Jacobi Diagrams in Handlebodies

• Jacobi diagrams in handlebodies are diagrammatic objects combining uni-trivalent graphs with the topology of genus‑m handlebodies, enriched by free‑group data.
• They form the universal symmetric monoidal PROP generated by a Casimir Hopf algebra, linking finite‑type topology, free‑group functor theory, and PROP‑based algebra.
• Their structured framework underpins extensions of the Kontsevich integral and enables explicit computations of module structures and Ext‑groups relevant to quantum invariants.

Jacobi diagrams in handlebodies are diagrammatic objects that combine the usual Jacobi-diagram calculus of uni-trivalent graphs with the topology of a genus-$m$ handlebody. In the Habiro–Massuyeau framework, they form a $\Bbbk$-linear category $\mathbf A$ (also denoted $\mathbb A$ in part of the literature) whose objects are nonnegative integers and whose morphisms are Jacobi diagrams on oriented arc systems, enriched by free-group data recording how the diagram runs through the handles. The subject lies at the intersection of finite-type topology, free-group functor theory, and PROP-based algebra: $\mathbf A$ is the universal symmetric monoidal category generated by a Casimir Hopf algebra, its degree-zero part identifies with $\Bbbk\mathbf{gr}^{op}$, its degree-completion receives an extension of the Kontsevich integral for bottom tangles in handlebodies, and its module theory connects it to the PROP for Casimir Lie algebras and to explicit $Aut(F_n)$- and $IA_n$-modules built from low-degree diagram spaces (Habiro et al., 2017, Vespa, 2022, Katada, 14 Sep 2025, Katada, 23 Sep 2025, Katada, 2023).

1. Diagrammatic definition and ambient topology

For $m\ge 0$, let $V_m$ or $\Bbbk$0 denote the genus-$\Bbbk$1 handlebody obtained from a cube by attaching $\Bbbk$2 handles, and let $\Bbbk$3 be the free group generated by the stretched handle cores. For $\Bbbk$4, let $\Bbbk$5 be the oriented $\Bbbk$6-manifold consisting of $\Bbbk$7 oriented arc components. A Jacobi diagram on $\Bbbk$8 is a uni-trivalent graph whose trivalent vertices carry cyclic orientations, whose univalent vertices are attached to the interior of $\Bbbk$9, and whose connected components each meet the support $\mathbf A$0. Two such diagrams are identified when there is a homeomorphism respecting the cyclic orientations and isotopic to the identity on the support (Habiro et al., 2017, Katada, 14 Sep 2025).

The handlebody enhancement enters in two equivalent ways. In the topological description, a morphism $\mathbf A$1 is a Jacobi diagram on $\mathbf A$2 together with a homotopy class, relative to the boundary, of maps into the genus-$\mathbf A$3 handlebody. In the colored or beaded description, the same object is encoded by a Jacobi diagram on $\mathbf A$4 whose oriented dashed edges, and in some formulations also the support arcs, carry labels in $\mathbf A$5. The resulting morphism space is written

$\mathbf A$6

or equivalently $\mathbf A$7, depending on notation (Habiro et al., 2017, Vespa, 2022).

The bead calculus expresses the ambient fundamental-group information through local relations. Two consecutive beads $\mathbf A$8 and $\mathbf A$9 on an oriented segment are equivalent to a single bead $\mathbb A$0; a bead labelled by the unit is deleted; reversing the orientation of an edge changes a bead $\mathbb A$1 to $\mathbb A$2; and there are compatibility moves at trivalent vertices. These relations are imposed together with the STU relation. In this setting, STU implies AS and IHX, so the usual Jacobi-diagram identities remain valid after incorporating the handlebody labels (Vespa, 2022, Katada, 14 Sep 2025).

A recurring point in the literature is that the handlebody is not a passive support. The source object $\mathbb A$3 records the ambient genus, while the target object $\mathbb A$4 records the number of bottom arcs. Consequently, a morphism is not merely a graph attached to a one-manifold; it is a graph considered up to homotopy inside a $\mathbb A$5-dimensional handlebody, with the free-group labels serving as the algebraic shadow of that homotopy data (Katada, 14 Sep 2025).

