Crosscap numbers and the Jones polynomial

Abstract

We give sharp two-sided linear bounds of the crosscap number (non-orientable genus) of alternating links in terms of their Jones polynomial. Our estimates are often exact and we use them to calculate the crosscap numbers for several infinite families of alternating links and for several alternating knots with up to twelve crossings. We also discuss generalizations of our results for classes of non-alternating links.

Crosscap numbers of alternating knots via unknotting splices

“…[1, 4, 7], [2, 1, 3], [5,4,2,3,6], [5,6,7] Crossings around B-faces: [4,5,7], [5,6], [6,3,1,7] Edges around A-faces: [14,4,10], [8, 1, 3], [12,5,9,2,7], [13,6,11] Edges around B-faces: [4, 1, 9], [2, 8], [10,5,13], [6,12], [11,7,3,14] D G [n] and find a Gauss code D G [n][K] for a reduced alternating diagram of K . Then we clean up this data by replacing each Gauss code with its reduced form.…”

“…Currently, knotinfo lists crosscap numbers for 174 of the 367 prime alternating knots with 11 crossings and for 316 of the 1288 with 12 crossings [1]. Most of these values, and the upper and lower bounds for the remaining 11-and 12-crossing knots, come from either Burton-Ozlen, using normal surfaces [5], or from Kalfagianni-Lee, using properties of the colored Jones polynomial [11]. Interestingly, every new crosscap number we compute through 12 crossings matches the upper bound currently given on knotinfo.…”

Crosscap number and knot projections

“…(2) If m ≤ 2, then we apply the splice(s) to the crossing(s) so that the mgon becomes a state circle. If m > 2, then m = 3 by a simple Euler characteristic argument on the knot projection (see, e.g., [13,Lemma 3.1] or [10,Lemma 2]). Then, choose a triangle of D P .…”

“…For a given n crossing alternating knot diagram, consider 2 n − 1 non-orientable state surfaces (Definition 9); some of these surfaces achieve the crosscap number of the knot. By using coefficients of the colored Jones polynomials to establish two sided bounds on crosscap numbers, Kalfagianni and Lee [13] improve the efficiency of these computations. They apply this improved efficiency in order to calculate hundreds of crosscap numbers explicitly and rapidly.…”

Crosscap number and epimorphisms of two-bridge knot groups

“…The crosscap number has also been computed for any torus knot [21], any alternating knot [14], or any pretzel knot, [11]. For alternating links, K, lower and upper bounds for γ(K) that depend on the Jones polynomial of K are given in [12]. For alternating knots with 12 or fewer crossings, these bounds are often sufficient to determine γ(K).…”