Current research on inverse knot equivalence?

Question

What is the current status of the open problem in knot theory 'When is a knot equivalent to its inverse?'

Additionally, I would like to know what work has been done on this problem (I cannot find anything on it besides the problem name upon google searching), and any information including explanations about it. Thank you.

Answer 1

A knot is called amphichiral if it is isotopic to its mirror and chiral otherwise.

http://mathworld.wolfram.com/AmphichiralKnot.html is perhaps good place to start. Jones' result at the bottom is quite a strong obstruction.

Kauffman, Murasugi (independently) and Thistlewaite (also independently) in 1987 proved that for an alternating knot $K$ with an odd number of crossings the Jones polynomial of $K$ is not equal to the Jones polynomial of its mirror $-K$.

In general, the only examples of knots known to be smoothly or topologically concordant to their mirrors are either amphichiral or slice and many concordance invariants can naturally obstruct amphichirality. Recall that a knot $K$ is concordant to $K'$ iff $K\#-K'$ bounds a smoothly slice disk in the 4-ball, and that $K \# -K$ always bounds such a disk (in fact this disk is ribbon). So a necessary condition for amphichirality is for $K\#K$ to be smoothly slice. So if $\nu$ is a concordance invariant which is additive under connect sum and a lower bound on smooth four genus ($\nu(K) \leq g_4(K)$), then if $\nu(K)>0$, $K$ must be chiral.

e.g. if a knot admits a Legendrian representative with $tb(K)+|r(K)|+1>0$ then $K$ cannot even be concordant to its mirror otherwise $K\#K$ would be slice yet violate the adjunction inequality (in other words the maximum of $(tb(K)+|r(K)|+1)/2$ is such a $\nu$). Similar obstructions can be obtained from other additive concordance invariants such as Rassmussen's $s$-invariant (suitably normalized) and Ozvath-Szabo's $\tau$-invariant, and these all prove for instance that any non-trivial torus knot is chiral.

Answer 2

It's also useful to look at braids here. For example, the maximal self-linking number of an amphichiral link equals the negative of the minimal braid index.

(Note: The self-linking number of a braid $\beta$ is $sl(\beta)=w(\beta)-n(\beta)$, where $w$ is the writhe of the braid diagram and $n$ is the braid index. Then we get a knot invariant $\overline{sl}(K)$ by taking the largest $sl(\beta)$ among all braid representatives of $K$.)

By somewhat recent work of Dynnikov-Prasolov (and independent work of LaFountain-Menasco using braid theory), it turns out that a minimal braid index representative of a link always achieves the maximal self-linking number. If $\beta$ realizes the minimal braid index for an amphichiral knot $K$, then so does the mirror $m(\beta)$. This implies \begin{align*}\overline{sl}(K)&= sl(\beta)=w(\beta)-n(\beta)\\ \overline{sl}(K)&=sl(m(\beta))=w(m(\beta))-n(m(\beta))=-w(\beta)-n(\beta).\end{align*} It follows that $w(\beta)$ equals zero or, equivalently, that $\overline{sl}(K)$ equals $-n(\beta)$.