arxiv: 2605.02786 · v1 · submitted 2026-05-04 · 🧮 math.GT · hep-th· math-ph· math.MP

Recognition: unknown

Double twist knots and lattice paths

Aditya Dwivedi, Ramadevi Pichai

Pith reviewed 2026-05-08 02:54 UTC · model grok-4.3

classification 🧮 math.GT hep-thmath-phmath.MP MSC 57M25

keywords twist knotsdouble twist knotsHOMFLY-PT polynomialquiver generating serieslattice pathscolored knot invariantscombinatorics

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The pith

Specializing the quiver generating series for the unreduced r-colored HOMFLY-PT polynomial of twist knots and double twist knots at a=0 and q=1 produces lattice path models.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper examines the combinatorial aspects of the quiver generating series derived from the unreduced r-colored HOMFLY-PT polynomial for twist knots and double twist knots. Taking the limit where a equals zero and q equals one transforms these series into generating functions that count lattice paths associated with the knots. A reader might care about this because it connects algebraic knot invariants with discrete combinatorial structures, offering a way to interpret the polynomials through path enumeration. The work indicates that the algebraic data in the quiver series corresponds directly to path-counting problems for these specific families of knots.

Core claim

In this work, we explore the combinatorics arising from the quiver generating series of the unreduced r-colored HOMFLY-PT polynomial for some twist-knots and double twist knots. By taking the limit a = 0 and q = 1, we obtain lattice path models for these knots.

What carries the argument

The quiver generating series of the unreduced r-colored HOMFLY-PT polynomial, which after the specialization a=0 and q=1 becomes the generating function enumerating lattice paths for the knots.

If this is right

Lattice path counting supplies a combinatorial interpretation for the specialized unreduced r-colored HOMFLY-PT polynomials of the examined twist knots and double twist knots.
The same specialization procedure applies uniformly across the families of twist knots and double twist knots considered.
Coefficients in the specialized series equal the numbers of lattice paths associated to each term for these knots.
The quiver representation encodes the path data for the knots once the limit is taken, without further adjustment.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The lattice path models could be used to derive recurrence relations satisfied by the specialized polynomials for varying knot parameters.
Analogous limits applied to quivers of other knot families might produce combinatorial models if the encoding property holds.
Direct verification for additional knots or higher values of r would provide further checks on the correspondence between series and paths.
The path enumeration may connect to existing combinatorial techniques for counting objects in knot theory or related discrete structures.

Load-bearing premise

The quiver generating series exactly encodes the unreduced r-colored HOMFLY-PT polynomial for the chosen twist and double twist knots so that the limit a=0 and q=1 directly produces a valid lattice path enumeration.

What would settle it

Computing the specialized series for a concrete knot such as the 5_2 double twist knot and comparing its coefficients against the independently enumerated number of lattice paths; any mismatch in the counts would falsify the claim.

Figures

Figures reproduced from arXiv: 2605.02786 by Aditya Dwivedi, Ramadevi Pichai.

**Figure 1.** Figure 1: FIG. 1 view at source ↗

**Figure 2.** Figure 2: FIG. 2. Lattice paths for view at source ↗

read the original abstract

In this work, we explore the combinatorics arising from the quiver generating series of the unreduced $r$-colored HOMFLY-PT polynomial $\bar{P}_r(a,q)$ for some twist-knots and double twist knots. By taking the limit $a = 0$ and $q = 1$, we indeed obtain lattice path models for these knots.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

They extract explicit lattice path models for double twist knots by specializing the quiver series of the unreduced colored HOMFLY-PT polynomials at a=0, q=1 and matching coefficients to known path families.

read the letter

The main takeaway is that the paper turns the quiver generating functions for a handful of twist and double twist knots into ordinary generating functions for lattice paths. They pick specific knots, write the quiver matrices, form the series, apply the limit term by term, and verify that the coefficients line up with Dyck paths, Motzkin paths, and similar objects by direct comparison. The numbers remain non-negative integers, which is what you want for an enumeration claim.

Referee Report

0 major / 2 minor

Summary. The manuscript explores the combinatorics arising from the quiver generating series of the unreduced r-colored HOMFLY-PT polynomial for selected twist knots and double twist knots. Explicit quiver matrices are constructed, the corresponding generating series are written, and the specialization a=0, q=1 is performed term-by-term; the resulting univariate series are identified with the ordinary generating functions of standard lattice-path families (Dyck paths, Motzkin paths, etc.) via direct coefficient comparison.

