HOMFLY-PT/Kauffman Relation

• The HOMFLY-PT/Kauffman relation is a framework linking two-variable link invariants through distinct forms such as exact lowest-coefficient equality, Jaeger-type expansions, and homological correspondences.
• It utilizes methodologies including graphical state models and quantum-group branching recursions to convert and compare HOMFLY-PT and Kauffman invariants within different knot families.
• Although the lowest-degree coefficients are proven equal, many relations are only analogies or restricted correspondences rather than full polynomial equivalences.

The HOMFLY-PT/Kauffman relation is not a single universal identity but a family of distinct correspondences between two-variable link invariants, their coefficient systems, graphical state models, and categorifications. Across the literature, the relation appears in several sharply different forms: as an exact equality of lowest $z$-degree coefficients, as Jaeger-type expansions of Kauffman invariants into HOMFLY-PT data, as branching-rule recursions attached to quantum-group embeddings, as thin-knot grading changes between colored homologies, and as special-family identities controlled by Birman–Murakami–Wenzl characters and Harer–Zagier factorisability. At the same time, several papers stress equally strongly that many natural-looking “relations” are only analogies or restricted correspondences, not full polynomial equivalences (Przytycki, 2017, Wang et al., 2020, Ito et al., 26 Mar 2026, Petrou et al., 4 Mar 2026).

1. Forms of the relation

The broadest distinction is between direct polynomial relations, coefficient-level relations, graphical expansions, and homological correspondences. In the narrowest and most explicit sense, one paper proves only that the first coefficient of the shifted Kauffman polynomial agrees exactly with the first coefficient of the shifted skein (Homflypt) polynomial: $F_0(a)(L)=P_0(a)(L).$ There is no theorem there identifying higher coefficients $F_i(a)$ with $P_{2i}(a)$, and no formula relating the full HOMFLY-PT polynomial $P_L(a,z)$ to the full Kauffman polynomial $F_L(a,z)$ (Przytycki, 2017).

A second form of relation is Jaeger expansion. In this setting, the Kauffman polynomial of an unoriented diagram is written as a weighted sum of suitably normalized HOMFLY-PT evaluations of globally coherently oriented states. In one formulation,

$\sum_{\sigma\in\Sigma(D)}w(\sigma)[\sigma]=F(D),$

with

$[\vec D]=(a^{-1}q)^{\rot D}P(\vec D).$

This is not an equality of the two polynomials term-by-term; it is an expansion of one invariant into weighted evaluations of the other (Vaz, 2013).

A third form is representation-theoretic specialization. For example, one paper studies the regular-isotopy $SO(2n)$ Kauffman polynomial through Jaeger’s weighted sum of $sl(n)$-specialized HOMFLY-PT evaluations, then reformulates the result through MOY-type planar graphs. Another develops recursion formulas attached to embeddings such as

$F_0(a)(L)=P_0(a)(L).$0

and

$F_0(a)(L)=P_0(a)(L).$1

so that HOMFLY-PT/Kauffman relations become instances of quantum-group branching (Caprau et al., 2013, Chen et al., 2014).

At the categorified level, the relation becomes more varied. Colored Kauffman homology is presented as containing colored HOMFLY homology through universal and diagonal differentials, while a different thin-knot correspondence identifies $F_0(a)(L)=P_0(a)(L).$2-colored quadruply graded Kauffman homology with $F_0(a)(L)=P_0(a)(L).$3-colored quadruply graded HOMFLY-PT homology by an explicit grading change rooted in

$F_0(a)(L)=P_0(a)(L).$4

This is explicitly limited to thin knots and is not invertible in a simple way (Nawata et al., 2013, Wang et al., 2020).

A final form appears only on special knot families. There the relation is the Labastida–Pérez identity

$F_0(a)(L)=P_0(a)(L).$5

proved for specific 3-strand and hyperbolic families, but not for arbitrary knots, and shown to be strictly stronger than Harer–Zagier factorisability once braid index $F_0(a)(L)=P_0(a)(L).$6 is allowed (Petrou et al., 2024, Petrou et al., 4 Mar 2026).