2. The category $\mathbb A$6 as a symmetric monoidal PROP

The category of Jacobi diagrams in handlebodies has

$\mathbb A$7

Its monoidal structure is strict, with tensor product on objects given by addition,

$\mathbb A$8

and tensor product on morphisms given by horizontal juxtaposition. Symmetry is realized by the obvious permutations of tensor factors, written $\mathbb A$9 in arity $\mathbf A$0 and $\mathbf A$1 for general $\mathbf A$2 (Habiro et al., 2017, Katada, 23 Sep 2025).

Composition is geometric and is defined by cabling and gluing. If $\mathbf A$3 and $\mathbf A$4, the bead words of $\mathbf A$5 specify how copies of $\mathbf A$6 are cabled and inserted, and the result is a well-defined morphism $\mathbf A$7. This composition is associative, and the identity morphism $\mathbf A$8 is represented by the $\mathbf A$9 support arcs with the canonical free-group labels (Habiro et al., 2017). The same geometric composition is emphasized in later work as natural topologically but difficult to control directly in algebraic form, which motivates functorial and module-theoretic reformulations (Vespa, 2022).

A central structural theorem identifies $\Bbbk\mathbf{gr}^{op}$0 as the $\Bbbk\mathbf{gr}^{op}$1-linear PROP freely generated by a Casimir Hopf algebra

$\Bbbk\mathbf{gr}^{op}$2

Here $\Bbbk\mathbf{gr}^{op}$3 is the generating object, $\Bbbk\mathbf{gr}^{op}$4 and $\Bbbk\mathbf{gr}^{op}$5 are multiplication and unit, $\Bbbk\mathbf{gr}^{op}$6 and $\Bbbk\mathbf{gr}^{op}$7 are comultiplication and counit, $\Bbbk\mathbf{gr}^{op}$8 is the antipode, and $\Bbbk\mathbf{gr}^{op}$9 is a symmetric primitive ad-invariant Casimir tensor. In the PROP language, every object $Aut(F_n)$0 is $Aut(F_n)$1, and every morphism is generated from these structure maps and the symmetric monoidal operations (Habiro et al., 2017, Katada, 23 Sep 2025).

This universal property yields an explicit normal form. Any element of $Aut(F_n)$2 is a linear combination of morphisms of the form

$Aut(F_n)$3

where

$Aut(F_n)$4

$Aut(F_n)$5, and each $Aut(F_n)$6 (Katada, 23 Sep 2025). This formula isolates the degree $Aut(F_n)$7 contribution as the number of inserted Casimir tensors and is one of the key tools in the later study of modules and extensions.

The category is also enriched over cocommutative coalgebras, with comultiplication given by splitting a diagram into disjoint unions. In the completed setting this makes sense of group-like morphisms, a point that becomes fundamental in the construction of the handlebody Kontsevich integral (Habiro et al., 2017).

3. Gradings, free-group functoriality, and beaded open diagrams

Two gradings organize the internal structure of $Aut(F_n)$8. The first is the degree grading

$Aut(F_n)$9

where degree is the usual Jacobi-diagram degree, equivalently half the total number of vertices. Composition is additive in degree. In the PROP formulation this grading is also the number of copies of the Casimir tensor $IA_n$0 (Vespa, 2022, Katada, 23 Sep 2025).

The second grading is by homotopy class. For a morphism $IA_n$1, the bead labels along the $IA_n$2-th support arc determine a word in $IA_n$3, hence a homomorphism

$IA_n$4

Accordingly,

$IA_n$5

The $IA_n$6-summand corresponds to the trivial homomorphism; a morphism lies in $IA_n$7 precisely when it can be represented with no beads on the support $IA_n$8, so all bead data is carried only by the dashed graph (Vespa, 2022).

The degree-zero piece is especially important: $IA_n$9 Thus the category of finitely generated free groups appears as the degree-zero skeleton of the handlebody diagram category. Fixing $m\ge 0$0, the representable module $m\ge 0$1 therefore restricts to a contravariant functor on $m\ge 0$2, and each degree piece $m\ge 0$3 becomes a contravariant functor $m\ge 0$4 (Vespa, 2022, Katada, 23 Sep 2025).

