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An application of TQFT: determining the girth of a knot. (English) Zbl 1160.57005
This paper defines an invariant of knots called girth, which it suggests as a tool for knot tabulation. Girth minus one bounds from above the Heegaard genus of the double branched cover of the knot complement. Knots of girth \(2\) are \(2\)–bridge knots with (alternating) Conway notation \(pq\) or \(p1q\), and the the authors pose the interesting problem of classifying knots of girth \(3\).
The reader should ignore the “application of TQFT” of the title, which occurs in sections 4 and 5 which set out to show that \(9_{44}\) has girth \(4\). The girth of \(9_{44}\) is in fact \(3\), as may be seen for example from its knot diagram corresponding to DT notation \(4\;-10\;20\;14\;-2\;-16\;-18\;8\;-12\;6\) (pointed out to the reviewer by Alexander Stoimenow). The claimed proof “shows” that the Heegaard genus of the branched double cover of \(9_{44}\) is \(3\), whereas it is in fact equal to \(2\) by Theorem 5 of [J. S. Birman and H. M. Hilden, Trans. Am. Math. Soc. 213, 315–352 (1975; Zbl 0312.55004)] (see also [O. J. Viro, Math. USSR, Sb. 16, 223–236 (1972; Zbl 0248.55002)]).
MSC:
| 57M25 | Knots and links in the \(3\)-sphere (MSC2010) |