Handle number is not always realized by a minimal genus Seifert surface

Abstract

We construct genus one knots whose handle number is only realized by Seifert surfaces of non-minimal genus. These are counterexamples to the conjecture that the Seifert genus of a knot is its Morse-Novikov genus. As the Morse-Novikov genus may be greater than the Seifert genus, we define the genus $g$ Morse-Novikov number $MN_g(L)$ as the minimum handle number among Seifert surfaces for $L$ of genus $g$. Since, as we further show, the Morse-Novikov genus and the minimal genus Morse-Novikov number are additive under connected sum of knots, it then follows that there exists examples for which the discrepancies between Seifert genus and Morse-Novikov genus and between the Morse-Novikov number and the minimal genus Morse-Novikov number can be made arbitrarily large.

Figures (5)

• Figure 1: (a) A portion of $\Sigma$ near $J$ is shown with the annular neighborhood $A$ of $J$. (b) The portion $B\backslash A$ of the Hopf band $B$ is highlighted and the knot $K$ is indicated. (c) & (d) Attaching $B \backslash A$ to $A$ or $\Sigma \backslash A$ gives the Seifert surfaces $F$ and $G$.
• Figure 2: (a) The sutured manifold exterior $(M_A, \gamma_A)$ of the annulus $A$ is shown along with the surface $\Sigma_A$. (b) $\Sigma_A$ is tubed twice to form a Heegaard surface $\Sigma_A'$ for $(M_A, \gamma_A)$. (c) & (d) Two product disks for the decomposition of $M_A$ along $\Sigma_A'$. (e) The result of further decomposing along these product disks is a pair of simple compression bodies: handlebodies with one "spot".
• Figure 3: (a) The surface $G$ as a banding of $\Sigma \backslash A$. (b) The surface $G$ after isotoping the core arc of the band $B \backslash A$ into $A$. (c)& (d) A disk $D \subset \Sigma$ disjoint from $B\backslash A$ (above) and the boundary of a corresponding handlebody thickening $\bar{N}(G)$ of $G$ (below). (e) A view of the boundary of the sutured manifold $(M_G, \gamma_G)$ complementary to $G$ in $(\Sigma \backslash D) \times [-\epsilon, \epsilon]$.
• Figure 4: (a) The sutured manifold $(M_G, \gamma_G)$. (b) The Heegaard surface $S$ as it sits inside $M_G$. (c) The Heegaard surface $S$ on its own. (d) An 'exploded' view of $S$ for the pieces $S_+$, $S_0$, and $S_-$ of its construction.
• Figure 5: (a) The sutured manifolds $C_\pm$ resulting from decomposing $M_G$ along $S$. (b) A pair of product disks in each $C_\pm$. (c) The result of decomposing $C_\pm$ along these product disks. (d) An isotopy of the decompositions into more standard pieces.

Theorems & Definitions (16)

• Theorem 1.1
• Theorem 1.2
• Theorem 2.1
• proof
• Lemma 3.1
• proof
• Corollary 3.2
• Lemma 4.1: goda-murasugisum
• Theorem 4.2
• proof