Vespa showed that the full degree-$m\ge 0$5 functors are almost never polynomial: $m\ge 0$6 The correct polynomial replacement is the null-homotopy part

$m\ge 0$7

To describe it, the paper introduces $m\ge 0$8-beaded open Jacobi diagrams: uni-trivalent graphs with free labelled legs, no ambient support manifold, and edge labels in a group $m\ge 0$9, modulo the bead moves together with AS and IHX. Their degree-$V_m$0 spaces are denoted $V_m$1 (Vespa, 2022).

The crucial translation uses Powell’s equivalence between analytic functors on $V_m$2 and $V_m$3-linear functors on the Lie PROP $V_m$4. Under this equivalence,

$V_m$5

This identifies the null-homotopy sector of handlebody composition with a much simpler combinatorial model of $V_m$6-beaded open Jacobi diagrams. It also makes polynomial degree transparent: $V_m$7 and the polynomial filtration is given intrinsically by the number of trivalent vertices,

$V_m$8

In degree $V_m$9,

$\Bbbk$00

where $\Bbbk$01 is the second Passi functor (Vespa, 2022).

The same framework resolves the outer-functor question. The null-homotopy functor $\Bbbk$02 is outer if and only if $\Bbbk$03 or $\Bbbk$04. For $\Bbbk$05 and $\Bbbk$06, nontrivial bead labels survive the relevant $\Bbbk$07-operation and create a genuine conjugation-sensitive defect. In this sense, handlebody bead data remembers inner-automorphism information that disappears in the trivial-source case $\Bbbk$08 (Vespa, 2022).

4. Module theory, adjunctions, and extension classes

An $\Bbbk$09-module is a covariant $\Bbbk$10-linear functor

$\Bbbk$11

The category $\Bbbk$12 is abelian with enough projectives. Because $\Bbbk$13, there is an exact forgetful functor

$\Bbbk$14

and a fully faithful exact functor

$\Bbbk$15

induced by projection to degree $\Bbbk$16, with $\Bbbk$17. On $\Bbbk$18, every positive-degree morphism of $\Bbbk$19 acts by zero (Katada, 23 Sep 2025).

Katada enlarged $\Bbbk$20 to a mixed category $\Bbbk$21 whose objects are tensor words in two generators $\Bbbk$22 and $\Bbbk$23. It contains $\Bbbk$24 and the PROP $\Bbbk$25 for Casimir Lie algebras as full graded subcategories. The mixed hom-spaces $\Bbbk$26 are represented by extended Jacobi diagrams in handlebodies, with univalent vertices allowed on the upper line as well as on the support arcs. The category is generated by the Hopf algebra structure on $\Bbbk$27, the Lie algebra structure and Casimir $\Bbbk$28 on $\Bbbk$29, together with maps $\Bbbk$30 and $\Bbbk$31, and the original handlebody Casimir tensor is recovered by

$\Bbbk$32

A key basis theorem identifies the degree-zero mixed hom-space $\Bbbk$33 with chord diagrams running from the upper line to the bottom support (Katada, 14 Sep 2025).

From composition in $\Bbbk$34 one obtains an $\Bbbk$35-bimodule

$\Bbbk$36

and therefore an adjunction

$\Bbbk$37

This generalizes Powell’s adjunction for analytic functors on free groups. On the analytic subcategory $\Bbbk$38, defined by requiring $\Bbbk$39 to be analytic in Powell’s sense, the restricted adjunction is the mechanism by which module-theoretic information is transferred between the handlebody PROP and the PROP for Casimir Lie algebras; the paper notes that Minkyu Kim independently proved that this restricted adjunction is an equivalence (Katada, 14 Sep 2025). The same bridge underlies the later computation of Ext-groups (Katada, 23 Sep 2025).

