Hard to identify (non-)mutations

Math. Proc. Camb. Phil. Soc. (2006), 141, 281 c 2006 Cambridge Philosophical Society Graduate School of Mathematical Sciences, University of Tokyo, 3-8-1, Komaba, Tokyo 153-8914, Japan. WWW: http://www.ms.u-tokyo.ac.jp/∼stoimeno (Received 6 June 2005; revised 23 August 2005) We show examples that mutants may not be visible as such from minimal crossing diagrams, and of knots with the same polynomial invariants and same hyperbolicvolume, which are not mutants.
The term mutation was coined by Conway [3], and has become known as a way of creating difficult to distinguish knots. Using simple skein arguments, onecan prove that the various knot polynomials [2, 5, 10, 11, 18] do not distinguishmutants. (Here, and below, the term ‘knot polynomials’ refers to the HOMFLY andKauffman polynomial, and their variously popular special cases, as applied on theknots themselves, and not on any of their satellites. Each time we refer to the ap-plication of knot polynomials on a satellite we will specify it explicitly.) This wasextended by Lickorish and Lipson [17] and Przytycki [26] to polynomials of 2-cablesof mutants, and by Morton and Traczyk to the Jones polynomial of all cables [22].
The cabling formula for the Alexander polynomial (see for example [16, theorem6·15]) likewise establishes its failure in this regard. Ruberman [27] showed thatmutants have equal hyperbolic volume. This series of results excludes most of theefficient yet efficiently computable invariants as tools to distinguish mutants. Suc-cessful tools come from knot group representations, for example twisted Alexanderpolynomials [15]. Some (other) hyperbolic invariants do distinguish mutants too, aswell as (when computable) knot Floer homology [25].
Mutation is, as many other properties of knots, diagrammatic, that is, (can be) defined by the existence of proper diagrams of the knots. Seeking such diagrams, oneis naturally led to consider first minimal crossing diagrams. The question whether onecan decide on knot properties from minimal crossing diagrams has a long tradition.
A celebrated result [12, 23, 29] says that alternation is exhibited in minimal crossingdiagrams. For some other properties counterexamples are known. In particular,Bleiler [1] and Nakanishi [24] observed that minimal crossing diagrams may not havethe correct unknotting number. However, it remains an open question whether onecan obtain the correct unknotting number by interchangingly switching crossings † Supported by JSPS Postdoc grant P04300.
Fig. 1. Two mutant pairs, where the mutation is not visible from minimal crossing diagrams.
and going over to a proper minimal diagram of the intermediate knots. (This inparticular would imply that unknotting number one is achieved in a proper minimaldiagram.) Among the intuitive (but not generally directly decidable) knot properties and invariants, the effect of mutation is not entirely understood, but in the studiedcases not always trivial. Gabai [6] showed that the Kinoshita–Terasaka and Conwaymutants have distinct genera. His description of fiber pretzel surfaces [7] shows alsothat there are (same genus Montesinos) mutant knots, one of which is fibered and theother one not. Livingston’s result [19] implies in particular that the smooth 4-ballgenus is not always a mutation invariant (see also [13]). A long-standing question ofBoileau (see [14, problem 1·69 (C)]) asks if the unknotting number is invariant undermutation. (Gordon–Luecke announced recently a positive solution for unknottingnumber one.) Among certain classes of knots, possible mutations can be determined. Using the study of 2-branched covers, it was established [8] that 2-bridge knots (andin particular the unknot) have no mutants. Also, mutants of Montesinos knots areagain Montesinos knots. From the classification of Menasco [20] of Conway spheres inalternating link complements, it is not too hard to deduce that if an alternating knothas a mutant, this mutant is also alternating, and the mutation is visible (possibly asan iterated mutation) in the alternating diagrams. This, together with [12, 23, 29],shows that for alternating knots all minimal crossing diagrams exhibit the mutation.
In the following we will give examples of knots, where mutation cannot be exhibited Example 1. Figure 1 shows the knots 1446187 and !1443907 in the tables of [9]. (!K is the mirror image of K.) The 15 crossing diagrams above show that these knotsare mutants. However, both knots have unique 14 crossing diagrams shown below,which cannot display the mutation. Note that the diagrams show that these knots Fig. 2. A pair of difficult to prove non-mutants, and a pair where existence of mutation has not are semi-homogeneous fibered knots with no semi-homogeneous minimal crossingdiagrams.
