Knot Floer Torsion Order

• Knot Floer torsion order is defined as the minimum exponent that annihilates the torsion submodule in the minus version of knot Floer homology.
• Cobordism maps and doubling relations yield inequalities that provide lower bounds for invariants such as the bridge index, band-unlinking number, and ribbon distance.
• Refinements like the Upsilon torsion function and extensions to hyperbolic and cable knots illustrate the invariant's role in understanding concordance and complex knot operations.

Searching arXiv for recent and foundational papers on knot Floer torsion order and related developments. Knot Floer torsion order is the integer-valued measure of nilpotence of the polynomial-variable action on the torsion submodule of the minus version of knot Floer homology. In the formulation of Juhász, Miller, and Zemke, for a knot $K \subset S^3$ one considers $\HFK^-(K)$ as a finitely generated $\mathbb F_2[v]$-module and defines $\Ord_v(K)$ as the minimal $k$ such that $v^k$ annihilates the torsion submodule; this invariant is finite, computable from $\CFK^\infty$, and constrained by decorated cobordism maps in ways that produce lower bounds for bridge index, band-unlinking number, fusion number, ribbon distance, and related complexity measures (Juhász et al., 2019). Subsequent work reformulated the same construction with a $U$-variable, introduced parallel torsion orders in unoriented or equal-variables theories, interpolated them by a piecewise-linear Upsilon torsion function, and extended the computational and geometric range of the invariant to hyperbolic knots and L-space cables (Allen et al., 2022).

1. Algebraic definition and standard models

For a knot $K \subset S^3$, the minus version of knot Floer homology admits a noncanonical decomposition

$\HFK^-(K) \cong \mathbb F_2[v] \oplus \HFK^-_{\mathrm{red}}(K),$

where $\HFK^-(K)$0 is the $\HFK^-(K)$1-torsion submodule. If $\HFK^-(K)$2 is any $\HFK^-(K)$3-module, its $\HFK^-(K)$4-torsion order is

$\HFK^-(K)$5

and for knots one sets

$\HFK^-(K)$6

Because $\HFK^-(K)$7 is a finite direct sum of cyclic torsion modules $\HFK^-(K)$8, the invariant is always finite. In the conventions used by Juhász–Miller–Zemke, $\HFK^-(K)$9 has Alexander grading $\mathbb F_2[v]$0; in the symmetric $\mathbb F_2[v]$1 model over $\mathbb F_2[v]$2, the variable $\mathbb F_2[v]$3 has Maslov degree $\mathbb F_2[v]$4 and Alexander degree $\mathbb F_2[v]$5 (Juhász et al., 2019).

For L-space knots, $\mathbb F_2[v]$6 is given by the Ozsváth–Szabó staircase model, determined by the symmetrized Alexander polynomial

$\mathbb F_2[v]$7

If $\mathbb F_2[v]$8, then in the minus complex the torsion order is exactly the maximum of these exponents: $\mathbb F_2[v]$9 Equivalently, if

$\Ord_v(K)$0

then $\Ord_v(K)$1 in the later $\Ord_v(K)$2-based notation (Suchodoll, 11 Jun 2025).

Later literature uses several closely related torsion orders.

Theory	Module	Notation
Minus knot Floer homology	$\Ord_v(K)$3 over $\Ord_v(K)$4 or $\Ord_v(K)$5	$\Ord_v(K)$6, $\Ord_v(K)$7
Unoriented knot Floer homology	$\Ord_v(K)$8 over $\Ord_v(K)$9	$k$0
Equal-variables specialization	$k$1 over $k$2	$k$3, $k$4

The $k$5-based formulation used in later papers is explicitly identified with the Juhász–Miller–Zemke torsion order on $k$6, while $k$7 denotes the analogous invariant for unoriented or equal-variables theories (Himeno et al., 2024). This multiplicity of notation reflects changes of specialization and decoration rather than a single universal convention.

2. Cobordism inequalities and functorial structure

The central structural result is a doubling relation for decorated knot cobordism maps. If $k$8 is a connected, oriented cobordism from $k$9 to $v^k$0 with $v^k$1 minima, $v^k$2 saddles, and $v^k$3 maxima, and if $v^k$4 and $v^k$5 are the compatible decorations used in the construction of link cobordism maps, then

$v^k$6

From this, together with $v^k$7, one obtains the main inequality

$v^k$8

for any connected cobordism $v^k$9 (Juhász et al., 2019).

This estimate is the mechanism behind essentially all geometric applications of the invariant. It transfers torsion information across cobordisms while recording the contribution of local maxima and genus. In ribbon situations, where $\CFK^\infty$0, it becomes a monotonicity statement. In genus-zero situations, it measures the obstruction created purely by births, saddles, and deaths.

A non-orientable analogue exists in the unoriented knot Floer setting. If $\CFK^\infty$1 is a connected non-orientable cobordism from $\CFK^\infty$2 to $\CFK^\infty$3 with $\CFK^\infty$4 local maxima and non-orientable genus $\CFK^\infty$5, then

$\CFK^\infty$6

The proof uses disoriented cobordism maps and a composition identity

$\CFK^\infty$7

that mirrors the orientable relation (Gong et al., 2020).

