Inverted Habiro Series (IHS)

• Inverted Habiro Series (IHS) is a dual expansion built from negative-index cyclotomic bases that regularizes the Gukov–Manolescu knot series F_K(x,q).
• IHS admit a theta-function representation and are structured within a commutative ring (Ω), enabling systematic residue calculus and convergence control.
• Under technical conditions like the lower bound condition, IHS facilitate new Dehn surgery formulas and connect knot invariants with classical q-identities.

Searching arXiv for the cited paper and closely related background on inverted Habiro series and GM series. First, retrieving the primary paper by arXiv id. Now retrieving a foundational related paper on inverted Habiro series and inverted state sums. Optionally retrieving related background on Habiro-type arithmetic or asymptotics, to support contextual connections if needed. Inverted Habiro Series (IHS) are Park’s “dual” or “inverted” analogue of Habiro’s universal series, introduced to package and regularize the Gukov–Manolescu two-variable knot series $F_K(x,q)$. In Svoboda’s treatment, the IHS of a knot $K$ is denoted $P_K(x,q)$; it is built from negative-index analogues of Habiro’s cyclotomic basis, matches the formal Taylor expansion of the Gukov–Manolescu series at $x=0$, admits a theta-function representation under a lower bound condition, carries a natural commutative ring structure, and supports a residue calculus with applications to Dehn surgery and root-of-unity asymptotics (Svoboda, 26 Sep 2025). The construction continues and systematizes the perspective introduced by Park in the context of inverted state sums, inverted Habiro series, and regularized surgery formulas for $F_K$ and $\hat Z$ (Park, 2021).

1. Origin, definition, and relation to Habiro’s universal series

The starting point is the Gukov–Manolescu series of a knot $K$,

$F_K(x,q)=\sum_{n\ge 0} f_n(q)\,x^n,$

with coefficients $f_n(q)\in \mathbb{Z}((q))$. Conceptually, $F_K$ is presented as a knot-complement counterpart of the closed 3-manifold invariant $K$0, and is conjecturally a lift of the Melvin–Morton–Rozansky expansion from the regime $K$1 to a two-variable series in the Alexander variable $K$2 (Svoboda, 26 Sep 2025).

Habiro’s universal invariant $K$3 lies in the completed ring

$K$4

and has a unique expansion

$K$5

where $K$6. Park’s proposal is to extend the same basis to negative indices and thereby produce an “inverted” expansion adapted to the $K$7 regime (Svoboda, 26 Sep 2025, Park, 2021).

For all integers $K$8, Svoboda defines

$K$9

with recurrence

$P_K(x,q)$0

For $P_K(x,q)$1, the normalized basis element is $P_K(x,q)$2, and an inverted Habiro series is a formal sum

$P_K(x,q)$3

For a knot $P_K(x,q)$4,

$P_K(x,q)$5

where $P_K(x,q)$6 are the inverted Habiro coefficients (Svoboda, 26 Sep 2025).

Object	Basis regime	Role
Habiro universal series $P_K(x,q)$7	$P_K(x,q)$8 for $P_K(x,q)$9	Universal cyclotomic expansion
Inverted Habiro series $x=0$0	$x=0$1 for $x=0$2, normalized to $x=0$3	“Dual” expansion adapted to $x=0$4
Gukov–Manolescu series $x=0$5	Power series in $x=0$6	Knot-complement two-variable invariant

The essential matching condition is that the term-by-term Taylor expansion of $x=0$7 at $x=0$8 recovers the Gukov–Manolescu series. Writing

$x=0$9

one requires

$F_K$0

Park and Svoboda prove that the coefficient sequences $F_K$1 and $F_K$2 determine each other uniquely: $F_K$3

$F_K$4

Accordingly, $F_K$5 is algebraically equivalent to $F_K$6 (Svoboda, 26 Sep 2025).

2. Theta-function representation and regularization of the GM series

A central result is an explicit formula expressing $F_K$7 in terms of the GM coefficients $F_K$8 and truncated theta functions. Svoboda uses the Jacobi theta function

$F_K$9

and defines truncated theta functions $\hat Z$0 (Svoboda, 26 Sep 2025).

