Non-Orientable 4-Genus in Knot Theory

• Non-orientable 4-genus is defined as the minimal first Betti number of a non-orientable surface in the 4-ball bounding a knot, capturing its topological complexity.
• Floer-theoretic techniques, including Heegaard Floer d-invariants and the Upsilon invariant, provide sharp lower bounds and demonstrate that this invariant can be arbitrarily large.
• Band-move constructions and classical invariants like the signature and Casson–Gordon invariants interlock to refine the geography of (e, b₁) pairs and enhance our understanding of knot concordance.

The non-orientable four-genus of a knot, denoted $\gamma_4(K)$, is the minimal first Betti number $b_1(F)$ of a non-orientable surface $F \subset B^4$ smoothly embedded in the 4-ball $B^4$ with boundary $\partial F = K \subset S^3$. This invariant quantifies the complexity of non-orientable surfaces bounding $K$ in four-dimensional topology and is fundamental for studying knot concordance, slicing obstructions, and the interplay between classical and modern gauge-theoretic invariants. The non-orientable 4-genus is strictly greater than zero for any non-slice knot and can be arbitrarily large, as shown by infinite families of knots.

1. Definition and Basic Properties

Given a smooth knot $K \subset S^3$, a non-orientable surface $F \subset B^4$ with $\partial F = K$ has first Betti number $b_1(F)$. The non-orientable 4-genus is defined by: $$\gamma_4(K) = \min \{\, b_1(F) \mid F \subset B^4 \text{ smoothly embedded, non-orientable}, \ \partial F = K \,\}$$ If $K$ is slice (bounds a disk), then $\gamma_4(K) = 1$, as the disk may be modified into a Möbius band with $b_1=1$ (Batson, 2012). The invariant satisfies $1 \leq \gamma_4(K) \leq \gamma_3(K)$, where $\gamma_3(K)$ is the non-orientable genus in $S^3$ (Jabuka et al., 2019).

2. Fundamental Lower Bounds and Floer-Theoretic Techniques

The initial lower bounds for $\gamma_4(K)$ are provided by classical signature and quadratic form constraints. The Gordon–Litherland signature theorem gives: $|\, \sigma(K) - e(F)/2\,| \leq b_1(F)$ where $e(F)$ is the normal Euler number of $F$ (which satisfies $e(F) \equiv 2 b_1(F) \pmod{4}$) (Gilmer et al., 2010). Building on this, Batson employed Heegaard Floer correction terms $d(S^3_{-1}(K))$: $$\gamma_4(K) \geq \frac{1}{2}\,\sigma(K) - d(S^3_{-1}(K))$$ This bound is sharp for many families and implies that $\gamma_4$ is unbounded across knots; for $T(2k,2k-1)$ one gets $\gamma_4(T(2k,2k-1)) = k-1$ (Batson, 2012). More generally, Floer-theoretic obstructions prove that there exist infinitely many knots with arbitrarily large non-orientable 4-genus (Sato, 2014).

Recent advances utilize the Upsilon invariant $\Upsilon_K(1)$: $$|\, \Upsilon_K(1) - \sigma(K)/2\,| \leq \gamma_4(K)$$ which often aligns with the signature for alternating and quasi-alternating knots, but in general yields nontrivial lower bounds (Sabloff, 2022, Jabuka et al., 2018).

3. Band-Move Constructions, Pinch Moves, and Exact Computations

Non-orientable surfaces in $B^4$ are often constructed by sequences of non-orientable band moves, called pinch moves in the context of torus knots. For $T(p,q)$, repeated pinch moves reduce the knot to the unknot, each band increasing $b_1(F)$ by at most 1, so the minimal number of pinch moves $n$ gives an upper bound $\gamma_4(T(p,q)) \leq n$ (Jabuka et al., 2018). Batson conjectured that this upper bound is often sharp, i.e., $\gamma_4 = n$ for infinite subfamilies of torus knots, though explicit counterexamples exist (such as $T(4,9)$, where $\gamma_4 = 1$ despite $n=2$) (Sabloff, 2022, Sinha, 16 Jul 2025).

Recent results established the exact values for infinite families of torus knots:

• $\gamma_4(T_{4n, (2n\pm1)^2+4n-2}) = 2n-1$, with non-orientable genus strictly less than the pinch-move number $\vartheta=2n$ (Sinha, 16 Jul 2025).
• $\gamma_4(T_{2k,2k-1}) = k-1$ by explicit surface construction matching the lower bound (Batson, 2012).

Counterexamples to Batson's conjecture demonstrate that $\gamma_4$ can differ from the pinch-move norm by at most 1 in known cases (Sinha, 16 Jul 2025, Binns et al., 2021).

4. Geography Problem: $(e(F), b_1(F))$ Pairs

The relationship between the normal Euler number $e(F)$ and the first Betti number $b_1(F)$ leads to the geography problem: characterizing all possible pairs $(e(F), b_1(F))$ for non-orientable surfaces bounding a knot $K$ (Sabloff, 2022, Allen, 2020, Feller et al., 2020). Massey's parity constraint enforces $e(F) \equiv 2 b_1(F) \pmod{4}$. The possible pairs lie in the intersection of "wedges" determined by signature and Upsilon inequalities: $$|\,\sigma(K) - e/2\,| \leq b_1(F), \quad |\,\Upsilon_K(1) - e/2\,| \leq b_1(F)$$ Heegaard Floer $d$-invariants for double branched covers further eliminate portions of the admissible region, refining the geography and producing strict inequalities for infinite families (Allen, 2020). In select subfamilies (notably "JVC-knots"), one can completely classify the realizable $(e,b_1)$ pairs.