Another pair are 1442947 and !1443476 shown on the right. They both have a few 14 crossing diagrams, but none admits a non-trivial tangle decomposition (i.e. as a sumof two non-rational tangles). Both knots have trivial Alexander polynomial. 1442947has genus 3 but no minimal crossing diagrams of minimal genus, while !1443476 has such diagrams and is of genus (and canonical genus) 3.
Example 2. The knots 1441721 and 1442125 on Figure 2 may be potentially another “hidden” mutant pair. They have the same polynomial invariants, the same volume(at least up to 10−10), and the same (arbitrarily framed) 2-cable HOMFLY polyno-mials. (In order to verify the coincidence of polynomials for all framings, it suffices tocheck the blackboard framed 2-parallel of some diagram with one half-twist, and theblackboard framed 2-parallel of the mirrored diagram with a half-twist of the samesign; see [28].) But up to 16 crossings I could not find diagrams of any of them witha non-trivial tangle decomposition. Finally, calculation of the (blackboard framed)Whitehead double HOMFLY polynomial showed that these knots are not mutants.
Another such pair is 1441763 and 1442021. By a simple skein argument one sees thatWhitehead double HOMFLY polynomials are essentially equivalent in distinguish-ing knots as are HOMFLY polynomials of the reverse 2-strand cable link (with thesame framing). Thus Whitehead double HOMFLY polynomials will fail distinguish-ing mutants in the same way the reverse 2-cable polynomials do [17]. (Whiteheaddoubles were given preference here for merely technical reasons.) Previously there have been found pairs of 12 crossing knots [28] where the polyno- mials and the 2-cable HOMFLY polynomials coincide but the volumes differ. TheirWhitehead double HOMFLY polynomials also differ in the pairs for which theywere calculated. (This calculation was suggested to me by Taizo Kanenobu.) Thus itseems, peculiarly, that the efficiency of the volume to identify mutants lies somewherebetween that of the HOMFLY polynomial of the 2-cable and the HOMFLY poly-nomial of the Whitehead double. The 2-cable (or equivalently, Whitehead double)Kauffman polynomial distinguishes too 1441721 and 1442125, but its calculation effortcan hardly be considered reasonable.
Example 3. Even more problematic is the pair 1441739 and 1442126. It shares the features of the two pairs in the previous example 2, except that now also (arbitrarily framed) Whitehead double HOMFLY polynomials coincide. So far I could not cal-culate the 2-cable Kauffman (or 3-cable Jones) polynomials. So it seems extremelyhard to decide if these knots are mutants or not. I tried out some 15-crossing pairsof similar status (no mutated diagrams of FLY polynomials coincide). I could calculate 2-cable Kauffman polynomials (for oneframing at least) for two pairs, (15148731, 15156433) and (15210638, 15213436), and thepolynomials coincided.
The calculation of polynomial invariants uses the skein algorithm of Millett–Ewing (provided in KnotScape [9]). It could be completed in decent time except for the 2-cable Kauffman polynomials, where, if it was feasible1, it typically took between halfand one week for each knot (and so makes it difficult to examine a larger number ofpairs as in Example 3).
It is interesting to also compare Jones polynomials of higher cables. Unfortunately, so far I have found no efficient general program to do this. (The skein method is quitecertainly out of question.) Four pairs of 15 crossing knots of the status of Example 2and eight of the status of Example 3 have braid index 4, and I could use Morton’srepresentation theoretic program [21] to obtain the 3-cable Jones polynomials, whichall coincided.
The above presented examples came about in attempts to obtain a list of mutant groups among the knots tabulated in [9]. A pre-selection shows among the about1.7 million knots ≈ 200 000 groups of knots with equal volume, Alexander polynomialand Jones polynomial. Using a self-written code in C++, that generates mutationsand allows tangle decompositions to be found, I determined the list of all mutantgroups of prime knots through 13 crossings. (The result is available on my webpage.
There are in total 865 groups, consisting of 789 pairs, 43 triples, 32 groups of 4 andone group of 6.) It confirms, at least in length, a compilation up to 12 crossings doneshortly previously by hand by David De Wit, see [4]. Up to 13 crossings, in all thepre-selected groups of knots the mutations could be explicitly exhibited (in minimalcrossing diagrams). But, as seen, a careful identification or exclusion of mutations ingroups of more crossings would require considerable effort. A natural question is: Question 1. Are there mutants with different crossing numbers? Clearly, if such knots exist, they would be a sharpening of examples of the above type. As remarked, there are no such alternating mutants. The groups so far obtainedshow also that there are no such pairs among prime Acknowledgements. I wish to thank to David De Wit, Taizo Kanenobu, and the referee for some helpful remarks, and my JSPS host Prof. T. Kohno for his support.
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