To ensure convergence as a $\hat Z$1-series, Svoboda imposes the lower bound condition (LBC). If $\hat Z$2 denotes the minimal $\hat Z$3-power in a Laurent series, a sequence $\hat Z$4 satisfies LBC if

$\hat Z$5

for some constant $\hat Z$6. This guarantees that the expansion of $\hat Z$7 at $\hat Z$8 is a genuine formal Laurent series in $\hat Z$9. Computational evidence indicates that LBC holds for many “nice” knots, including homogeneous braid knots up to 13 crossings (Svoboda, 26 Sep 2025).

Under LBC, Svoboda proves the theta formula

$K$0

In product form,

$K$1

This formula makes the analytic role of IHS explicit. The denominator $K$2 produces simple poles at $K$3, $K$4, so it encodes the entire pole structure of $K$5. At the same time, for fixed $K$6, $K$7 has positive $K$8-degree at least $K$9, and this compensates for the arbitrarily negative powers of $F_K(x,q)=\sum_{n\ge 0} f_n(q)\,x^n,$0 that can occur in $F_K(x,q)=\sum_{n\ge 0} f_n(q)\,x^n,$1. In this precise sense, IHS are a regularization of the GM series (Svoboda, 26 Sep 2025).

This regularization viewpoint is consistent with Park’s earlier use of inverted Habiro series as a mechanism for re-expressing the GM invariant in a basis of inverted cyclotomic factors adapted to the $F_K(x,q)=\sum_{n\ge 0} f_n(q)\,x^n,$2 regime (Park, 2021).

3. Algebraic structure: multiplication and the ring $F_K(x,q)=\sum_{n\ge 0} f_n(q)\,x^n,$3

Svoboda constructs a ring $F_K(x,q)=\sum_{n\ge 0} f_n(q)\,x^n,$4 of inverted Habiro series in direct analogy with Habiro’s ring $F_K(x,q)=\sum_{n\ge 0} f_n(q)\,x^n,$5. Writing $F_K(x,q)=\sum_{n\ge 0} f_n(q)\,x^n,$6 for the Taylor expansion of $F_K(x,q)=\sum_{n\ge 0} f_n(q)\,x^n,$7 at $F_K(x,q)=\sum_{n\ge 0} f_n(q)\,x^n,$8, the fundamental multiplication formula is

$F_K(x,q)=\sum_{n\ge 0} f_n(q)\,x^n,$9

where the coefficients $f_n(q)\in \mathbb{Z}((q))$0 are given explicitly in the paper. This extends Habiro’s product formula from the range $f_n(q)\in \mathbb{Z}((q))$1 to all integers $f_n(q)\in \mathbb{Z}((q))$2 (Svoboda, 26 Sep 2025).

The ring $f_n(q)\in \mathbb{Z}((q))$3 consists of formal sums

$f_n(q)\in \mathbb{Z}((q))$4

with the condition that the coefficients $f_n(q)\in \mathbb{Z}((q))$5 satisfy LBC. Multiplication is defined by lifting the formula above to formal sums. Svoboda proves that the map

$f_n(q)\in \mathbb{Z}((q))$6

embeds $f_n(q)\in \mathbb{Z}((q))$7 as a $f_n(q)\in \mathbb{Z}((q))$8-subalgebra of $f_n(q)\in \mathbb{Z}((q))$9. In particular, $F_K$0 is a well-defined commutative $F_K$1-algebra, and the IHS of any knot satisfying LBC lies in $F_K$2 (Svoboda, 26 Sep 2025).

The structural analogy with Habiro’s framework is explicit. Habiro’s $F_K$3 is generated by $F_K$4 for $F_K$5, whereas $F_K$6 is generated by $F_K$7 for $F_K$8. Svoboda further records the expectation that $F_K$9 is the center of a suitable integral form of $K$00, extending Habiro’s algebraic setting from $K$01 to $K$02 and $K$03. This expectation is not proved in the paper (Svoboda, 26 Sep 2025).

4. Residues, pole expansions, and recovery of GM data

Because $K$04 has poles at $K$05, its residues become intrinsic invariants. For $K$06, the basic building block $K$07 is meromorphic on $K$08 with simple poles at $K$09 for $K$10, and an additional pole at $K$11 when $K$12 (Svoboda, 26 Sep 2025).

