Band-Unknotting Number

• Band-unknotting number is defined as the minimum count of band surgeries required to transform a knot into the unknot, with variations for orientable and non-orientable bands.
• It bridges classical knot invariants with 4-manifold topology by employing surface and homological techniques, such as Betti number evaluations, to derive bounds.
• Computational methods and diagrammatic strategies, including broken surface diagrams and Floer-theoretic invariants, provide practical insights into estimating these invariant values.

The band-unknotting number and its generalizations quantify how knots in the 3-sphere can be simplified or transformed to the unknot using band surgeries, a class of local operations that generalize classical Reidemeister moves and crossing changes. These invariants, and their associated constructions, play a central role in knot theory, especially in the study of surfaces in 4-manifolds, double slicing, and the topological properties of knot complements.

1. Definitions and Basic Properties

Let $K \subset S^3$ be a knot. Band surgery refers to the process of cutting $K$ at two points and reconnecting via a band—either orientable or non-orientable—yielding a new knot or link.

• Band-unknotting number ($u_b(K)$): The minimum number of (possibly orientable or unoriented) band surgeries needed to transform $K$ into the unknot. For oriented bands (oriented saddle moves), $u_b(K)$ is the minimal such count subject to producing a connected ribbon surface with a single disk as its 0-handle. This is equivalently the minimum number of bands in an oriented band-presentation of $K$ with a single disk (McDonald, 2019, Abe et al., 2011).

$u_b(K) = \min\{\text{number of oriented bands in a presentation of %%%%6%%%% with a single disk}\}$

• Band number ($b(K)$): A generalization of $u_b(K)$ allowing ribbon surfaces in $B^4$ with multiple disks (0-handles) in their handle decomposition. It is the minimal number of ribbon bands in any handle decomposition whose boundary is $K$ (McDonald, 2019):

$b(K) = \min\{\,b\,|\,\exists\,\text{ribbon–surface for }K\text{ with %%%%11%%%% bands}\}$

Always $b(K) \le u_b(K)$, since $u_b(K)$ corresponds to the case with a single disk.

• Non-orientable band-unknotting number ($u_{nb}(K)$): The smallest number of non-orientable (half-twisted) band moves needed to unknot $K$ (Ghanbarian et al., 6 Dec 2025).
• H(2)-unknotting number ($u_2(K)$): The minimal number of twisted band moves (so-called H(2)-moves: special band surgeries that preserve the number of components) required to unknot $K$ (Bao, 2010, Abe et al., 2011).

For torus knots, the unoriented band-unknotting number coincides with the "pinch number," defined as the minimal number of Batson's pinch moves (special non-orientable band surgeries) needed to unknot the knot (Himeno, 20 Feb 2025).

2. Surface-Theoretic and Algebraic Characterizations

Band-unknotting numbers admit a surface-theoretic and homological interpretation. For any two knots $J, K \subset S^3$, the band-Gordian distance $d_b(J,K)$ is defined as the minimal number of band surgeries required to convert $J$ to $K$. The following characterization holds (Abe et al., 2011):

$$d_b(J, K) = \min\{\,b_1(F)\,|\, F \subset S^3,\, \partial F = J \cup K,\, F \text{ connected}\,\} - 1$$

Here, $b_1(F)$ denotes the first Betti number of the spanning surface $F$. For the unknot $U$,

$$u_b(K) = \min\{\,b_1(F)\,|\, \partial F = K \cup U,\, F \text{ connected}\} - 1$$

There is a close relationship between band-unknotting numbers and 4-dimensional knot invariants. Specifically:

• Every sequence of $n$ non-orientable band moves from $K$ to the unknot yields a non-orientable surface in $B^4$ bounding $K$ with Betti number $n+1$ (Ghanbarian et al., 6 Dec 2025).
• The band-unknotting number is always at least the maximum of the classical unknotting number $u(K)$ and twice the genus $2g_3(K)$ of the minimal-genus Seifert surface for $K$ (McDonald, 2019). For non-orientable band moves:

$$\mu_{an}(K) \leq \gamma_{4,t}(K) \leq \gamma_{4,s}(K) \leq u_{nb}(K)$$

where $\mu_{an}(K)$ is the minimal number of generators of the anisotropic part of the linking form, and $\gamma_{4,t}$, $\gamma_{4,s}$ are the minimal topological and smooth non-orientable genera in $D^4$ (Ghanbarian et al., 6 Dec 2025).

3. Relationships with Other Knot Invariants

Band-unknotting numbers are bounded above by and interact with various classical and 4-dimensional invariants:

Quantity	Definition/Relation	Source
$u_b(K)$	$\min\{\textrm{oriented bands to unknot$K$}\}$	(McDonald, 2019)
$u_{nb}(K)$	$\min\{\textrm{non-orientable bands to unknot$K$}\}$	(Ghanbarian et al., 6 Dec 2025)
$b(K)$	Band number, allowing multiple disks	(McDonald, 2019)
$u(K)$	Classical unknotting number
$g_3(K)$	Minimal genus of orientable Seifert surface in $S^3$
$g_{ds}(K)$	Double slice genus: minimal genus of an unknotted closed surface in $S^4$ with cross-section $K$	(McDonald, 2019)
$u_2(K)$	Minimum H(2)–moves to unknot $K$	(Abe et al., 2011)
$\gamma_4(K)$	Non-orientable 4-genus in $B^4$
$\mu_{an}(K)$	Minimal number of anisotropic generators in the linking form of $\Sigma_2(K)$	(Ghanbarian et al., 6 Dec 2025)

