---
title: Band-Unknotting Number
url: https://www.emergentmind.com/topics/band-unknotting-number
type: topic
---

# Band-Unknotting Number

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 [1901.07625, 1112.2449].
  
  \[
    u_b(K) = \min\{\text{number of oriented bands in a presentation of $K$ with a single disk}\}
  \]
  
- **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 $K$ [1901.07625]:

  \[
    b(K) = \min\{\,b\,|\,\exists\,\text{ribbon–surface for }K\text{ with $b$ bands}\}
  \]
  
  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 $K$ [2512.06299].

- **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 $K$ [1009.3411, 1112.2449].

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 [2502.14304].

## 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 [1112.2449]:

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

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 $n+1$ [2512.06299].
- 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 $K$ [1901.07625]. 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 $D^4$ [2512.06299].

## 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 $K$}\}$                                                                                                         | [1901.07625]   |
| $u_{nb}(K)$         | $\min\{\textrm{non-orientable bands to unknot $K$}\}$                                                                                                   | [2512.06299]   |
| $b(K)$              | Band number, allowing multiple disks                                                                                                                    | [1901.07625]   |
| $u(K)$              | 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 $K$                                                        | [1901.07625]   |
| $u_2(K)$            | Minimum H(2)–moves to unknot $K$                                                                                                                        | [1112.2449]    |
| $\gamma_4(K)$       | Non-orientable 4-genus in $B^4$                                                                                                                         |                |
| $\mu_{an}(K)$       | Minimal number of anisotropic generators in the linking form of $\Sigma_2(K)$                                                                           | [2512.06299]   |

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 $u_b(K) = u_2(K)$ [1112.2449]
- $u_b(K) \le u(K)$ if $u(K)$ is even; $\le u(K)+1$ if $u(K)$ is odd [1112.2449]
- For every $K$, $g_{ds}(K) \le \min\{2g_3(K), 2u(K), u_b(K), b(K)\}$ [1901.07625]
- 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 [2502.14304]

## 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) [1901.07625].
- **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 [1112.2449, 2512.06299].
- **Floer-theoretic invariants:** Torsion order of the unoriented knot Floer homology provides an exact computation of $u_b(T(p,q))$ for torus knots [2502.14304].

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 [2512.06299].

## 5. Explicit Examples and Formulas

Several instructive examples illustrate the invariants' values and bounds:

- The unknot $U$: $u_b(U) = b(U) = 0$, so $g_{ds}(U) = 0$ [1901.07625].
- 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 [1901.07625].
- Untwisted Whitehead double $Wh(K)$ (for ribbon $K$): $u_b(Wh(K)) \leq 3$, so $g_{ds}(Wh(K)) \leq 3$ [1901.07625].
- For torus knots $T(p,q)$, once $\frac{q}{p}$ is written in the specified continued-fraction expansion, the number of layers $n$ yields $u_b(T(p,q)) = n$ [2502.14304].

## 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 $\#_{n} S^2 \times S^2$ [1901.07625].
- **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 [1901.07625].
- **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 $u_{nb}(K_1\#K_2) \neq u_{nb}(K_1\#(-\bar{K_2}))$ [2512.06299].

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 [1901.07625, 2512.06299, 2502.14304, 1112.2449, 1009.3411].

Source: https://www.emergentmind.com/topics/band-unknotting-number