Narrow Results

Looking at the structure of minimal prime knot presentations, one can notice that there are often, perhaps always, segments that present either the trefoil or the figure-eight knot. This note explores the question as to whether this is always the case, reporting on conversations with Jablan Slavik that began at a $2009$ conference in Trieste, Italy and which never reached a conclusion. Evidence supporting this conjectured presence is reported and potential consequences are described.

The structure of classical minimal prime knot presentations suggests that there are often, perhaps always, subsegments that present either the trefoil or the figure-eight knot. A comprehensive study of the subknots of the minimal prime knot presentations through 15 crossings shows that this is always the case for these knot presentations. Among this set of 313, 258 prime knot presentation, there are only 547, or 0.17%, that do not contain a trefoil subknot. Thus, 99.83% of minimal prime knot presentations through 15 crossings contain trefoil subknots. We identify several infinite minimal alternating prime knot families that do not contain trefoil subknots but always contain figure-eight knots. We discuss the statistics of subknots of prime knots and, using knot presentation fingerprints, illustrate the complex character of the subknots of these classic minimal prime knot presentations. We conclude with a discussion the conjectures and open questions that have grown out of our research.