Allen and Livingston’s Upsilon torsion function repackages these cobordism constraints into a one-parameter family. Their piecewise-linear function $\CFK^\infty$8 satisfies

$\CFK^\infty$9

and yields bounds for local maxima, local minima, and genus in concordances and slice disks through the normalized quantity $U$0 (Allen et al., 2022). This suggests that the original torsion order is the $U$1 endpoint of a broader filtration-sensitive theory rather than an isolated numerical invariant.

3. Bridge index, band moves, ribbon distance, and local simplification

From the main cobordism inequality, Juhász–Miller–Zemke derive several lower bounds for classical and ribbon-theoretic complexity measures. The most basic is the bridge index estimate

$U$2

They also prove

$U$3

where $U$4 is the band-unlinking number, and for a ribbon knot $U$5,

$U$6

where $U$7 is the fusion number. If $U$8 is a slice disk for $U$9 with $K \subset S^3$0 local minima of the radial function, then

$K \subset S^3$1

For a ribbon concordance $K \subset S^3$2 with $K \subset S^3$3 saddles, one has

$K \subset S^3$4

so a difference in torsion orders forces a lower bound on the number of saddles (Juhász et al., 2019).

The same paper introduces a refined cobordism distance

$K \subset S^3$5

and proves that $K \subset S^3$6 is a metric. It satisfies

$K \subset S^3$7

where $K \subset S^3$8 is the minimal number of oriented band moves required to pass from one knot to the other. As a special case, differences in torsion order obstruct short band-move sequences.

A further refinement compares torsion to Sarkar’s ribbon distance. If

$K \subset S^3$9

then

$\HFK^-(K) \cong \mathbb F_2[v] \oplus \HFK^-_{\mathrm{red}}(K),$0

where $\HFK^-(K) \cong \mathbb F_2[v] \oplus \HFK^-_{\mathrm{red}}(K),$1 is the ribbon distance. Since $\HFK^-(K) \cong \mathbb F_2[v] \oplus \HFK^-_{\mathrm{red}}(K),$2, one gets

$\HFK^-(K) \cong \mathbb F_2[v] \oplus \HFK^-_{\mathrm{red}}(K),$3

For $\HFK^-(K) \cong \mathbb F_2[v] \oplus \HFK^-_{\mathrm{red}}(K),$4, the equality

$\HFK^-(K) \cong \mathbb F_2[v] \oplus \HFK^-_{\mathrm{red}}(K),$5

produces ribbon knots with arbitrarily large ribbon distance from the unknot (Juhász et al., 2019).

Later work extended this local-simplification perspective from bands to tangles. If $\HFK^-(K) \cong \mathbb F_2[v] \oplus \HFK^-_{\mathrm{red}}(K),$6 is the minimum $\HFK^-(K) \cong \mathbb F_2[v] \oplus \HFK^-_{\mathrm{red}}(K),$7 such that an oriented $\HFK^-(K) \cong \mathbb F_2[v] \oplus \HFK^-_{\mathrm{red}}(K),$8-tangle replacement changes $\HFK^-(K) \cong \mathbb F_2[v] \oplus \HFK^-_{\mathrm{red}}(K),$9 to $\HFK^-(K)$00, then

$\HFK^-(K)$01

and in particular

$\HFK^-(K)$02

The non-oriented analogue uses $\HFK^-(K)$03 instead (Eftekhary, 2024). A related rational-tangle framework defines $\HFK^-(K)$04 and $\HFK^-(K)$05, proves $\HFK^-(K)$06, and establishes

$\HFK^-(K)$07

for proper and general rational unknotting numbers (Eftekhary, 2022). These results suggest that torsion order is best understood as a robust obstruction to efficient local simplification, not only to cobordism complexity.

4. Explicit calculations, sharpness, and concordance-theoretic consequences

For positive coprime $\HFK^-(K)$08, the torus knot $\HFK^-(K)$09 satisfies

$\HFK^-(K)$10

by Schubert’s formula, and the staircase model gives

$\HFK^-(K)$11

Hence

$\HFK^-(K)$12

so the bridge-index bound is sharp on the entire torus-knot family (Juhász et al., 2019).

Sharpness also holds for fusion number on a standard ribbon family. For

$\HFK^-(K)$13

the standard ribbon disk has

$\HFK^-(K)$14

and connected sum together with mirror invariance gives

$\HFK^-(K)$15

Thus the estimate $\HFK^-(K)$16 is sharp on these ribbon knots.

The invariant also interacts with concordance obstructions. Using the Dai–Hom–Stoffregen–Truong invariant $\HFK^-(K)$17, which satisfies

$\HFK^-(K)$18

and

$\HFK^-(K)$19

for L-space knots $\HFK^-(K)$20, Juhász–Miller–Zemke prove that if $\HFK^-(K)$21 is concordant to $\HFK^-(K)$22, then

$\HFK^-(K)$23

Accordingly, the bridge index of $\HFK^-(K)$24 is minimal within its concordance class.