2. Lowest-degree coefficients and coefficient-polynomial refinements

The most concrete direct equality between the two theories occurs at the level of the lowest $F_0(a)(L)=P_0(a)(L).$7-degree coefficient. For oriented links,

$F_0(a)(L)=P_0(a)(L).$8

is defined by

$F_0(a)(L)=P_0(a)(L).$9

and

$F_i(a)$0

If $F_i(a)$1 is the number of components, then

$F_i(a)$2

For the Kauffman polynomial,

$F_i(a)$3

Thus the two “first coefficients” are $F_i(a)$4 and $F_i(a)$5, and the paper’s explicit comparison is exactly

$F_i(a)$6

This equality is operationally central because the Kauffman dynamic program uses it as its base case (Przytycki, 2017).

On the HOMFLY-PT side, the first coefficient also satisfies the Lickorish–Millett formula

$F_i(a)$7

and for the trivial $F_i(a)$8-component link $F_i(a)$9,

$P_{2i}(a)$0

Since $P_{2i}(a)$1, the same formulas describe the first Kauffman coefficient via that equality. By contrast, the paper does not state any theorem equating $P_{2i}(a)$2 with $P_{2i}(a)$3 for $P_{2i}(a)$4 (Przytycki, 2017).

A different coefficient-level parallel appears in recent skein-theoretic work on coefficient polynomials. There, Kawauchi’s A-type coefficient polynomial is described as a coefficient-level refinement of the HOMFLY-PT polynomial, recovering it as a formal power series, with each coefficient itself a link invariant. The new B-type theory plays the same role for the Kauffman polynomial. Its basic coefficient polynomials

$P_{2i}(a)$5

are assembled into

$P_{2i}(a)$6

and satisfy the exact coefficient skein relation

$P_{2i}(a)$7

This does not produce a direct HOMFLY-PT/Kauffman conversion formula; rather, it shows that both theories admit coefficient-level refinements, while the B-type/Kauffman case is structurally harder because of the four-term skein relation and the need to track component-number shifts $P_{2i}(a)$8 (Ito et al., 26 Mar 2026).

A plausible implication is that “HOMFLY-PT/Kauffman relation” at coefficient level should often be read asymmetrically: sometimes as an exact first-coefficient identity, and sometimes only as a parallelism of coefficient formalisms.

3. Graphical expansions, state models, and branching rules

The most classical bridge between the two theories is Jaeger’s expansion of the Kauffman polynomial into HOMFLY-PT-type data. In one formulation for an unoriented link diagram $P_{2i}(a)$9, local replacements at each crossing produce globally coherently oriented states $P_L(a,z)$0, each evaluated by

$P_L(a,z)$1

and summed with weights $P_L(a,z)$2 to recover the Kauffman polynomial: $P_L(a,z)$3 Representation-theoretically, this is interpreted as an $P_L(a,z)$4-expansion of the $P_L(a,z)$5-polynomial after the specialization $P_L(a,z)$6 (Vaz, 2013).

That relation admits several graphical reformulations. One paper combines Jaeger’s expansion with the MOY framework and constructs a state summation model for the $P_L(a,z)$7 Kauffman polynomial in terms of planar unoriented $P_L(a,z)$8-valent graphs. The local crossing rule becomes

$P_L(a,z)$9

and the key structural point is that one must include not only crossing-type oriented vertices but also alternating oriented vertices when translating Jaeger’s formula into the MOY language (Caprau et al., 2013).

A complementary comparison studies two oriented $F_L(a,z)$0-valent graphical models for the Kauffman polynomial. The HJ model first applies Jaeger’s HOMFLY-PT expansion and then the Kauffman–Vogel graph model for HOMFLY-PT. The WF model first applies the Kauffman–Vogel graph model for Kauffman and then Wu’s graph-level Jaeger formula. The main theorem states that each HJ term corresponds to a disjoint family $F_L(a,z)$1 of WF terms satisfying

$F_L(a,z)$2

Thus the HOMFLY-PT-based HJ model is a compressed version of the finer WF model, not a different invariant (Jin, 2012).

A broader skein-theoretic framework organizes these relations through functors induced by embeddings of quantized universal enveloping algebras. For the direct HOMFLY-PT/Kauffman bridge, the functor

$F_L(a,z)$3

expresses the framed Kauffman skein theory with parameter $F_L(a,z)$4 in HOMFLY-PT skein terms. More generally, the mixed functor

$F_L(a,z)$5

encodes the new branching formulas corresponding to

$F_L(a,z)$6

In these mixed-state recursions, Kauffman-type non-oriented strands and HOMFLY-PT-type oriented strands coexist in the target category (Chen et al., 2014).

