A knot diagram is alternating if the crossings alternate over, under, over, under, etc. as one travels along the knot. A knot is alternating if it has an alternating diagram. Alternating knots have many important properties, but their invariants are not completely understood.
In this post, I would like to consider two questions.
What are some nice properties of alternating knots?
What are some open conjectures about alternating knots?
For the first question, "nice" can mean that invariants are easier to compute or structure is easier to describe than in the general case. For the second question, I am looking for either conjectures about alternating knots specifically, or conjectures about a class of knots containing alternating knots, but where the answer could somehow be easier or special for alternating knots.
I'll start by listing some answers to both questions. My list of open conjectures is much more meager than my list of nice properties.
Nice properties of alternating knots.
The Seifert genus of an alternating link is the degree of the symmetrized Alexander polynomial and is the genus of the Seifert surface obtained from Seifert's algorithm applied to an alternating diagram. [Murasugi 1958, Crowell 1959]
An alternating knot is either hyperbolic or a $(2,q)$-torus knot. [Menasco 1984]
An alternating diagram has the fewest number of crossings among all diagrams of the knot. [Kauffman 1987, Murasugi 1987, Thistlethwaite 1987]
Any two diagrams of the same alternating knot are related by a sequence of flypes. [Menasco, Thistlethwaite 1993]
The knot Floer homology of an alternating knot is determined by its Alexander polynomial and its signature. [Ozsvath and Szabo 2003]
The Khovanov homology of an alternating knot is determined by its Jones polynomial and its signature. [Lee 2005]
Alternating knots have topological characterizations in terms of their spanning surfaces. Consequently, there is an algorithm to decide whether a knot is alternating. [Greene 2017, Howie 2017]
Open conjectures about alternating knots.
- Fox's trapezoidal conjecture. Fox conjectured that the coefficients of the Alexander polynomial of an alternating knot are unimodal. See this MathOverflow post for progress on the conjecture. There is a strengthening of the conjecture asserting that the coefficients of the Alexander polynomial of an alternating knot are log-concave (see [Stoimenow 2014]).
Crowell, Richard. Genus of alternating link types. Ann. of Math. (2) 69 1959, 258-275.
Greene, Joshua Evan. Alternating links and definite surfaces. With an appendix by András Juhász and Marc Lackenby. Duke Math. J. 166 (2017), no. 11, 2133–2151.
Howie, Joshua. A characterisation of alternating knot exteriors. Geom. Topol. 21 (2017), no. 4, 2353–2371.
Kauffman, Louis. State models and the Jones polynomial, Topology 26 (1987), no. 3, 395–407.
Lee, Eun Soo. An endomorphism of the Khovanov invariant. Adv. Math. 197 (2005), no. 2, 554–586.
Menasco, W. Closed incompressible surfaces in alternating knot and link complements, Topology 23 (1984), no. 1, 37–44.
Menasco, William and Thistlethwaite, Morwen. The classification of alternating links. Ann. of Math. (2) 138 (1993), no. 1, 113–171.
Murasugi, Kunio. On the genus of the alternating knot, I, II.J. Math. Soc. Japan 10 1958, 94–105, 235–248.
Murasugi, Kunio. Jones polynomials and classical conjectures in knot theory, Topology 26 (1987), no. 2, 187–194.
Ozsváth, Peter and Szabó, Zoltán. Heegaard Floer homology and alternating knots. Geom. Topol. 7 (2003), 225–254.
Stoimenow, Alexander.Log-concavity and zeros of the Alexander polynomial. Bull. Korean Math. Soc. 51 (2014), no. 2, 539–545.
Thistlethwaite, Morwen B. A spanning tree expansion of the Jones polynomial, Topology 26 (1987), no. 3, 297–309.
Thistlethwaite, Morwen B. Kauffman's polynomial and alternating links, Topology 27 (1988), no. 3, 311–318.