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$0 (McDonald, 2019):

$K$1

Always $K$2, since $K$3 corresponds to the case with a single disk.

• Non-orientable band-unknotting number ($K$4): The smallest number of non-orientable (half-twisted) band moves needed to unknot $K$5 (Ghanbarian et al., 6 Dec 2025).
• H(2)-unknotting number ($K$6): 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$7 (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 $K$8, the band-Gordian distance $K$9 is defined as the minimal number of band surgeries required to convert $u_b(K)$0 to $u_b(K)$1. The following characterization holds (Abe et al., 2011):

$u_b(K)$2

Here, $u_b(K)$3 denotes the first Betti number of the spanning surface $u_b(K)$4. For the unknot $u_b(K)$5,

$u_b(K)$6

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

• Every sequence of $u_b(K)$7 non-orientable band moves from $u_b(K)$8 to the unknot yields a non-orientable surface in $u_b(K)$9 bounding $K$0 with Betti number $K$1 (Ghanbarian et al., 6 Dec 2025).
• The band-unknotting number is always at least the maximum of the classical unknotting number $K$2 and twice the genus $K$3 of the minimal-genus Seifert surface for $K$4 (McDonald, 2019). For non-orientable band moves:

$K$5

where $K$6 is the minimal number of generators of the anisotropic part of the linking form, and $K$7, $K$8 are the minimal topological and smooth non-orientable genera in $K$9 (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)$0	$u_b(K)$1K$u_b(K)$2	(McDonald, 2019)
$u_b(K)$3	$u_b(K)$4K$u_b(K)$5	(Ghanbarian et al., 6 Dec 2025)
$u_b(K)$6	Band number, allowing multiple disks	(McDonald, 2019)
$u_b(K)$7	Classical unknotting number
$u_b(K)$8	Minimal genus of orientable Seifert surface in $u_b(K)$9
$K$0	Double slice genus: minimal genus of an unknotted closed surface in $K$1 with cross-section $K$2	(McDonald, 2019)
$K$3	Minimum H(2)–moves to unknot $K$4	(Abe et al., 2011)
$K$5	Non-orientable 4-genus in $K$6
$K$7	Minimal number of anisotropic generators in the linking form of $K$8	(Ghanbarian et al., 6 Dec 2025)

The following inequalities hold (selected):

• $K$9
• $u_b(K) = \min\{\text{number of oriented bands in a presentation of %%%%6%%%% with a single disk}\}$0
• $u_b(K) = \min\{\text{number of oriented bands in a presentation of %%%%6%%%% with a single disk}\}$1 is either $u_b(K) = \min\{\text{number of oriented bands in a presentation of %%%%6%%%% with a single disk}\}$2 or $u_b(K) = \min\{\text{number of oriented bands in a presentation of %%%%6%%%% with a single disk}\}$3, and $u_b(K) = \min\{\text{number of oriented bands in a presentation of %%%%6%%%% with a single disk}\}$4 odd iff $u_b(K) = \min\{\text{number of oriented bands in a presentation of %%%%6%%%% with a single disk}\}$5 (Abe et al., 2011)
• $u_b(K) = \min\{\text{number of oriented bands in a presentation of %%%%6%%%% with a single disk}\}$6 if $u_b(K) = \min\{\text{number of oriented bands in a presentation of %%%%6%%%% with a single disk}\}$7 is even; $u_b(K) = \min\{\text{number of oriented bands in a presentation of %%%%6%%%% with a single disk}\}$8 if $u_b(K) = \min\{\text{number of oriented bands in a presentation of %%%%6%%%% with a single disk}\}$9 is odd (Abe et al., 2011)
• For every $b(K)$0, $b(K)$1 (McDonald, 2019)
• For torus knots, $b(K)$2, where $b(K)$3 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 $b(K)$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 $b(K)$5—particularly, minimal $b(K)$6 for an embedding in $b(K)$7—are intimately related to $b(K)$8 and thus to the band number.
• Link invariants: The Jones polynomial at special values, the $b(K)$9-polynomial, and linking forms furnish lower bounds for $u_b(K)$0 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(K)$1 for torus knots (Himeno, 20 Feb 2025).

Remarkable phenomena, such as strict subadditivity (i.e., $u_b(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_b(K)$3: $u_b(K)$4, so $u_b(K)$5 (McDonald, 2019).
• Stevedore knot $u_b(K)$6: $u_b(K)$7, $u_b(K)$8, with minimal-band ribbon surface using fewer bands than required for minimal-genus ribbon disk (McDonald, 2019).
• Untwisted Whitehead double $u_b(K)$9 (for ribbon $B^4$0): $B^4$1, so $B^4$2 (McDonald, 2019).
• For torus knots $B^4$3, once $B^4$4 is written in the specified continued-fraction expansion, the number of layers $B^4$5 yields $B^4$6 (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: $B^4$7 bounds the minimal number $B^4$8 so that $B^4$9 embeds in $K$00 (McDonald, 2019).
• Ribbon fusion number: $K$01 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 $K$02: The knot group does not determine $K$03; for infinitely many alternating $K$04, one has $K$05 but $K$06 (Ghanbarian et al., 6 Dec 2025).

Open questions remain regarding:

• The possible strict decrease of $K$07 under iterated connected sum,
• Analogues for other band-type operations (e.g., $K$08-twisted band moves),
• The construction of knots for which $K$09 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 $K$10, 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).