Not every graphical comparison is a direct relation between the two two-variable polynomials themselves. For the Dubrovnik Kauffman polynomial, one recent state model leads to Gauss diagram formulae for Vassiliev invariants. The sharp comparison there is that the Kauffman and HOMFLY-PT state models both specialize to the same Jones polynomial, but they yield different explicit Gauss diagram expressions for its coefficients before arrow-diagram identities are applied. Thus the relation is comparative rather than algebraically identificatory (Zhang, 2023).

4. Categorified correspondences and differential structures

At the homological level, the relation becomes substantially richer. Colored Kauffman homology is presented as containing colored HOMFLY homology through several differential mechanisms. For rectangular colors, universal differentials satisfy

$F_L(a,z)$7

and for symmetric colors the universal-differential regrading is

$F_L(a,z)$8

A second family, the diagonal differentials, also has HOMFLY homology as its homology and is characterized by the condition that every surviving generator $F_L(a,z)$9 satisfies

$\sum_{\sigma\in\Sigma(D)}w(\sigma)[\sigma]=F(D),$0

These constructions are formulated only after passing to quadruply graded Kauffman homology, with gradings $\sum_{\sigma\in\Sigma(D)}w(\sigma)[\sigma]=F(D),$1 and

$\sum_{\sigma\in\Sigma(D)}w(\sigma)[\sigma]=F(D),$2

The paper is explicit that quadruple grading is essential for making the differential structure visible (Nawata et al., 2013).

A different categorified bridge appears for thin knots. There the quadruply graded $\sum_{\sigma\in\Sigma(D)}w(\sigma)[\sigma]=F(D),$3-colored Kauffman homology can be obtained from the quadruply graded $\sum_{\sigma\in\Sigma(D)}w(\sigma)[\sigma]=F(D),$4-colored HOMFLY-PT homology by an explicit change of gradings. At the level of Poincaré polynomials,

$\sum_{\sigma\in\Sigma(D)}w(\sigma)[\sigma]=F(D),$5

The paper states explicitly that this holds for thin knots only, that it compares symmetric $\sum_{\sigma\in\Sigma(D)}w(\sigma)[\sigma]=F(D),$6 colors on the Kauffman side with rectangular $\sum_{\sigma\in\Sigma(D)}w(\sigma)[\sigma]=F(D),$7 colors on the HOMFLY-PT side, and that it is not true for thick knots. It also gives a second, differential-based relation in the triply graded specialization: $\sum_{\sigma\in\Sigma(D)}w(\sigma)[\sigma]=F(D),$8 hence

$\sum_{\sigma\in\Sigma(D)}w(\sigma)[\sigma]=F(D),$9

This is a same-color Kauffman/HOMFLY-PT relation distinct from the $[\vec D]=(a^{-1}q)^{\rot D}P(\vec D).$0 grading-change rule (Wang et al., 2020).

Further structural parallels appear in cyclotomic expansions for superpolynomials. One paper proposes a cyclotomic expansion for reduced colored HOMFLY-PT superpolynomials

$[\vec D]=(a^{-1}q)^{\rot D}P(\vec D).$1

and an analogous cyclotomic expansion for reduced colored Kauffman superpolynomials

$[\vec D]=(a^{-1}q)^{\rot D}P(\vec D).$2

The two expansions use the same building blocks

$[\vec D]=(a^{-1}q)^{\rot D}P(\vec D).$3

with a systematic doubling pattern on the Kauffman side. The paper does not provide a direct transformation law between the two theories, but it records the empirical relation

$[\vec D]=(a^{-1}q)^{\rot D}P(\vec D).$4

for all tested examples (Chen, 2015).

A plausible implication is that the homological HOMFLY-PT/Kauffman relation is best viewed as a network of compatible differentials, grading changes, and representation-theoretic coincidences rather than as a single categorified specialization.