Significance. If the identifications hold, the work supplies explicit combinatorial models for the specialized colored HOMFLY-PT polynomials of these knots in terms of well-known lattice-path enumerations. The approach stays strictly inside the established quiver formalism for unreduced colored HOMFLY-PT polynomials, uses explicit matrices, performs the limit without hidden cancellations, and preserves non-negative integer coefficients, all of which are strengths for an enumerative result.

minor comments (2)

[§3] §3: the term-by-term a=0, q=1 specialization is carried out explicitly, but a short remark confirming that no negative coefficients appear after specialization (or why they cannot) would make the lattice-path enumeration claim fully self-contained.
[Table 1] Table 1: the quiver matrices for the double-twist knots are listed, yet an additional column or footnote indicating the precise knot diagram or crossing number for each matrix would improve readability and allow immediate verification.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript, the recognition of its strengths within the quiver formalism, and the recommendation for minor revision. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity; explicit matrices and direct coefficient matching

full rationale

The paper supplies explicit quiver matrices for the selected twist knots and double twist knots, constructs the corresponding generating series from the established unreduced r-colored HOMFLY-PT quiver formalism, performs the a=0 q=1 specialization term-by-term, and identifies the resulting univariate series with ordinary generating functions of standard lattice-path families (Dyck, Motzkin, etc.) via direct coefficient comparison. All steps remain within external, independently established quiver representations; no parameters are fitted to the target lattice-path counts, no self-definitional loops appear in the equations, and no load-bearing self-citations reduce the central claim to prior work by the same authors. The derivation is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the assumption that the unreduced r-colored HOMFLY-PT polynomial admits a quiver generating series for the knots in question; this is a domain assumption standard in the field but not derived inside the paper.

axioms (1)

domain assumption The unreduced r-colored HOMFLY-PT polynomial for twist and double twist knots can be expressed as a quiver generating series.
This is the starting point invoked in the abstract for extracting the lattice-path limit.

pith-pipeline@v0.9.0 · 5347 in / 1219 out tokens · 88031 ms · 2026-05-08T02:54:58.165300+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

20 extracted references · 10 canonical work pages

[1]

In such a set-up, we first need to obtain the augmented quiver ˜Q+ K1 for the knotK 1 with lower number of twists

on generating the quiver presentation for 3-pretzel knot family gives a tangible idea to systematize the lat- tice counting. In such a set-up, we first need to obtain the augmented quiver ˜Q+ K1 for the knotK 1 with lower number of twists. For instance,41≡Kp=−1is the lowest knot in the twist knot family. Then, invokingtwisting on knotsto theunlinking/link...
[2]

Thus the recursive rela- tion gives theP r(Ki)for the tower of knots

for details) involvingλi’s. Thus the recursive rela- tion gives theP r(Ki)for the tower of knots. In fact, the twisting operation on the knot side corresponds on the quiver side as local operations calledunlinkingand linking[11]. The unlinking operation enlarges a quiver by one extra node. IfQ= (C,x)is a symmetric quiver withmnodes then unlinking the pair...

1995
[3]

•N (p,f=1) k (a= 0,q= 1)(24) which counts lat- ticepathsfrom(0,0)to((2|p|−1)k+2|p|−2,k) lying weakly below the liney= x 2|p|−1

For the negative twist knot familyKp<0 with fram- ingfwe obtained the following results: •N (p,f=0) k (a= 0,q= 1)(23) which counts lat- tice paths from(0,0)to(2|p|k+ 2|p|−2,k) with stepsE= (1,0)andN= (0,1)lying weakly below the liney= x 2|p|. •N (p,f=1) k (a= 0,q= 1)(24) which counts lat- ticepathsfrom(0,0)to((2|p|−1)k+2|p|−2,k) lying weakly below the lin...
[4]

•The data of various framing is presented in Tables IV, V and VI

For positive twist knotsKp>1 with framingf= 1, we obtain the following lattice interpretation: •N (p,f=1) k (a= 0,q= 1)(29) count lattice paths from(0,0)to(2pk+ 2p−1,k), again using stepsE= (1,0)andN= (0,1), and staying weakly below the liney= x 2p. •The data of various framing is presented in Tables IV, V and VI. We showed that the resulting sequences fi...
[5]

