$\omega$-triviality of knots? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T01:46:31Z http://mathoverflow.net/feeds/question/41611 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/41611/omega-triviality-of-knots $\omega$-triviality of knots? Jim Conant 2010-10-09T17:07:47Z 2010-10-10T15:39:03Z <p>From the theory of finite type invariants of knots comes the concept of $n$-triviality. A knot is said to be $n$-trivial if there is some projection of the knot and $n$ pairwise-disjoint sets of crossings $S_1,S_2,\ldots,S_n$, such that changing the crossings in every nontrivial subset of <code>$S=\{S_1,\ldots, S_n\}$</code> turns the knot into the unknot. </p> <p>An easy example is the trefoil knot with a three-crossing projection. Let $S_1$ and $S_2$ each contain just one crossing from this projection. Than changing $S_1$, $S_2$ or $S_1\cup S_2$ yields the unknot. </p> <p>This notion of $n$-triviality is quite interesting because Goussarov first proved that a knot is $n$-trivial if and only if all finite type invariants vanish up to degree $n-1$. My question is about generalizing this notion to $\omega$-triviality. For this purpose, it is probably better to regard the crossing changes in $S_i$ as homotopies supported in neighborhoods of arcs connecting the knot to itself (finger moves.) Then one can define a knot to be $\omega$-trivial if there is a pairwise disjoint collection <code>$\{S_1,S_2,....\}$</code> where each $S_i$ is a set of finger moves on $K$, such that doing any nonempty subcollection <code>$\{S_i: i\in J\}$</code> for $\emptyset\neq J\subset \mathbb N$, of the finger moves gives you the unknot.</p> <p><strong>Question:</strong> If a knot is $\omega$-trivial, must it be the unknot?</p> <p>A knot which is $\omega$-trivial would have vanishing finite type invariants of all degrees, so it shouldn't exist, but this question should be a lot easier than the question of whether finite type invariants detect knottedness. Does anybody have any ideas on this question?</p> http://mathoverflow.net/questions/41611/omega-triviality-of-knots/41641#41641 Answer by Daniel Moskovich for $\omega$-triviality of knots? Daniel Moskovich 2010-10-10T00:53:12Z 2010-10-10T15:39:03Z <p>I would guess that you would need to understand the transfinite lower central series of the mapping class group. A knot complement which could not be distinguished from a solid torus by any finite lower central series quotient of the <strike>mapping class group</strike> Torelli group of a Heegaard surface would be n-trivial for all n, which if I'm not mistaken follows from work of [a subset of] Ted Stanford, Habiro, and Garoufalidis-Goussarov-Polyak. I don't see offhand why such an example could not exist.<br> Therefore, I think this question is open, and I can't see why it would be easy. </p>