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Baguenaudy, Baguenaudier, Baguenaudiest

Danny Calegari

A crochet row of chain stitches is formed by repeatedly pulling a folded over loop of yarn (a “bight”) through the previous loop. Each bight encircles the waist of the bight in front, like a row of elephants, each holding the tail of the next in its trunk.

Figure 1.

A chain stitch row of jute.

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So that the whole row does not come loose, we must “tie off” the left-hand side with some sort of knot. In a more schematic picture (Figure 2) this is accomplished by adding an arc in the lower left to create a trivalent vertex:

Figure 2.

An abstracted crochet line.

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Taking the fundamental group of the complement translates topology into algebra; if denotes a meridian loop around the initial loop (in red), and denotes a meridian loop around the free end (in green), then a meridian around the first bight is the conjugate , and meridians around successive bights are the conjugates of by the meridian around the previous bight; i.e., they form a sequence

The meridian around the last bight (in blue) represents the result of -fold iterated conjugation, where is the number of chains in the row.

If one cuts Figure 2 along the base and rejoins it as in Figure 3 one obtains an unlink, so that the short meridian loops (in red) become algebraically independent free variables , and now the meridian around the last bight (in blue) represents the result of free iterated conjugation; i.e., the element

Figure 3.

An interesting embedding of an unlink.

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The exponential growth of the word length of this element is at the heart of a classical puzzle, known (amongst other names) as a baguenaudier (Figure 4)—literally “time waster.” The red hook in the figure (which by implication extends indefinitely far to the left) is made of some rigid material and must be extricated from the rings, by sliding it left or right (when the rings are not in the way) and by moving the rings on their side through the hook to remove them or put them back.

Figure 4.

Can you remove the red hook from the rings?

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After some experimentation, one learns that it is possible to perform the following two moves (or their inverses):

(1)

one can always take the rightmost ring off the red hook (if it is on); or

(2)

if the th and st rings are on the red hook but no for , then one can take off the th ring.

Either move is accomplished physically by tipping the ring on its side and slipping it between the parallel bars of the red hook. The second move is illustrated in Figure 5.

Figure 5.

Can you see how to perform this move?

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If one thinks of the configuration of the puzzle as a binary string (with denoting on and denoting off) the initial configuration is , the target configuration (the hook removed from the rings) is , and the two legal moves are of the form and where denotes an arbitrary initial string of s and s. Using these operations one may solve the puzzle by induction.

In complexity terms, this puzzle may be unentangled by a procedure that takes up resources that are linear in space, but exponential in time (this fact is not obvious, but a rigorous argument is beyond the scope of this article). In an appendix to a paper of Hastings, Freedman shows that this phenomenon may be used to give simple examples of quantum systems with simply-connected configuration space for which the numerical quantum Monte Carlo algorithm fails to find the ground state of the system in polynomial time.

Some closely related objects appear in knot theory, specifically in the theory of Habiro claspers.

Two knots or links in the 3-sphere are said to be related by a -move if they differ as indicated in Figure 6 (the figure shows ; the convention in knot theory to illustrate a “move” is to show a fragment of two link diagrams, with the implication that the two diagrams are identical outside the fragments shown). They are -equivalent if they are related by a finite sequence of -moves.

Figure 6.

A -move relating two knots or links.

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These equivalence relations become successively finer as increases: -equivalence implies -equivalence for any . This is easy to see—Figure 7 shows how a move may be obtained as the composition of two moves and an isotopy.

Figure 7.

A move as the composition of two moves and an isotopy.

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Habiro showed that knots in the 3-sphere are equivalent if and only if they have the same finite type (i.e., Vassiliev) invariants of degree . This latter condition is known to have several geometric avatars:

(1)

by Stanford, it is identical to the relation on knots obtained by a finite sequence of moves of the following form: grab a family of parallel strands, and replace it by a pure braid in the th term of the lower central series of the pure braid group on strands for some ;

(2)

by Conant–Teichner, it is equivalent to the relation on knots of capped group cobordism of class .

A major open problem in knot theory is whether the entirety of all finite type invariants together is sufficient to distinguish knots—i.e., whether it is the case that if two knots and have the same finite type invariants of all degrees, they must necessarily be isotopic. Gropes of infinite order (with geometric control on the ends) may be thickened (in four dimensions) into Casson handles, wild geometric beasts that turn out to be homeomorphic—though typically not diffeomorphic!—to ordinary 2-handles, and play a key role in Freedman’s proof of the (topological) 4-dimensional Poincaré conjecture. Might grope cobordisms of infinite order between knots with the same finite type invariants provide a 4-dimensional approach to this 3-dimensional open problem?

You may choose to work on this problem; successful or not, I hope you do not consider the effort to be time wasted.

References

[1]
J. Conant and P. Teichner, Grope cobordism of classical knots, Topology 43 (2004), 119–156.
[2]
K. Habiro, Claspers and finite type invariants of knots and links, Geometry and Topology, 4 (2000), 1–83.
[3]
M. Hastings (with an appendix by M. Freedman), Obstructions To Classically Simulating The Quantum Adiabatic Algorithm, Quantum Information & Computation, 13 (2013), no. 11–12, 1038–1076.
[4]
T. Stanford, Braid commutators and Vassiliev invariants, Pac. J. Math. 174 (1996), 269–276.

Credits

All images are courtesy of Danny Calegari.