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The notion of quandle is known to be closely related to knot theory. The three axioms in the definition of quandle correspond to the Reidemeister moves.

Recently I learned that there are higher analogues of Reidemeister moves. For surface knots. I've heard that seven moves of Roseman determine isotopy classes.

Is there an algebraic structure, which can be regarded as a 2-dimensinal analogue of quandle, corresponding to Roseman moves?

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How analogous would you like it to be? Isomorphism of the object is equivalent to isotopy of knots, like that? That'd be a decidedly more complicated type of object. Some kind of "homotopy-algebra quandle" instead of the classical "fundamental group quandle". –  Ryan Budney Nov 28 '09 at 23:50
    
I would be glad if there is an algebraic structure described by a set of seven equations corresponding to Roseman moves. Since rack or quandle is obtained by abstracting group conjugations, my guess is we might be obtain such an algebraic structure from conjugations in 2-groups or crossed modules or categorical groups or some kinds of 2-categorical structure. –  Dai Tamaki Nov 29 '09 at 1:24

2 Answers 2

The definition of strict 2-quandle and examples thereof has not be written down in a public forum yet. Crans, Elhamdadi, Saito, and I have a notion and examples. I think that Crans spoke about the idea in Riverside, and I will give a talk about it at Knots in Washington next weekend.

I don't reckon that I will have slides ready before then. If I do, I will post them on my web page. I am considering blackboard talks for the up-coming conference.

It is still conjectural that a weak 2-quandle is an axiomatization of the Roseman moves. I am playing with a couple of diagrammatic schemata to make sure everything works. Also after your question, I started puzzling about the tetrahedral move. This should be a theorem, and right now it appears to be an axiom.

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Thank you very much for mentioning your work in progress here. I look forward to seeing a definition of 2-quandle. –  Dai Tamaki Nov 29 '09 at 20:21

For any codimension 2 embedding that is locally flat, there is a notion of the fundamental rack (not every element is idempotent). In Euclidean space, there is a fundamental quandle. It is defined as in the classical case: homotopy classes of paths that start on a tubular neighborhood and that end at the base point. The top of the homotopy is required to remain at the base point, but the bottom stays on the tubular neighborhood. More specifically, two paths $a_0$ and $a_1$ with $a_j(0) \in \partial (N(K))$ and $a_j(1)= {\mbox{ pt.}}$ are equivalent if $\exists: A:[0,1]\times [0,1] \rightarrow {\bf R}^{n+2}$ such that $A(j,t)=a_j(t)$ for $j=0,1$, while $A(0,s)\in N(K)$ and $A(1,s)={\mbox{pt.}}$

The quandle operation is to follow $a$ to the basepoint, go down $b$ around the meridian at $b$ (in an oriented fashion), and return to the base point at $b$. See this manuscript (in particular the illustration therein) to see what is going on.

In additional to the fundamental quandle, there are quandle cocycle invariants that detect a lot of things about classical knots and knotted surfaces. For classical knots, the 2-cocycle invariant is related to a choice of longitude. See M. Eisermann. Homological characterization of the unknot. J. Pure Appl. Algebra, 177(2):131–157, 2003 for this result.

Fenn and Rourke showed that the 3-cocycle invariant for classical knots could detect the chirality of the trefoil.

The 3-cocycle invariant for knotted surfaces (and its generalizations) are known to give strong results about non-invertibility, bounds on triple point numbers, and in the case of symmetric quandle homology, they give bounds on the triple point numbers of non-orientable surfaces.

In higher dimensions, one can prove that higher cocycle invariants exist --- there are generalizations of the Roseman moves that are given via multi-germs (due to Mond and another author), and the quandle $n$-cocycle condition corresponds to the $n$-simplex move in this context. There is a lot of work that can be done in this regard. I think most of it is straight-forward.

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My question is how to extend or modify the notion of quandle to include seven Roseman moves, not a construction of a quandle from higher dimensional knots. It would be nice if there is an algebraic structure defined by seven equations corresponding to Roseman moves. –  Dai Tamaki Nov 29 '09 at 1:19
    
I thought that I posted this comment: The structure that you seek is probably a weak (or coherent) 2-quandle. This would be a category in which the quandle axioms hold up to certain isomorphisms. –  Scott Carter Nov 29 '09 at 6:16
    
Thank you for the word "2-quandle". Has "strict 2-quandle" been defined by anybody? I googled a little bit but the only document containing 2-quandle seems to be the thesis of Crans, which does not give a definition. As you suggested, what I seek seems to be a version of 2-quandle. –  Dai Tamaki Nov 29 '09 at 7:04

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