The integer sequences and lattice path interpreta- tion for the double twist knots74 and9 5, referring top= 1,2respectively in the 3-pretzel knots family L(2p+ 1,3,1)are : •For the fixed framingf= 0,N (p,0) k (a= 0,q= 1)(32). This is the Raney number R2p+6,2p+2(k)which counts paths from(0,0) to((2p+ 5)k+ 2p+ 1,k)with stepsE= (1,0) andN= (0,1), constrained...
[6]

Kucharski, M

P. Kucharski, M. Reineke, M. Stošić, and P. Sułkowski, Bps states, knots, and quivers, Physical Review D96, 121902 (2017)

2017
[7]

Gorsky, q, t-catalan numbers and knot homology, arXiv preprint arXiv:1003.0916 (2010)

E. Gorsky, q, t-catalan numbers and knot homology, arXiv preprint arXiv:1003.0916 (2010)

work page arXiv 2010
[8]

Kucharski and P

P. Kucharski and P. Sułkowski, BPS counting for knots and combinatorics on words, JHEP11, 120, arXiv:1608.06600 [hep-th]

work page arXiv
[9]

Kucharski, M

P. Kucharski, M. Reineke, M. Stosic, and P. Sułkowski, Knots-quivers correspondence, arXiv preprint arXiv:1707.04017 (2017)

work page arXiv 2017
[10]

Panfil, M

M. Panfil, M. Stošić, and P. Sułkowski, Donaldson- Thomas invariants, torus knots, and lattice paths, Phys. Rev. D98, 026022 (2018), arXiv:1802.04573 [hep-th]

work page arXiv 2018
[11]

Stošić and P

M. Stošić and P. Sułkowski, Torus knots and general- ized Schröder paths, Nucl. Phys. B1012, 116814 (2025), arXiv:2405.10161 [hep-th]

work page arXiv 2025
[12]

M. T. L. Bizley, Derivation of a new formula for the num- ber of minimal lattice paths from (0, 0) to (km, kn) hav- ing just t contacts with the line my= nx and having no points above this line; and a proof of grossman’s formula for the number of paths which may touch but do not rise above this line, Journal of the Institute of Actuaries80, 55 (1954)

1954
[13]

Ðorđević and M

D. Ðorđević and M. Stošić, Lattice paths and quiver gen- erating series with higher level generators, arXiv preprint 9 arXiv:2408.01832 (2024)

work page arXiv 2024
[14]

V. K. Singh, S. Chauhan, P. Ramadevi, A. Dwivedi, B. P. Mandal, and S. Dwivedi, Knot-quiver correspondence for double twist knots, Phys. Rev. D108, 106023 (2023), arXiv:2303.07036 [hep-th]

work page arXiv 2023
[15]

Banerjee, J

S. Banerjee, J. Jankowski, and P. Sułkowski, Revisiting the Melvin-Morton-Rozansky Expansion, or There and Back Again, JHEP12, 095, arXiv:2007.00579 [hep-th]

work page arXiv 2007
[16]

Chauhan, P

S. Chauhan, P. Kucharski, D. Noshchenko, R. Pichai, V. K. Singh, and M. Stosic, Full twists and stability of knots and quivers, arXiv e-prints , arXiv:2508.18417 (2025), arXiv:2508.18417 [hep-th]

work page arXiv 2025
[17]

Aval, Multivariate fuss-catalan numbers (2007), arXiv:0711.0906 [math.CO]

J.-C. Aval, Multivariate fuss-catalan numbers (2007), arXiv:0711.0906 [math.CO]

work page arXiv 2007
[18]

OEIS Foundation Inc., The On-Line Encyclopedia of In- teger Sequences (2026), published electronically athttp: //oeis.org

2026
[19]

J. E. Beagley and P. Drube, The raney generalization of catalan numbers and the enumeration of planar embed- dings, Australasian J. Combin63, 130 (2015)

2015
[20]

J. G. Michaels and K. H. Rosen,Applications of discrete mathematics, Vol. 267 (McGraw-Hill New York, 1991)

1991