3-braid knots with maximal 4-genus

Abstract

We classify 3-braid knots whose topological 4-genus coincides with their Seifert genus, using McCoy's twisting method and the Xu normal form. In addition, we give upper bounds for the topological 4-genus of positive and strongly quasipositive 3-braid knots.

Figures (5)

• Figure 1: The generators $a$ and $b$ of the braid group $B_3$ and the elements $x=a^{-1}ba$, $\delta=ba$ and $\Delta=aba$. The latter is used for the Garside normal form (further down in the text).
• Figure 2: Isotopy (denoted $\approx$) from the closure of the braid $a^{u_1}b^{u_2}x^{u_3}$ to the pretzel knot $P(u_1,1,u_2,u_3)\approx P(u_1,u_2,u_3,1)$; here $(u_1,u_2,u_3)=(4,3,5)$.
• Figure 3: Two examples of twists, at the locations marked by $\star$. Here, the boundary of the respective disc $D$ is drawn in blue; in subsequent figures, it will be omitted.
• Figure 4: Left: Saddle move. Middle: How to use isotopy and a saddle move to add or remove a braid crossing. Right: Example of a cobordism between $\mathop{\mathrm{cl}}\nolimits(\alpha)$ and $\mathop{\mathrm{cl}}\nolimits(\alpha\cdot abx)$ that consists of three saddle moves, for some $3$-braid $\alpha$.
• Figure 7: In blue, the graph of the Levine--Tristram signature $\sigma_{e^{2\pi i t}}(K)$ for $t \in \left[0,\frac{1}{2}\right]$ and $K$ the closure of the 3-braid $\left(a^2b^2\right)^8\left(a^5b^5\right)^4 \sim \delta^{12}\left(a^4b^4x^4\right)^2a^4b^4(xab)^5x$. In black, the linear approximation by GG for $t \in [0,\frac{1}{3}]$. The maximum absolute value $\widehat{\sigma}(K)$ of $\sigma_{e^{2\pi i t}}(K)$, which equals $|\sigma(K)| + 4 = |\sigma_{e^{2\pi i/3}}(K)| + 4$, is assumed between $0.3599$ and $0.3826$ (rounded). This graph was drawn with sage Sage, by computing the signatures of the Hermitian matrices $(1-\omega)A + (1-\overline{\omega})A^{\top}$, for $A$ a Seifert matrix of $K$, and $\omega$ between the roots of $\Delta_K$ on $S^1$.

Theorems & Definitions (30)

• Theorem 1.1
• Definition 2.1
• Theorem 2.2
• Lemma 2.3
• proof : Proof of the 'if' direction of \ref{['lem:bkl']}
• Theorem 2.4: BM1BM2
• Corollary 2.5
• proof
• Proposition 2.6: T
• Lemma 2.7