Day convolution endows $\Bbbk$40, $\Bbbk$41, and $\Bbbk$42 with symmetric monoidal structures. In this setting, the graded algebra $\Bbbk$43 is quadratic: it is generated in degree $\Bbbk$44 by the Casimir element $\Bbbk$45, and its relations are generated by explicit quadratic elements $\Bbbk$46 and $\Bbbk$47. Transporting this structure yields a quadratic presentation of the corresponding graded algebra $\Bbbk$48, generated by $\Bbbk$49, with quadratic relations $\Bbbk$50 and $\Bbbk$51; the latter is identified with the $\Bbbk$52-relation (Katada, 14 Sep 2025).

The representable module

$\Bbbk$53

plays a distinguished role. It is indecomposable, and for $\Bbbk$54 and $\Bbbk$55 the finite-degree subquotients

$\Bbbk$56

are also indecomposable. The degree-$\Bbbk$57 layer splits as

$\Bbbk$58

with

$\Bbbk$59

and with $\Bbbk$60 carrying the nontrivial extension-theoretic complexity. In low degrees,

$\Bbbk$61

while $\Bbbk$62 has a unique composition series with factors

$\Bbbk$63

(Katada, 14 Sep 2025).

In the category of $\Bbbk$64-modules, the first Ext-groups between simple modules induced from Schur functors are explicitly known. If $\Bbbk$65 are partitions, $\Bbbk$66, and $\Bbbk$67, then

$\Bbbk$68

The case $\Bbbk$69 is the degree-zero free-group contribution; the case $\Bbbk$70 is the genuinely handlebody-specific positive-degree contribution arising from a single Casimir insertion. For one-row partitions this specializes to

$\Bbbk$71

By contrast, in the mixed Schur/exterior case no new positive-degree $\Bbbk$72-extensions appear beyond those already present in degree zero (Katada, 23 Sep 2025).

5. Degree-two arc modules and automorphism-group actions

A complementary direction studies Jacobi diagrams on $\Bbbk$73-component oriented arcs as explicit $\Bbbk$74-modules. For $\Bbbk$75, let $\Bbbk$76 be the $\Bbbk$77-vector space spanned by degree-$\Bbbk$78 Jacobi diagrams on $\Bbbk$79 oriented arcs, modulo STU. In degree $\Bbbk$80, the descending filtration by the number of trivalent vertices is

$\Bbbk$81

This filtration is adapted to handlebody considerations because automorphisms are represented by $\Bbbk$82-component arc systems in a genus-$\Bbbk$83 handlebody, and the action is defined by composition in the category $\Bbbk$84 of Jacobi diagrams in handlebodies (Katada, 2023).

If $\Bbbk$85 and $\Bbbk$86, the action is

$\Bbbk$87

where $\Bbbk$88 is the arc system representing $\Bbbk$89. The action is explicitly nontrivial: for the Nielsen shear

$\Bbbk$90

the paper gives an example in which a basis vector $\Bbbk$91 is sent to

$\Bbbk$92

At the same time,

$\Bbbk$93

so the $\Bbbk$94-action factors through $\Bbbk$95 (Katada, 2023).

For the IA-subgroup, the action raises the filtration: $\Bbbk$96 Hence $\Bbbk$97 acts trivially on the associated graded quotients $\Bbbk$98, and the nontriviality is carried by the extension data between successive filtration levels. In degree $\Bbbk$99, the module splits as

$\mathbf A$00

where $\mathbf A$01 is irreducible and $\mathbf A$02 is indecomposable with unique composition series

$\mathbf A$03

and factors

$\mathbf A$04

This separates the degree-$\mathbf A$05 arc-diagram module into a summand on which $\mathbf A$06 is trivial and an indecomposable summand governed by the trivalent-vertex filtration (Katada, 2023).

The first homology of $\mathbf A$07 with coefficients in the indecomposable summand is computed explicitly: $\mathbf A$08 Since $\mathbf A$09 acts trivially on $\mathbf A$10,

$\mathbf A$11

and therefore

$\mathbf A$12

The paper also computes the corresponding homology groups for the intermediate subquotients $\mathbf A$13 and $\mathbf A$14, using explicit boundary maps and, in rank $\mathbf A$15, explicit $\mathbf A$16-cycles built from Day–Putman relators (Katada, 2023).

