Discussion and Questions
In this final chapter, we state some questions that arose from this work and speculate about future directions related to this project. In Sect. 10.1, we discuss modifications of the diagram D that preserve A-adequacy. In Sect. 10.2, we speculate about us
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Discussion and Questions
In this final chapter, we state some questions that arose from this work and speculate about future directions related to this project. In Sect. 10.1, we discuss modifications of the diagram D that preserve A-adequacy. In Sect. 10.2, we speculate about using normal surface theory in our polyhedral decomposition of MA to attack various open problems, for example the Cabling Conjecture and the determination of hyperbolic A-adequate knots. In Sect. 10.3, we discuss extending the results of this monograph to states other than the all-A (or all-B) state. Finally, in Sect. 10.4, we discuss a coarse form of the hyperbolic volume conjecture.
10.1 Efficient Diagrams To motivate our discussion of diagrammatic moves, recall the well-known Tait conjectures for alternating links: .1/ Any two reduced alternating projections of the same link have the same number of crossings. .2/ A reduced alternating diagram of a link has the least number of crossings among all the projections of the link. .3/ Given two reduced, prime alternating diagrams D and D 0 of the same link, it is possible to transform D to D 0 by a finite sequence of flypes. Statements (1) and (2) where proved by Kauffman [55] and Murasugi [74] using properties of the Jones polynomial. A shorter proof along similar lines was given by Turaev [95]. Statement (3), which is known as the “flyping conjecture” was proven by Menasco and Thistlethwaite [67]. Note that the Jones polynomial is also used in that proof. One can ask to what extend the statements above can be generalized to semiadequate links. It is easy to see that statements (1) and (2) are not true in this case: For instance, the two diagrams in Example 5.3 on p. 74 are both A-adequate, but have different numbers of crossings. Nonetheless, some information is known about D. Futer et al., Guts of Surfaces and the Colored Jones Polynomial, Lecture Notes in Mathematics 2069, DOI 10.1007/978-3-642-33302-6 10, © Springer-Verlag Berlin Heidelberg 2013
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10 Discussion and Questions
crossing numbers of semi-adequate diagrams: Stoimenow showed that the number of crossings of any semi-adequate projection of a link is bounded above by a link invariant that is expressed in terms of the 2-variable Kauffman polynomial and the maximal Euler characteristic of the link. As a result, he concluded that each semiadequate link has only finitely many semi-adequate reduced diagrams [91, Theorem 1.1]. In view of his work, it seems natural to ask for an analogue of the flyping conjecture in the setting of semi-adequate links. Problem 10.1. Find a set of diagrammatic moves that preserve A-adequacy and that suffice to pass between any pair of reduced, A-adequate diagrams of a link K. A solution to Problem 10.1 would help to clarify to what extent the various quantities introduced in this monograph actually depend on the choice of A-adequate diagram D.K/. Recall the prime polyhedral decomposition of MA D 0 S 3 nnSˇA introduced above, and let ˇK and K be as in Definition 9.15 on p. 149. ˇ 0 ˇ 0 ˇ Since ˇK
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