The following inequalities hold (selected):

• $u_b(K) \ge \max\{u(K), 2g_3(K)\}$
• $g_{ds}(K) \le b(K) \le u_b(K)$
• $u_b(K)$ is either $u_2(K)$ or $u_2(K) - 1$, and $u_b(K)$ odd iff $u_b(K) = u_2(K)$ (Abe et al., 2011)
• $u_b(K) \le u(K)$ if $u(K)$ is even; $\le u(K)+1$ if $u(K)$ is odd (Abe et al., 2011)
• For every $K$, $g_{ds}(K) \le \min\{2g_3(K), 2u(K), u_b(K), b(K)\}$ (McDonald, 2019)
• For torus knots, $u_b(T(p,q)) = P(p,q)$, where $P(p,q)$ is the pinch number derived from a special continued fraction expansion (Himeno, 20 Feb 2025)

4. Computational and Diagrammatic Aspects

Explicit computation of the band-unknotting number for specific knots involves both diagrammatic strategies and algebraic bounds:

• Broken surface diagrams: Used extensively for analyzing surfaces in $S^4$ and verifying that double constructions or band-surgeries yield unknotted surfaces (i.e., those bounding handlebodies) (McDonald, 2019).
• Use of covering spaces: Bounds from the homology of the double-branched cover $\Sigma_2(K)$—particularly, minimal $n$ for an embedding in $\#_n S^2 \times S^2$—are intimately related to $g_{ds}(K)$ and thus to the band number.
• Link invariants: The Jones polynomial at special values, the $Q$-polynomial, and linking forms furnish lower bounds for $u_b(K)$ and related quantities (Abe et al., 2011, Ghanbarian et al., 6 Dec 2025).
• Floer-theoretic invariants: Torsion order of the unoriented knot Floer homology provides an exact computation of $u_b(T(p,q))$ for torus knots (Himeno, 20 Feb 2025).

Remarkable phenomena, such as strict subadditivity (i.e., $u_{nb}(K_1 \# K_2) < u_{nb}(K_1) + u_{nb}(K_2)$), have been established, notably for two-bridge knots and double-twist knots (Ghanbarian et al., 6 Dec 2025).

5. Explicit Examples and Formulas

Several instructive examples illustrate the invariants' values and bounds:

• The unknot $U$: $u_b(U) = b(U) = 0$, so $g_{ds}(U) = 0$ (McDonald, 2019).
• Stevedore knot $6_1$: $b(6_1) \leq 1$, $g_{ds}(6_1) = 1$, with minimal-band ribbon surface using fewer bands than required for minimal-genus ribbon disk (McDonald, 2019).
• Untwisted Whitehead double $Wh(K)$ (for ribbon $K$): $u_b(Wh(K)) \leq 3$, so $g_{ds}(Wh(K)) \leq 3$ (McDonald, 2019).
• For torus knots $T(p,q)$, once $\frac{q}{p}$ is written in the specified continued-fraction expansion, the number of layers $n$ yields $u_b(T(p,q)) = n$ (Himeno, 20 Feb 2025).

6. Applications, Embeddings, and Open Problems

Band-unknotting numbers have meaningful consequences for 3- and 4-manifold topology and knot concordance:

• Embeddings and slicing: $g_{ds}(K)$ bounds the minimal number $n$ so that $\Sigma_2(K)$ embeds in $\#_{n} S^2 \times S^2$ (McDonald, 2019).
• Ribbon fusion number: $b(K)$ may be strictly less than the minimal-genus ribbon disk's band count. McDonald conjectures the existence of such knots, establishing the non-coincidence of band number and ribbon fusion number (McDonald, 2019).
• Knots with isomorphic groups but differing $u_{nb}$: The knot group does not determine $u_{nb}$; for infinitely many alternating $K_1, K_2$, one has $$\pi_1(S^3 \setminus (K_1 \# K_2)) \cong \pi_1(S^3 \setminus (K_1 \# (-\bar{K_2})))$$ but $u_{nb}(K_1\#K_2) \neq u_{nb}(K_1\#(-\bar{K_2}))$ (Ghanbarian et al., 6 Dec 2025).

Open questions remain regarding:

• The possible strict decrease of $u_{nb}$ under iterated connected sum,
• Analogues for other band-type operations (e.g., $n$-twisted band moves),
• The construction of knots for which $b(K)$ is strictly less than minimal genus ribbon disk band counts,
• The realization of the theoretical bounds in infinite families.

7. Broader Significance and Research Directions

The study of band-unknotting numbers and generalizations provides profound connections between knot theory, 3-manifold topology, and 4-manifold theory. These invariants:

• Bridge the combinatorial complexity of knot diagrams with the smooth and topological properties of surfaces in 3- and 4-manifolds.
• Furnish new bounds and embedding theorems for branched covers, impacting the study of $S^4$, slice genus, and double slicing.
• Reveal structural phenomena such as strict subadditivity and the independence of the band-unnoting number from group-theoretic knot invariants.
• Motivate the development of new invariants (e.g., torsion order in Floer homology) and computational approaches for classical and satellite knots.

Continued research involves fine-tuning lower and upper bounds, exploring relationships with quantum invariants, and constructing explicit families illustrating extremal behavior and exceptional cases among these band-type unknotting invariants (McDonald, 2019, Ghanbarian et al., 6 Dec 2025, Himeno, 20 Feb 2025, Abe et al., 2011, Bao, 2010).