In rank $\mathbf A$17, the indecomposable module $\mathbf A$18 admits a particularly rigid structure. It has basis

$\mathbf A$19

dimension $\mathbf A$20, and an $\mathbf A$21-equivariant self-duality

$\mathbf A$22

The induced quotient-level isomorphism

$\mathbf A$23

identifies the middle stages of the trivalent filtration as dual $\mathbf A$24-modules. The same paper notes that $\mathbf A$25 as an $\mathbf A$26-module in the sense of Turchin–Willwacher, and hence

$\mathbf A$27

(Katada, 2023)

6. Kontsevich integral, finite-type theory, and relation to the LMO functor

The topological source category for the handlebody theory is the category $\mathbf A$28 of bottom tangles in handlebodies. Its objects are nonnegative integers, and a morphism $\mathbf A$29 is an isotopy class of framed oriented $\mathbf A$30-component bottom tangles in the genus-$\mathbf A$31 handlebody. After non-strictification to $\mathbf A$32, Habiro and Massuyeau constructed a functor

$\mathbf A$33

where $\mathbf A$34 is the degree-completion of the category of Jacobi diagrams in handlebodies. This extends the ordinary Kontsevich integral to the handlebody setting and is modeled on the extension of the Kontsevich integral to tangles in handlebodies due to Andersen, Mattes, and Reshetikhin (Habiro et al., 2017).

For a bottom $\mathbf A$35-tangle $\mathbf A$36, the construction uses a cube presentation $\mathbf A$37 together with cabling anomalies $\mathbf A$38: $\mathbf A$39 With a chosen Drinfeld associator $\mathbf A$40, one obtains a braided monoidal functor

$\mathbf A$41

The image of every bottom tangle is group-like in the completed coalgebra structure and lies in the component determined by its handlebody homotopy class (Habiro et al., 2017).

The finite-type content is encoded by the identification of filtrations. The Vassiliev–Goussarov filtration on $\mathbf A$42 is generated by crossing-change and framing-change plots, while the target degree filtration on $\mathbf A$43 is generated by the Casimir element. The associated graded categories are isomorphic: $\mathbf A$44 Equivalently, Jacobi diagrams in handlebodies are the associated-graded receptacle for finite-type invariants of bottom tangles in handlebodies. In the completed setting, $\mathbf A$45 becomes an isomorphism of filtered braided monoidal categories (Habiro et al., 2017).

The same paper constructs from any Drinfeld associator a ribbon quasi-Hopf algebra inside $\mathbf A$46. The defining elements are

$\mathbf A$47

together with the associated quasi-antipode data. The resulting quasi-Hopf algebra $\mathbf A$48 is universal in the diagrammatic setting, and the paper proves that the canonical Hopf algebra in $\mathbf A$49 is sent by $\mathbf A$50 to the transmutation of $\mathbf A$51 (Habiro et al., 2017).

Jacobi diagrams in handlebodies also refine the LMO formalism. There is a monoidal “hair” or Magnus functor

$\mathbf A$52

obtained by replacing each bead $\mathbf A$53 with the exponential $\mathbf A$54. Under the identification of bottom tangles with special Lagrangian cobordisms, the handlebody Kontsevich integral factors through the LMO functor via $\mathbf A$55. However, for $\mathbf A$56, this map is not injective. Thus the handlebody target $\mathbf A$57 retains noncommutative $\mathbf A$58-information that is lost after passage to the hair expansion (Habiro et al., 2017).

Taken together, these developments place Jacobi diagrams in handlebodies at a precise crossroads. They are simultaneously a geometric encoding of graphs up to homotopy in handlebodies, the universal PROP of Casimir Hopf algebras, the associated graded of handlebody bottom-tangle theory, the source of polynomial and outer-functor phenomena on free groups, and the basis for a module theory whose low degrees are already rich enough to detect indecomposability, nontrivial Ext-classes, and explicit $\mathbf A$59-homology (Habiro et al., 2017, Vespa, 2022, Katada, 14 Sep 2025, Katada, 23 Sep 2025, Katada, 2023).