Batson's Conjecture on Torus Knots

• Batson's conjecture is a proposal relating the non-orientable four-ball genus of a torus knot to its pinch number, defined by the minimal sequence of pinch moves to unknot it.
• Counterexamples by Lobb, Longo, and Sinha show that the standard pinch-move construction can overestimate the non-orientable genus by exactly one in several infinite families.
• Floer-theoretic techniques provide sharp lower bounds on γ4, indicating that while the pinch number remains a useful first approximation, more refined invariants are needed for exact computations.

Searching arXiv for Batson's conjecture and related torus-knot/nonorientable 4-ball genus papers. Batson's conjecture is a conjectural formula for the smooth non-orientable four-ball genus of torus knots. For a knot $K\subset S^3=\partial B^4$, the invariant

$$\gamma_4(K) = \min\Bigl\{\dim_{\mathbb Z/2}H_1(F;\mathbb Z/2)\ \Bigm|\ F\hookrightarrow B^4\text{ smooth, non-orientable, }\partial F=K\Bigr\}$$

is the smallest first $\mathbb Z/2$-Betti number of any smooth non-orientable surface in $B^4$ bounded by $K$. Batson conjectured that for every torus knot $T_{p,q}$ with relatively prime $p,q$, this invariant equals the pinch number, i.e. the length of the shortest sequence of pinch moves needed to convert $T_{p,q}$ to the unknot. The conjecture was motivated as a non-orientable analogue of Milnor's conjecture for orientable slice genus, but it is now known to be false: Lobb exhibited the first counterexample $T_{4,9}$, Longo constructed infinite families of counterexamples, and Sinha later produced new infinite families with explicitly computed values of $\gamma_4$ and an exact gap of one from the pinch number (Lobb, 2019, Longo, 2020, Sinha, 16 Jul 2025).

1. Definition and formal statement

Batson's conjecture concerns two invariants of a torus knot $$\gamma_4(K) = \min\Bigl\{\dim_{\mathbb Z/2}H_1(F;\mathbb Z/2)\ \Bigm|\ F\hookrightarrow B^4\text{ smooth, non-orientable, }\partial F=K\Bigr\}$$0. The first is the non-orientable four-ball genus $$\gamma_4(K) = \min\Bigl\{\dim_{\mathbb Z/2}H_1(F;\mathbb Z/2)\ \Bigm|\ F\hookrightarrow B^4\text{ smooth, non-orientable, }\partial F=K\Bigr\}$$1. The second is the pinch number, denoted $$\gamma_4(K) = \min\Bigl\{\dim_{\mathbb Z/2}H_1(F;\mathbb Z/2)\ \Bigm|\ F\hookrightarrow B^4\text{ smooth, non-orientable, }\partial F=K\Bigr\}$$2 in some sources and $$\gamma_4(K) = \min\Bigl\{\dim_{\mathbb Z/2}H_1(F;\mathbb Z/2)\ \Bigm|\ F\hookrightarrow B^4\text{ smooth, non-orientable, }\partial F=K\Bigr\}$$3 in others, defined as the minimal number of pinch moves—also described as band surgeries between adjacent strands, or as surgery on non-orientable bands all lying on the standard torus in $$\gamma_4(K) = \min\Bigl\{\dim_{\mathbb Z/2}H_1(F;\mathbb Z/2)\ \Bigm|\ F\hookrightarrow B^4\text{ smooth, non-orientable, }\partial F=K\Bigr\}$$4—required to obtain the unknot. These moves provide an elementary upper bound

$$\gamma_4(K) = \min\Bigl\{\dim_{\mathbb Z/2}H_1(F;\mathbb Z/2)\ \Bigm|\ F\hookrightarrow B^4\text{ smooth, non-orientable, }\partial F=K\Bigr\}$$5

since a minimal pinch-move sequence can be capped off by a disk in $$\gamma_4(K) = \min\Bigl\{\dim_{\mathbb Z/2}H_1(F;\mathbb Z/2)\ \Bigm|\ F\hookrightarrow B^4\text{ smooth, non-orientable, }\partial F=K\Bigr\}$$6 to produce a non-orientable spanning surface (Sinha, 16 Jul 2025).

