On the Non-Orientable $3$- and $4$-Genera of a Knot: Connections and Comparisons

TL;DR

The paper introduces the Euler-normalized non-orientable genus $\hat{\gamma}_n^{\pm}$ as a unifying framework for studying non-orientable spanning surfaces in knot theory, connecting 3- and 4-dimensional genera, geography, and boundary-slope data. Using a geography perspective with boundary slopes, normal Euler numbers, and Gordon–Litherland and upsilon bounds, it derives sharp results for alternating knots and ties these to the Turaev genus, including a bound $\overline{\gamma}_3(K) \le g_T(K)$. It then analyzes genus gaps through both state-surface and edgepath techniques, obtaining exact and bound results for pretzel knots and torus knots, and demonstrates that Euler-normalized gaps can be controlled even when ordinary gaps are large. Finally, it shows additivity under connected sum, yielding families of knots with arbitrarily large gaps between $\hat{\gamma}_3^-$ and $\hat{\gamma}_4^-$, thereby highlighting the distinct behavior of Euler-normalized invariants. Together, these results illuminate the interplay between non-orientable surfaces, knot invariants, and 3- and 4-dimensional topology, with potential connections to the Jones polynomial and the Slope Conjecture.

Abstract

We define a new quantity, the Euler-normalized non-orientable genus, to connect a variety of ideas in the theory of non-orientable surfaces bounded by knots. This quantity is used to reframe non-orientable slice-torus bounds on the non-orientable $4$-genus, to bound below the Turaev genus as a measure of distance to an alternating knot, and to understand gaps between the $3$- and $4$-dimensional non-orientable genera of pretzel knots. Further, we make connections to essential surfaces in knot complements and the Slope Conjecture.

Figures (22)

• Figure 1: The addition of $\pm$-twisted band to a spanning surface or filling $F$ of $K$ raises the first Betti number by $1$ and changes the normal Euler number by $\pm 2$, but leaves invariant $\Gamma^\pm(F)$ and the isotopy class of the boundary. The sign convention for the normal Euler number means that the sign of the crossing is the opposite of the sign of the twist.
• Figure 2: The geography $R_*(K)$ is constrained by the Gordon-Litherland signature bound. The Euler-normalized first Betti number measures the vertical distance from the $(e,b)$ coordinates of a surface to the signature bound.
• Figure 3: A trefoil knot $K$ bounding a Möbius strip $F$ with $e_3(F) = -6$. Each crossing of $K$ induces four crossings (all positive) between $K$ and the pushoff $K'$.
• Figure 4: The three-dimensional geography $R_3(K)$ for the trefoil knot $K$, with each solid dot denoting a point realized by a spanning surface. The vertex at $(-6,1)$ is the Möbius strip in Figure \ref{['fig:trefoil-mobius']}, while the vertex $(2,3)$ is the positive twist of the Seifert surface. The Seifert surface itself is denoted by an open dot at $(0,2)$.
• Figure 5: A crossing of a knot diagram can be split with an $A$-resolution or a $B$-resolution.

Theorems & Definitions (47)

• Definition 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Corollary 1.5
• Conjecture 1.6
• Proposition 1.7
• Proposition 1.8
• Theorem 1.9
• Definition 2.1: Adapted from allen:geography