Order Necklaces

Higher-order necklaces appear in this paper of Baas – New States of Matter Suggested by New Topological Structures.

Posted by: David Corfield on February 17, 2011 5:26 PM | Permalink | Reply to this

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