5. Link to Other Knot Invariants and Non-Orientable Sliceness

$\gamma_4$ is often strictly larger than the orientable slice genus $g_4(K)$. For torus knots, Seifert and Kronheimer–Mrowka proved $g_4(T(p,q)) = \frac{(p-1)(q-1)}{2}$, but the non-orientable genus can grow much faster, with the gap $\gamma_3-\gamma_4$ (where $\gamma_3$ is the crosscap number) arbitrarily large for $T(p,q)$ with even $p$ and odd $q$ (Jabuka et al., 2019). For double-twist knots $C(m,n)$, explicit constructions show all possible values $\gamma_4 = 0,1,2,3$ occur, with thorough tables for small $|m|, |n|$ (Hoste et al., 2022).

The classical invariants, particularly the signature and Arf invariant, obstruct non-orientable sliceness. If $\sigma(K)+4\operatorname{Arf}(K) \equiv 4 \pmod{8}$, then $\gamma_4(K) \geq 2$, forbidding Möbius band fillings (Gilmer et al., 2010).

6. Modern Obstructions: Linking Forms, $d$-invariants, and Casson-Gordon Theory

The Murakami–Yasuhara criterion uses the linking form $\lambda$ on the double branched cover $\Sigma_2(K)$ to obstruct Möbius band fillings: for $H_1(\Sigma_2(K)) \simeq \mathbb{Z}_n$, the generator $a$ must have $\lambda(a,a) = \pm 1/n$ (Gilmer et al., 2010, Hoste et al., 2022). Casson–Gordon invariants provide further linear lower bounds by examining characters on $H_1(\Sigma_2(K))$, producing infinite families of knots with arbitrarily large non-orientable ribbon genus (Gilmer et al., 2010).

Heegaard Floer $d$-invariants for (–1)-surgery or for $\Sigma_2(K)$ obstruct small non-orientable genus for both smooth and ribbon cases (Batson, 2012, Allen, 2020).

7. Extensions, Equivariant and Topological Variants

Generalization to punctured 4-manifolds and periodic settings reveals further phenomena. For any closed, simply-connected spin 4-manifold $X$, the null-homologous non-orientable 4-genus $\gamma_X^0(K)$ is unbounded; explicit lower bounds involve knot and manifold signatures (Sato, 2014). In the topological (locally-flat) category, the non-orientable genus can be smaller due to Freedman’s machinery, which allows capping curves with disks absent smooth constraints. For locally-flat Möbius bands, subtle number-theoretic criteria control fillability (Feller et al., 2020).

Equivariant non-orientable 4-genus $\gamma_4^{G}(K)$ for periodic knots can exceed the classical $\gamma_4(K)$, reflecting symmetry constraints (Grove et al., 2021).

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Table: Lower Bounds for $\gamma_4(K)$

Bound Type	Formula/Condition	Reference
Signature/Euler number (Gordon–Litherland)	$|\,\sigma(K) - e(F)/2\,| \leq b_1(F)$	(Gilmer et al., 2010)
Signature plus $d$-invariant (Batson)	$$\frac{1}{2}\sigma(K) - d(S^3_{-1}(K)) \leq \gamma_4(K)$$	(Batson, 2012)
Arf, signature congruence (Yasuhara, Gilmer–Livingston)	$$\sigma(K)+4\,\operatorname{Arf}(K) \equiv 4 \pmod{8}$$ need $\gamma_4(K)\geq 2$	(Gilmer et al., 2010)
Upsilon invariant (Ozsváth–Stipsicz–Szabó)	$|\,\Upsilon_K(1) - \sigma(K)/2\,| \leq \gamma_4(K)$	(Jabuka et al., 2018)
Casson–Gordon invariants	$$x\cdot\tau_{\max} + (*)\cdot\tau_{\min} \leq 2h + 1 + b_1(M)$$	(Gilmer et al., 2010)

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8. Summary and Research Directions

The non-orientable 4-genus of knots is deeply sensitive to smooth topology, Floer-theoretic invariants, and number-theoretic subtleties. It is unbounded across knots, with sharp lower and upper bounds now accessible via Floer homology. All known constructions and obstructions (band-moves, linking forms, $d$-invariants, Upsilon) interlock to yield a comprehensive, yet intricate, picture for classes such as torus and double-twist knots.

Current research includes:

• Classification of $(e,b_1)$ geography for wider knot families (Allen, 2020, Sabloff, 2022).
• Systematic identification of counterexamples to genus bounds (Sinha, 16 Jul 2025).
• Extension to equivariant and locally-flat categories (Grove et al., 2021, Feller et al., 2020).
• Application of higher gauge-theoretic and Floer-theoretic obstructions.

Open questions remain about the precise geography, sharpness of genus bounds for broader classes, and the full reach of gauge-theoretic invariants in detecting non-orientable slicing complexity.