Given

$K$13

Svoboda defines the residue at $K$14 term-by-term: $K$15 At infinity,

$K$16

The resulting residue theorem is

$K$17

Thus the sum of all finite residues equals minus the residue at infinity (Svoboda, 26 Sep 2025).

The residue data and the GM coefficients determine each other explicitly. For any $K$18,

$K$19

Conversely, the theta representation expresses each $K$20 in terms of the full family $K$21. Hence the GM series can be reconstructed from the discrete spectral data $K$22, and vice versa (Svoboda, 26 Sep 2025).

In examples, the residue calculus recovers classical $K$23-identities. For the left-handed trefoil $K$24, one has

$K$25

and the residues satisfy

$K$26

In this case the residue theorem becomes Euler’s pentagonal number theorem. For $K$27,

$K$28

whose coefficients are partition numbers (Svoboda, 26 Sep 2025).

5. Dehn surgery formulas and root-of-unity asymptotics

The principal geometric application is to Dehn surgery. For integer $K$29 and a $K$30 structure labelled by $K$31, the Gukov–Manolescu surgery formula may be written in terms of the GM coefficients as

$K$32

with $K$33 (Svoboda, 26 Sep 2025).

Using the residue formula for $K$34, Svoboda rewrites the surgery expression as a double sum in $K$35, and under convergence assumptions interchanges sums to obtain a residue-only formula: $K$36 The inner sum is a finite polynomial in $K$37, so the coefficient of each $K$38 is a Laurent polynomial. This shows that the Dehn surgery invariants $K$39 can be computed from residues of the inverted Habiro series $K$40, without directly using the GM series in regimes where convergence is problematic. Svoboda presents this as a unification of Gukov–Manolescu’s original surgery formula and Park’s regularized surgery formula, with the distinction arising from order-of-summation issues (Svoboda, 26 Sep 2025, Park, 2021).

The paper also isolates a relation to the Kashaev invariant in the figure-eight case. For $K$41, Svoboda reports

$K$42

while the Kashaev invariant satisfies

$K$43

A remarkable empirical observation is that

$K$44

has integer coefficients, beginning

$K$45

The paper does not formulate this as a theorem, but presents it as part of the “curious relation” between GM asymptotics at roots of unity and the Kashaev invariant (Svoboda, 26 Sep 2025).

6. Examples, scope, and current limitations

Several examples are especially explicit. For the right-handed trefoil $K$46,

$K$47

and

$K$48

Ramanujan’s identity then separates this expression into a term identified with $K$49 and a term encoding residues, leading to alternate residue formulas involving partial theta sums and Hecke–Rogers identities (Svoboda, 26 Sep 2025).

For the figure-eight knot $K$50, the inverted Habiro coefficients are trivial: $K$51 The GM coefficients become

$K$52

Its residue theorem yields a nontrivial double-sum identity, and the special residue

$K$53

is identified with a series already known from the analysis of Kashaev invariants for $K$54 (Svoboda, 26 Sep 2025).

The current scope of the theory is controlled by technical hypotheses. LBC is essential for interpreting $K$55 as a $K$56-series or meromorphic function and for justifying residue calculus. It holds for many homogeneous braid knots but fails for some fibered knots, including $K$57, $K$58, $K$59, and $K$60. The surgery formulas depend on convergence of double sums and may have to be treated formally outside certain surgery ranges. Further, the full definition of $K$61 for all knots, the lift $K$62, and the interpretation of $K$63 as the center of an integral form of $K$64 remain conjectural or only partially established. The paper also notes nonabelian branches $K$65, for which Park’s IHS involve additional factors $K$66 and have intriguing relations to descendant series and hyperbolic invariants (Svoboda, 26 Sep 2025).

In this sense, IHS occupy a distinct position in the landscape of quantum knot invariants. They invert Habiro’s original construction in two senses explicitly emphasized by Svoboda: they use negative indices of the same cyclotomic basis, and they move from the $K$67, $K$68 regime of Habiro’s universal series to the $K$69, $K$70-generic regime of the Gukov–Manolescu framework. A plausible implication is that they provide the most natural common language for discussing algebraic regularization, residue calculus, Dehn surgery, and root-of-unity asymptotics within a single two-variable formalism (Svoboda, 26 Sep 2025).