Batson observed that for the infinite family $$\gamma_4(K) = \min\Bigl\{\dim_{\mathbb Z/2}H_1(F;\mathbb Z/2)\ \Bigm|\ F\hookrightarrow B^4\text{ smooth, non-orientable, }\partial F=K\Bigr\}$$7,

$$\gamma_4(K) = \min\Bigl\{\dim_{\mathbb Z/2}H_1(F;\mathbb Z/2)\ \Bigm|\ F\hookrightarrow B^4\text{ smooth, non-orientable, }\partial F=K\Bigr\}$$8

and conjectured that

$$\gamma_4(K) = \min\Bigl\{\dim_{\mathbb Z/2}H_1(F;\mathbb Z/2)\ \Bigm|\ F\hookrightarrow B^4\text{ smooth, non-orientable, }\partial F=K\Bigr\}$$9

In the language of Lobb's exposition, the conjecture asserts that the canonical non-orientable surface obtained from the Euclidean-algorithm procedure is optimal. A common misunderstanding is to identify the pinch number with the minimal number of arbitrary non-orientable band surgeries. The conjecture is instead about a very specific torus-diagrammatic construction, and the later counterexamples exploit the difference between that constrained procedure and more flexible non-orientable band attachments (Lobb, 2019).

2. Relation to Milnor's conjecture and early evidence

The orientable prototype is Milnor's conjecture for torus knots, proved by Kronheimer and Mrowka, which states that

$\mathbb Z/2$0

In the orientable case, the standard Seifert-algorithm surface realizes the slice genus. Batson's conjecture proposed an analogous picture for non-orientable surfaces: the most obvious torus-based band construction would realize the true non-orientable genus as well (Longo, 2020).

Before any counterexample was known, there was substantial evidence supporting the conjecture. Batson's own work showed that non-orientable slice genera can be arbitrarily large, while Van Cott and Jabuka checked the conjectural equality for numerous small torus knots. Lobb's summary records that the equality had been verified for all torus knots with $\mathbb Z/2$1 and for many additional cases up to moderate parameters (Lobb, 2019).

This early evidence helps explain why the conjecture was regarded as a plausible non-orientable analogue of Milnor's theorem. The orientable formula is exact and uniform across torus knots, and the pinch construction is combinatorially natural. The later failure of the non-orientable version therefore marked a genuine structural difference between orientable and non-orientable four-dimensional knot theory.

3. The first counterexample: $\mathbb Z/2$2

Lobb's counterexample showed that the torus knot $\mathbb Z/2$3 bounds a smooth Möbius band in $\mathbb Z/2$4. Since a Möbius band has first Betti number $\mathbb Z/2$5, this gives

$\mathbb Z/2$6

By contrast, the pinch number is $\mathbb Z/2$7: one must perform two pinch moves to reach the unknot. Thus

$\mathbb Z/2$8

contradicting Batson's conjecture (Lobb, 2019).

The construction proceeds by adding a single blackboard-framed $\mathbb Z/2$9-handle between an appropriate pair of adjacent strands in the standard braid diagram of $B^4$0. The resulting knot is the Stevedore knot $B^4$1, which is slice. Gluing the slice disk for $B^4$2 to the added band yields a properly embedded non-orientable surface in $B^4$3 with boundary $B^4$4 and $B^4$5, hence a Möbius band (Lobb, 2019).

This example clarified two points. First, the pinch-number surface need not be genus-minimizing. Second, non-orientable band surgeries not tied to the standard torus can outperform the canonical pinch sequence. In the later literature, $B^4$6 became the base case for broader infinite families of counterexamples.

4. Longo's infinite families

Longo showed that Lobb's phenomenon is not isolated but occurs in an infinite family of torus knots. For $B^4$7, define

$B^4$8

and for $B^4$9, define

$K$0

He proved that

$K$1

Hence each family violates Batson's conjecture by at least one, and in the examples under discussion the gap is exactly one (Longo, 2020).

A key input is the explicit behavior of a single pinch move on a torus knot: $K$2 where $K$3 and $K$4 are the least nonnegative solutions of

$K$5

Iterating this lemma shows that the pinch number of these families is $K$6. The upper bound on $K$7 comes from an explicit collection of $K$8 non-orientable bands. Longo views $K$9 as the closure of a $T_{p,q}$0-strand braid with $T_{p,q}$1 full twists, labels the strands $T_{p,q}$2, and attaches bands between strand $T_{p,q}$3 and strand $T_{p,q}$4 for $T_{p,q}$5. After surgery and ambient isotopy, the result is a $T_{p,q}$6-bridge knot with continued-fraction expansion $T_{p,q}$7, and classical work of Casson–Gordon–Conway, later subsumed by Lisca's classification, implies that this knot is slice (Longo, 2020).

