The nonorientable 4–genus for knots with 8 or 9 crossings

Abstract

The non-orientable 4-genus of a knot in the 3-sphere is defined as the smallest first Betti number of any non-orientable surface smoothly and properly embedded in the 4-ball, with boundary the given knot. We compute the non-orientable 4-genus for all knots with crossing number 8 or 9. As applications we prove a conjecture of Murakami's and Yasuhara's, and give a new lower bound for the slicing number of knot.2010 Mathematics Subject Classification. 57M25 and 57M27.

The non-orientable 4-genus for knots with 10 crossings

“…Given a knot K, the non-orientable 4-genus or 4-dimensional crosscap number of K, denoted γ 4 (K) and defined by Murakami and Yasuhara [6], is the minimum first Betti number of non-orientable surfaces smoothly and properly embedded in the 4-ball D 4 and bounded by K. Among low-crossing knots, this knot invariant is currently only known for knots with crossing number up to 9, see [1,2]. The main result of this paper is a complete calculation of γ 4 for all 165 knots with 10 crossings.…”

“…In this section we review needed background material for computing γ 4 . We largely follow the outline from [1] and refer the intersted reader to consult said reference for more details.…”

“…Definition 2.1. (Definition 2.3. in [1]) A non-oriented band move on an oriented knot K is the operation of attaching an oriented band h = [0, 1]×[0, 1] to K along [0, 1]×∂[0, 1] in such a way that the orientation of the knot agrees with that of [0, 1]×0 and disagrees with that of [0, 1] × ∂1 (or vice versa), and then performing surgery on h, that is replacing the arcs [0, 1] × ∂[0, 1] ⊆ K by the arcs ∂[0, 1] × [0, 1]. Proposition 2.2.…”

“…Proposition 2.2. (Proposition 2.4. in [1]) If the knots K and K are related by a non-oriented band move then…”

On a nonorientable analogue of the Milnor conjecture

“…The knot invariant γ 4 (K) was introduced by Murakami and Yasuhara [16] only in the year 2000, and relatively little is known about it (for an overview of existing results on γ 4 see [6,8]). Two of the main tools for computing γ 4 are lower bounds coming from Heegaard Floer homology.…”