Once-punctured Klein bottles in knot complements

Seifert Klein bottles for knots with common boundary slopes

“…Proof of Corollary 1.6 That any crosscap number two torus knot is of the given form follows from the proof of Theorem 1.5. The uniqueness of the slope bounded by a Seifert Klein bottle in each case follows from [19], and that no such surface is π 1 -injective also follows from the proof of Theorem 1.5. For a knot K of the form T (•, 4n), any Seifert Klein bottle P is disjoint from the cabling annulus and can be constructed on only one side of the cabling annulus.…”

“…For any pretzel knot p(a, b, c) with a even, the black surface of its standard projection shown in Figure 18 is an algorithmic Seifert Klein bottle with meridian circle m, which has integral boundary slope ±2(b + c) by Lemma 2.1; an algorithmic Seifert surface is always unknotted. By [19], with the exception of the knots p(2, 1, 1) (which is the only knot that has two algorithmic Seifert Klein bottles of distinct slopes produced by the Now let F be the free group on x, y . If w is a cyclically reduced word in x, y which is primitive in F then, by [5] (cf [9]), the exponents of one of x or y , say x, are all 1 or all −1, and the exponents of y are all of the form n, n + 1 for some integer n. Finally, a word of the form x m y n is a proper power in F iff {m, n} = {0, k} for some |k| ≥ 2.…”

“…For any pretzel knot p(a, b, c) with a even, the black surface of its standard projection shown in Figure 18 is an algorithmic Seifert Klein bottle with meridian circle m, which has integral boundary slope ±2(b + c) by Lemma 2.1; an algorithmic Seifert surface is always unknotted. By [19], with the exception of the knots p(2, 1, 1) (which is the only knot that has two algorithmic Seifert Klein bottles of distinct slopes produced by the…”

“…In fact, by [13], any knot K ⊂ S 3 has at most two boundary slopes r 1 , r 2 corresponding to essential (incompressible and boundary incompressible, in the geometric sense) Seifert Klein bottles, and if so then ∆(r 1 , r 2 ) = 4 or 8, the latter distance occurring only when K is the figure-8 knot. The knots for which two such slopes exist were classified in [19] and, with the exception of the (2, 1, 1) and (−2, 3, 7) pretzel knots (the figure-8 knot and the Fintushel-Stern knot, respectively), all are certain satellites of 2-cable knots. Also, a minimal crosscap number Seifert surface for a knot need not even be essential in the knot exterior (cf [13]).…”

Klein slopes on hyperbolic 3-manifolds

“…In the following: We may note that there is a homeomorphism of W that sends a boundary component onto the other, thus M 2 , M 3 and M 4 do not depend on the choice of the boundary component. By [1,12] ∂ M 1 and ∂ M 2 both contain two Klein slopes with distance apart 8, and M 3 contains two Klein slopes with distance apart 6. By [10, figure 2], ∂ M 4 contains two Klein slopes with distance apart 5.…”