Longo's theorem established that the conjecture fails uniformly for arbitrarily large pinch number. The family $T_{p,q}$8 recovers Lobb's original example, so the first counterexample is the initial term of a systematic construction rather than an isolated anomaly.

5. Floer-theoretic lower bounds and partial sharpness

Subsequent work strengthened lower bounds on $T_{p,q}$9 using Floer-theoretic methods. Binns, Kang, Simone, and Truöl combined involutive knot Floer homology with unoriented knot Floer homology to produce a new lower bound on the smooth non-orientable four-ball genus of any knot. In the $p,q$0-space knot case, and hence for torus knots, they derived the theorem: $p,q$1 In particular, $p,q$2 bounds no smooth Möbius band unless $p,q$3 (Binns et al., 2021).

For torus knots this lower bound is sharp in several families. The paper states that Batson's pinch-move construction gives

$p,q$4

while Alexander-polynomial and semigroup computations yield

$p,q$5

Combined with a diagrammatic construction of a non-orientable surface of first Betti number $p,q$6, this implies that for even $p,q$7,

$p,q$8

The same paper also gives a two-parameter extension: $p,q$9 Thus Floer-theoretic obstructions do not merely disprove the conjecture abstractly; they can force exact values on infinite counterexample families (Binns et al., 2021).

The same work includes a topological Möbius-band obstruction. If $T_{p,q}$0 is even and $T_{p,q}$1 is not a perfect square, then the proportion of relatively prime $T_{p,q}$2 for which $T_{p,q}$3 bounds a locally flat Möbius band tends to zero as $T_{p,q}$4. This shows that, even in the locally flat category, Möbius bands are asymptotically exceptional for many fixed even $T_{p,q}$5.

6. Sinha's new infinite families and the current picture

Sinha extended the counterexample phenomenon to new infinite families whose non-orientable four-ball genus is computed exactly. For every integer $T_{p,q}$6,

$T_{p,q}$7

satisfies

$T_{p,q}$8

These are therefore genuine counterexamples to Batson's conjecture (Sinha, 16 Jul 2025).

The band-surgery argument is explicit. Starting from the standard $T_{p,q}$9-strand diagram of $T_{4,9}$0, one performs exactly $T_{4,9}$1 carefully chosen non-orientable band moves joining the strand pairs $T_{4,9}$2 for $T_{4,9}$3. After a sequence of isotopies through the partial twists, one obtains a knot $T_{4,9}$4 identical to the knot $T_{4,9}$5 arising in Longo's earlier construction for $T_{4,9}$6. Longo showed that $T_{4,9}$7 is the two-bridge knot

$T_{4,9}$8

which is slice by Lisca's classification. This gives the upper bound $T_{4,9}$9, and the Floer-theoretic lower bound forces equality (Sinha, 16 Jul 2025).

Sinha also generalized the construction. For all $\gamma_4$0 and $\gamma_4$1,

$\gamma_4$2

pinch down, one move at a time, to the base cases $\gamma_4$3 or $\gamma_4$4. Applying $\gamma_4$5 such moves and then the same lower-bound argument yields

$\gamma_4$6

while

$\gamma_4$7

Each of these infinitely many torus knots therefore violates Batson's conjecture by an exact gap of one (Sinha, 16 Jul 2025).

The current state of the subject is therefore clear in one respect and open in another. The conjectural equality $\gamma_4$8 is false, and false in several explicit infinite families. At the same time, many known families satisfy the sharp alternative

$\gamma_4$9

especially when

$$\gamma_4(K) = \min\Bigl\{\dim_{\mathbb Z/2}H_1(F;\mathbb Z/2)\ \Bigm|\ F\hookrightarrow B^4\text{ smooth, non-orientable, }\partial F=K\Bigr\}$$00

This suggests that the pinch number remains a robust first approximation, but not a complete invariant. The later counterexamples show concretely that the pinch-move bound can overestimate the true non-orientable four-genus by exactly one, and they indicate that accurate computation of $$\gamma_4(K) = \min\Bigl\{\dim_{\mathbb Z/2}H_1(F;\mathbb Z/2)\ \Bigm|\ F\hookrightarrow B^4\text{ smooth, non-orientable, }\partial F=K\Bigr\}$$01 requires more delicate Floer-theoretic or gauge-theoretic input than the naive count of torus-embedded bands (Sinha, 16 Jul 2025).