Several structural identities make explicit calculation feasible. The invariant satisfies

$\HFK^-(K)$25

and therefore

$\HFK^-(K)$26

Algorithmically, one may compute $\HFK^-(K)$27 and then apply Smith normal form over the PID $\HFK^-(K)$28 to read off torsion exponents; the paper states that this is effective for knots up to at least $\HFK^-(K)$29 crossings (Juhász et al., 2019). In practice, the staircase formula for L-space knots and the Smith-normal-form method for general complexes form the two principal computational routes.

5. Upsilon torsion, hyperbolic realization, and cabling behavior

Allen and Livingston define the Upsilon torsion function

$\HFK^-(K)$30

a piecewise-linear, nowhere negative function associated to the filtered complex $\HFK^-(K)$31. It is symmetric under $\HFK^-(K)$32, satisfies

$\HFK^-(K)$33

and interpolates between the two principal torsion orders through

$\HFK^-(K)$34

The function yields sharper lower bounds for cobordism complexity and signed Gordian distance than a single torsion order can provide (Allen et al., 2022).

A major later development is the realization of arbitrarily large torsion order by hyperbolic knots. For

$\HFK^-(K)$35

the twisted torus knot obtained from the torus knot of type $\HFK^-(K)$36 by twisting $\HFK^-(K)$37 adjacent strands by $\HFK^-(K)$38 full twist, one has

$\HFK^-(K)$39

for $\HFK^-(K)$40, and

$\HFK^-(K)$41

for $\HFK^-(K)$42. Moreover, if $\HFK^-(K)$43, then $\HFK^-(K)$44 is hyperbolic. Consequently, for any $\HFK^-(K)$45, there exist infinitely many hyperbolic knots with $\HFK^-(K)$46 and infinitely many hyperbolic knots with $\HFK^-(K)$47 (Himeno et al., 2024).

The same family illustrates a striking independence phenomenon. For each fixed $\HFK^-(K)$48, the infinite family

$\HFK^-(K)$49

shares a common Upsilon torsion function $\HFK^-(K)$50, independent of $\HFK^-(K)$51, while the genus varies according to

$\HFK^-(K)$52

This shows that within these families the torsion profile can remain fixed while the genus becomes arbitrarily large.

Cabling introduces a different form of structure. For all L-space cables of L-space knots with $\HFK^-(K)$53,

$\HFK^-(K)$54

For the trefoil $\HFK^-(K)$55, one has the explicit formula

$\HFK^-(K)$56

Thus $\HFK^-(K)$57, rather than $\HFK^-(K)$58, is the multiplicative quantity under cabling in the L-space regime (Suchodoll, 11 Jun 2025). A plausible implication is that the torsion order is sensitive not only to the staircase gaps of the companion knot but also to the endpoint combinatorics encoded by the cabling construction.

6. Variants, analogues, limitations, and open directions

The invariant admits refinements over $\HFK^-(K)$59, but the algebraic situation changes substantially because $\HFK^-(K)$60 is not a PID. Juhász–Miller–Zemke define refined torsion orders

$\HFK^-(K)$61

and prove a cobordism inequality for the chain version analogous to the main theorem. These refinements still bound $\HFK^-(K)$62 and $\HFK^-(K)$63, but they do not support a universal bridge bound: for torus knots,

$\HFK^-(K)$64

whereas

$\HFK^-(K)$65

The paper explicitly states that a naive bridge bound cannot hold over $\HFK^-(K)$66 (Juhász et al., 2019).

Even in the original one-variable theory, the invariant is not uniformly sharp. For torus knots,

$\HFK^-(K)$67

while

$\HFK^-(K)$68

This shows that the band-unlinking bound can be far from optimal. The paper interprets this as evidence that $\HFK^-(K)$69-torsion detects band moves efficiently when they are organized as ribbon or unlinking operations, but not necessarily the exact minimum for general band-unlinking (Juhász et al., 2019).

There is also an instanton-theoretic analogue. In the minus version of instanton knot homology, one defines

$\HFK^-(K)$70

constructs cobordism maps for specially decorated knot cobordisms, and proves

$\HFK^-(K)$71

The connected sum formula

$\HFK^-(K)$72

parallels the Heegaard Floer theory (Ghosh et al., 2023). This suggests that the bridge-index phenomenon is not peculiar to $\HFK^-(K)$73, although the papers do not assert a general equality of torsion orders across the two Floer theories.

Several open problems remain explicit. One question asks whether, given positive integers $\HFK^-(K)$74 and $\HFK^-(K)$75, there exists a knot $\HFK^-(K)$76 with

$\HFK^-(K)$77

(Himeno et al., 2024). Juhász–Miller–Zemke also leave open whether $\HFK^-(K)$78 detects deeper structure of non-ribbon slice knots, including possible constraints related to the slice–ribbon conjecture, and whether the metric $\HFK^-(K)$79 or the torsion distance $\HFK^-(K)$80 is sharp beyond torus and ribbon families (Juhász et al., 2019). These questions indicate that knot Floer torsion order is already effective as a computable obstruction, but its full range within concordance, ribbon theory, and satellite operations remains only partially understood.