Narrow Results

Ito–Takimura recently defined a splice-unknotting number $u^{-}(D)$ for knot diagrams. They proved that this number provides an upper bound for the crosscap number of any prime knot, asking whether equality holds in the alternating case. We answer their question in the affirmative. (Ito has independently proven the same result.) As an application, we compute the crosscap numbers of all prime alternating knots through at least 13 crossings, using Gauss codes.

For a torus knot K, we bound the crosscap number c(K) in terms of the genus g(K) and crossing number n(K) : c(K) ≤ ⌊(g(K)+9)/6⌋ and c(K) ≤ ⌊(n(K) + 16)/12⌋. The (6n - 2,3) torus knots show that these bounds are sharp.

This paper is concerned with the crosscap numbers of Montesinos knots. For a family of Montesinos knots, we give a lower bound of $-\chi /\#s$ among all essential surfaces $F$ embedded in the exterior, where $-\chi /\#s$ denotes the ratio of negative Euler characteristic of the surface and the number of sheets. From this result, we determine the crosscap numbers of a family of Montesinos knots. Our method relies on the algorithm of enumerating all essential surfaces for Montesinos knots given by Hatcher and Oertel.

We compare the values of the nonorientable three genus (or, crosscap number) and the nonorientable four genus of torus knots. In particular, let $T(p,q)$ be any torus knot with $p$ even and $q$ odd. The difference between these two invariants on $T(p,q)$ is at least $\frac{k}{2}$, where $p=qk+a$ and $0<a<q$ and $k\ge 0$. Hence, the difference between the two invariants on torus knots $T(p,q)$ grows arbitrarily large for any fixed odd $q$, as $p$ ranges over values of a fixed congruence class modulo $q$. This contrasts with the orientable setting. Seifert proved that the orientable three genus of the torus knot $T(p,q)$ is $\frac{1}{2}(p-1)(q-1)$, and Kronheimer and Mrowka later proved that the orientable four genus of $T(p,q)$ is also this same value.

We consider the relationship between the crosscap number $\gamma$ of knots and a partial order on the set of all prime knots, which is defined as follows. For two knots $K$ and $J$, we say $K\ge J$ if there exists an epimorphism $f:{\pi }_1(S^3-K)\to {\pi }_1(S^3-J)$. We prove that if $K$ and $J$ are 2-bridge knots and $K>J$, then $\gamma (K)\ge 3\gamma (J)-4$. We also classify all pairs $(K,J)$ for which the inequality is sharp. A similar result relating the genera of two knots has been proven by Suzuki and Tran. Namely, if $K$ and $J$ are 2-bridge knots and $K>J$, then $g(K)\ge 3g(J)-1$, where $g(K)$ denotes the genus of the knot $K$.

Given a knot in the 3-sphere, the non-orientable 4-genus or 4-dimensional crosscap number of a knot is the minimal first Betti number of non-orientable surfaces, smoothly and properly embedded in the 4-ball, with boundary the knot. In this paper, we calculate the non-orientable 4-genus of knots with crossing number 10.

We investigate the nonorientable 4-genus ${\gamma }_4$ of a special family of 2-bridge knots, the double twist knots $C(m,n)$. Because the nonorientable 4-genus is bounded by the nonorientable 3-genus, it is known that ${\gamma }_4(C(m,n))\le 3$. By using explicit constructions to obtain upper bounds on ${\gamma }_4$ and known obstructions derived from Donaldson’s diagonalization theorem to obtain lower bounds on ${\gamma }_4$, we produce infinite subfamilies of $C(m,n)$ where ${\gamma }_4=0,1,2,$ and $3$, respectively. However, there remain infinitely many double twist knots where our work only shows that ${\gamma }_4$ lies in one of the sets $\{1,2\},\{2,3\}$, or $\{1,2,3\}$. We tabulate our results for all $C(m,n)$ with $\left|m\right|$ and $\left|n\right|$ up to 50. We also provide an infinite number of examples which answer a conjecture of Murakami and Yasuhara.

The non-orientable 4-genus of a knot $K$ in $S^3$ is defined to be the minimum first Betti number of a non-orientable surface $F$ smoothly embedded in $B^4$ so that $K$ bounds $F$. We will survey the tools used to compute the non-orientable 4-genus, and use various techniques to calculate this invariant for non-alternating 11 crossing knots. We will also view obstructions to a knot bounding a Möbius band given by the double branched cover of $S^3$ branched over $K$.