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Integrality of genus-$g$ indices with adjoint Reidemeister torsions of twist knots

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Ryoto Tange, Yuji Terashima, Yoshikazu Yamaguchi

Geometric Topology · Submitted 25 Jun 2026 · 19 references

Pith's verdict: The paper proves integrality of the summed adjoint Reidemeister torsions for twist knots at the meridian and supplies explicit generating-function examples.

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arXiv ID2606.27006 (version 1)
Pith IDHOLRQY5Y, Pith's permanent record of this paper and its review
CitationRyoto Tange, Yoshikazu Yamaguchi, Yuji Terashima. "Integrality of genus-$g$ indices with adjoint Reidemeister torsions of twist knots." arXiv:2606.27006 (2026). https://pith.science/paper/HOLRQY5Y
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The paper

The paper claims

Sums of adjoint Reidemeister torsions are integers for twist knots at the meridian.

The reading

The paper proves that the sum of adjoint Reidemeister torsions is an integer for twist knots when using the meridian. A sympathetic reader would care because this provides an integrality result linking torsion invariants to integer quantities in knot theory. The authors also construct generating functions for these sums in concrete examples. This establishes a property that holds across the family of twist knots.

What carries the argument

The sum of the adjoint Reidemeister torsions for the meridian on twist knot complements.

Extended reading notes

Core claim

We consider the sum of the adjoint Reidemeister torsions and prove the integrality for twist knots and the meridian. We also give some concrete examples of the generating functions for these sums.

Load-bearing premise

The definitions and normalization conventions for the adjoint Reidemeister torsion are taken as standard and well-defined for the meridian of every twist knot.

Editorial extensions

If this is right

  • The sums yield integers for every twist knot.
  • Generating functions encode the sums for specific twist knots.
  • The integrality applies specifically to the meridian representation.
  • The result concerns genus-g indices in combination with these torsions.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The same approach might apply to other families of knots beyond twist knots.
  • Explicit generating functions could allow computation of higher order terms or asymptotics.
  • Integrality may connect these torsions to other integer invariants like Alexander polynomials in unexpected ways.
Original abstract

We consider the sum of the adjoint Reidemeister torsions and prove the integrality for twist knots and the meridian. We also give some concrete examples of the generating functions for these sums.

Figures
Figure 1. the diagram of J(−2, −n) We consider the topological invariants of twist knot exteriors which are de￾termined by the knot groups. Let K be a twist knot J(−2, −2l) in S 3 . The diagram below illustrates the case of l = 2, which corresponds to the knot 52
Figure 2. the diagrams of J(−2, 3) and the mirror image of J(−2, −4) Remark 1. Knot groups are isomorphic under mirror image. Since the twist knot J(−2, −2l + 1) is equivalent to the mirror image of J(−2, −2l), that is J(2, 2l), we focus on the twist knots J(−2, −2l). The twist knot K is hyperbolic except for l = 0 and 1. In the case of l = 0 and 1, K is unknot and the trefoil knot respectively. We denote by MK the knot exterior S 3 \ intN(K) where N(K) is a tubular neighbourhood of K. The knot group π1(MK) of K admits the presentation π1(MK) = ⟨g1, g2 | w l g1 = g2w l ⟩, 2
Figure 3. the diagram of J(−2, 2) = 41 Example 1. Let K = J(−2, 2) corresponding to the figure-eight knot 41. Then we have F(x, z) = z 2 − (x 2 − 1)z + (x 2 − 1) = z 2 − f1(x) · z + f2(x) = (z − a1)(z − a2). Note that fj (x) = ej (a1, a2) ∈ Z[x] for j = 1, 2. Since ∂F ∂z = 2z −f1(x), we have e1(b1, b2) = b1 + b2 = 2(a1 + a2) − 2f1(x) = 2e1(a1, a2) − 2e1(a1, a2) = 0, e2(b1, b2) = b1b2 = 4a1a2 − 2f1(x) · (a1 + a2) + f1(x) 2 = −
Figure 4. the diagram of J(−2, −4) We exclude zeros of ∂F/∂z on F(x, z) = 0 from the domain of x according to their resultant which determines the common roots of two polynomials. The resultant Res(F, Fz) of F and Fz = ∂F/∂z turns out to be Res(F, Fz) = (x 4 − 2x 3 − 5x 2 + 14x − 7)(x 4 + 2x 3 − 5x 2 − 14x − 7). We also exclude the case where the set Tr−1 µ (x) contains reducible characters, namely when 2x 2 − 7 = 0. The domain of x is defined as (2x 2 − 7)(x 4 − 2x 3 − 5x 2 + 14x − 7)(x 4 + 2x 3 − 5x 2 − 14x − 7) ̸= 0. Now we turn to compute E(x, t) and Z(x, t). Note that fj (x) = ej (a1,…

Figures reproduced from arXiv:2606.27006 by the authors.

References

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What this paper does with the arXiv papers it cites

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

What it cites · 12 sentences without a confident label

Cites as background 7 papers
  • It would be interesting to connect the recent relationships between gauge theory and quantum knot invariants, such as the colored Jones polynomial through the gauge–quiver correspondence [ref] and the gauge–knot correspondence [ref] , with the relationship between adjoint Reidemeister torsion and gauge theory studied in this paper, via refinements of the volume conjecture.

    Physics and geometry of knots-quivers correspondence · Introduction · 0.96

  • It would be interesting to connect the recent relationships between gauge theory and quantum knot invariants, such as the colored Jones polynomial through the gauge–quiver correspondence [ref] and the gauge–knot correspondence [ref] , with the relationship between adjoint Reidemeister torsion and gauge theory studied in this paper, via refinements of the volume conjecture.

    Knots-quivers correspondence · Introduction · 0.96

  • It would be interesting to connect the recent relationships between gauge theory and quantum knot invariants, such as the colored Jones polynomial through the gauge–quiver correspondence [ref] and the gauge–knot correspondence [ref] , with the relationship between adjoint Reidemeister torsion and gauge theory studied in this paper, via refinements of the volume conjecture.

    BPS states, knots and quivers · Introduction · 0.96

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