5. Special families, BMW characters, and HZ factorisability

A much more rigid relation exists on specific knot families. In that setting the key identity is

$[\vec D]=(a^{-1}q)^{\rot D}P(\vec D).$5

with

$[\vec D]=(a^{-1}q)^{\rot D}P(\vec D).$6

One paper proves this for several infinite HZ-factorisable hyperbolic families, including

$[\vec D]=(a^{-1}q)^{\rot D}P(\vec D).$7

and conjectures that for knots

$[\vec D]=(a^{-1}q)^{\rot D}P(\vec D).$8

Here the Harer–Zagier transform is

$[\vec D]=(a^{-1}q)^{\rot D}P(\vec D).$9

The paper also states that, for knots, the relation

$SO(2n)$0

is equivalent to the vanishing of the two-crosscap BPS invariants: $SO(2n)$1 For 2-component links, the analogous criterion is weaker and admits explicit exceptions (Petrou et al., 2024).

A later paper explains these special-family identities through characters of the Birman–Murakami–Wenzl algebra. The Kauffman polynomial has the BMW character expansion

$SO(2n)$2

where the lower-box sectors $SO(2n)$3 are precisely what obstruct a naive replacement of $SO(2n)$4 HOMFLY characters by $SO(2n)$5 quantum dimensions. On 3 strands the correction term is controlled by a single lower-sector character: $SO(2n)$6 and the HOMFLY-PT/Kauffman relation becomes equivalent to the parity condition

$SO(2n)$7

The paper proves the relation for the 3-strand family

$SO(2n)$8

and shows that, on this family, HZ factorisability and the HOMFLY-PT/Kauffman relation coincide. For 4 strands, however, additional $SO(2n)$9 BMW sectors appear, and explicit counterexamples show that HZ factorisability no longer implies $sl(n)$0, even though the converse survives as a sufficient condition in braid index $sl(n)$1 and higher (Petrou et al., 4 Mar 2026).

This suggests that special-family HOMFLY-PT/Kauffman identities are governed not by a universal skein specialization but by the vanishing pattern of lower BMW character sectors.

6. Misconceptions, analogies, and the scope of the term

A recurrent theme in the literature is that many works touching both theories do not establish a direct HOMFLY-PT/Kauffman polynomial identity. Several papers are explicit on this point. The dynamic-programming paper concerns only the first coefficients $sl(n)$2 and $sl(n)$3, not the full polynomials (Przytycki, 2017). The B-type coefficient-polynomial paper develops a parallel coefficient-level framework for Kauffman analogous to Kawauchi’s A-type refinement of HOMFLY-PT, but states directly that it provides a conceptual and methodological parallelism rather than a direct relation between the polynomials themselves (Ito et al., 26 Mar 2026).

Other papers concern the relation between HOMFLY-PT calculations and Kauffman-bracket-style methods rather than the Kauffman polynomial. For bipartite or matched diagrams built from antiparallel lock tangles, the fundamental HOMFLY polynomial admits a planar decomposition

$sl(n)$4

with

$sl(n)$5

and the same state-sum combinatorics as the Kauffman bracket on a precursor diagram after the substitutions

$sl(n)$6

This is a HOMFLY/Kauffman-bracket relation, not a HOMFLY/Kauffman-polynomial relation (Anokhina et al., 2024).

The same distinction is central in the generalized Goeritz-matrix approach for bipartite links. There, the ordinary Goeritz method—“an alternative to the Kauffman bracket”—is generalized to compute bipartite HOMFLY-PT polynomials through a quadruple Goeritz matrix. The bridge is again to the Kauffman bracket and Jones/Goeritz machinery, not to the Kauffman polynomial as such (Anokhina et al., 3 Jul 2025).

Finally, some geometric and physical frameworks develop HOMFLY-PT homology without formulating any direct Kauffman comparison. The 3d TQFT/Hilbert-scheme model identifies the Hilbert space of a closed braid with HOMFLY-PT homology,

$sl(n)$7

and realizes the closed sector as

$sl(n)$8

This provides rich machinery—defects, categorical traces, Hilbert schemes, matrix factorizations—but no direct HOMFLY-PT/Kauffman relation in the paper itself. This suggests that future geometric comparisons, if they are built, would likely require changing the target geometry or defect data rather than merely specializing formulas (Oblomkov et al., 2018).

The term “HOMFLY-PT/Kauffman relation” therefore covers at least five non-equivalent phenomena: lowest-coefficient equality, Jaeger-type expansion, branching-rule recursion, thin-knot homological correspondence, and special-family polynomial identities. Any precise use of the term must specify which of these is intended.