Crosscap numbers of 2-bridge knots

Abstract

We present a practical algorithm to determine the minimal genus of non-orientable spanning surfaces for 2-bridge knots, called the crosscap numbers. We will exhibit a table of crosscap numbers of 2-bridge knots up to 12 crossings (all 362 of them).

Crosscap numbers of pretzel knots

“…Now assume that F C is inessential. As claimed in Lemma 13 in [3], F C is in fact incompressible. Thus it must be boundary-compressible.…”

“…Thus it must be boundary-compressible. By boundary compressions, as argued in the proof of Theorem 1 in [3], an essential surface F E is obtained. Then it satisfies −χ (F E ) < −χ (F C ), and together with χ (F E ) 0 obtained in [4] and s(F C ) = 1…”

“…Next, in [7], he gave a formula for the crosscap numbers of torus knots. Recently, for two-bridge knots, Hirasawa and Teragaito established a practical algorithm to calculate their crosscap numbers in [3].…”

Crosscap Numbers of Two-component Links

“…Murakami and Yasuhara [6] brought a concrete calculation for the knot 7 4 which is the first example known to satisfy the equality above. It has been shown [3] that there exist numerous knots for which the equality holds.…”

Concordance crosscap number of a knot

“…For non-orientable surfaces, the first Betti number is taken as an invariant instead of the genus. The crosscap number γ(K), also known as the non-orientable genus, of K is defined to be the minimum first Betti number of non-orientable surfaces in S 3 bounding K. See [2,6,7,11,16] for studies on this invariant. The four-dimensional crosscap number γ * (K) is the minimum first Betti number of non-orientable surfaces in the 4-ball B 4 bounding K. Some results on four-dimensional crosscap numbers given by Murakami and Yasuhara can be found in [12].…”