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Alexander polynomial

The Alexander polynomial is a knot invariant in algebraic topology that assigns to each oriented knot or link in three-dimensional Euclidean space a Laurent polynomial with integer coefficients, unique up to multiplication by units in the ring \mathbb{Z}[t, t^{-1}] (specifically, \pm t^k for integer k).^[1] It was introduced by American mathematician J. W. Alexander II in his seminal 1928 paper, where it emerged from the study of the homology of the infinite cyclic cover of the knot complement as a module over \mathbb{Z}[t^{\pm 1}].^[2] As the first polynomial invariant for knots, it provided a major advance in distinguishing knot types under ambient isotopy, remaining the primary such tool until the discovery of the Jones polynomial in 1984.^[3] The polynomial can be computed algebraically from the knot group's Wirtinger presentation via Fox free calculus, yielding the Alexander matrix whose (n-1)×(n-1) minor determinants generate the first elementary ideal, with the polynomial as its gcd, up to units.^[1] Geometrically, it arises from a Seifert surface spanning the knot, where a basis for the first homology yields a Seifert matrix V whose symmetrized form t^{1/2}V - t^{-1/2}V^T has determinant equal to the normalized Alexander polynomial \Delta_K(t).^[3] An equivalent recursive definition via the Conway skein relation, \Delta_{L_+} - \Delta_{L_-} = (t^{1/2} - t^{-1/2})\Delta_{L_0}, was later developed by John Horton Conway in 1968, facilitating computations from knot diagrams.^[1] Key properties include symmetry \Delta_K(t) = \Delta_K(t^{-1}) (up to units), evaluation \Delta_K(1) = 1, and the fact that its degree bounds the Seifert genus of the knot from below.^[1] However, it is not a complete invariant, as it fails to distinguish a knot from its mirror image and many distinct knots share the same polynomial, for example, the Kinoshita–Terasaka knot and the Conway knot both have the trivial Alexander polynomial.^[3]^[4] Despite these limitations, the Alexander polynomial remains foundational in knot theory, influencing developments in quantum invariants and 3-manifold topology.^[1]

Fundamentals

Definition

The Alexander polynomial is a fundamental invariant in knot theory, assigning to each oriented knot K embedded in the three-sphere S^3 a Laurent polynomial \Delta_K(t) \in \mathbb{Z}[t, t^{-1}] with integer coefficients, defined up to multiplication by units \pm t^k for some integer k \in \mathbb{Z}.^[5]^[6] This polynomial captures topological properties of the knot and serves as a distinguishing feature among different knot types.^[7] To define it formally, consider the knot complement X = S^3 \setminus K, an open 3-manifold whose fundamental group \pi_1(X) abelianizes to \mathbb{Z}, with the generator corresponding to the homotopy class of a meridian curve around K. The infinite cyclic cover \tilde{X} \to X is the unique connected covering space with deck transformation group isomorphic to \mathbb{Z}, induced by the kernel of the abelianization map \pi_1(X) \to \mathbb{Z}. The Alexander module is then the first homology group H_1(\tilde{X}; \mathbb{Z}) viewed as a module over the Laurent polynomial ring \Lambda = \mathbb{Z}[t, t^{-1}], where the action of t arises from the generator of the deck group.^[8] The Alexander polynomial \Delta_K(t) is derived from this module as a generator of its first Fitting ideal (up to units in \Lambda), which encodes the torsion structure of the module over the principal ideal domain \Lambda.^[8] For knots, the module is torsion, ensuring the polynomial is well-defined and non-trivial in general. Although initially formulated for knots, the construction extends naturally to links with \mu components by replacing the infinite cyclic cover with the corresponding \mathbb{Z}^\mu-cover of the link complement, yielding a multi-variable Alexander polynomial \Delta_L(t_1, \dots, t_\mu) \in \mathbb{Z}[t_1^{\pm 1}, \dots, t_\mu^{\pm 1}], again defined up to units.^[9]

Historical development

The Alexander polynomial emerged as a pivotal invariant in knot theory through the work of James Waddell Alexander II, who introduced it in his 1928 paper "Topological Invariants of Knots and Links," building on presentations given as early as 1927.^[10] This polynomial, derived from the homology of the infinite cyclic cover of the knot complement, marked the first algebraic tool capable of distinguishing many non-trivial knots from the unknot and from each other.^[10] The development occurred amid the rapid advancement of algebraic topology in the early 20th century, where Henri Poincaré's 1895 introduction of the fundamental group in "Analysis Situs" provided essential machinery for analyzing the topology of spaces like knot complements. Kurt Reidemeister's 1926 formulation of equivalence moves for knot diagrams in "Knoten und Verkettungen" further enabled rigorous combinatorial studies of knot types, setting the stage for invariants like Alexander's. In 1927, Alexander collaborated with G. B. Briggs on "On Types of Knotted Curves," establishing a systematic notation for tabulating prime knots up to eight crossings, which facilitated early computations and classifications using the polynomial.^[11] Among the key milestones, Alexander computed the polynomial for the trefoil knot (yielding t^{-1} - 1 + t) and the figure-eight knot (yielding -t^{-1} + 3 - t) in his 1928 paper, illustrating its utility in distinguishing these knots.^[10] In the 1950s, Ralph Fox advanced the theory by developing the free differential calculus, a method for deriving Alexander invariants directly from group presentations via operations in the free group ring, as detailed in his series of papers beginning with "Free Differential Calculus. I" in 1954.^[12] This refinement simplified computations and extended the polynomial's applicability to links and groups.^[13] The Alexander polynomial's influence persisted into the 1980s, when Vaughan Jones introduced the Jones polynomial in 1984, revealing connections between classical invariants like the Alexander polynomial and quantum topology, thus inspiring further developments.

Computation

Presentation matrix method

The presentation matrix method computes the Alexander polynomial of a knot K \subset S^3 using a finite presentation of the knot group \pi_1(S^3 \setminus K). This approach begins with the Wirtinger presentation, derived from a diagram of the knot. In a knot diagram with n arcs, assign a generator x_i to each arc. At each crossing, the relation arises from the path along the under-arc being equal to the over-arc conjugating the incoming under-arc segment; specifically, if arcs a, b, and c meet at a crossing with b over and a incoming under to c outgoing under (oriented consistently), the relator is c = b a b^{-1}, or rearranged as b a b^{-1} c^{-1} = 1. This yields a presentation with n generators and n or more relators, though one relator is often redundant due to the group's structure.^[14]^[1] To obtain the Alexander matrix, apply Fox free differential calculus to the relators in the free group ring. The Fox derivative \partial / \partial x_j satisfies rules analogous to the product rule: \partial / \partial x_j (1) = 0, \partial / \partial x_j (x_i) = \delta_{ij}, \partial / \partial x_j (w z) = \partial / \partial x_j (w) + w \cdot \partial / \partial x_j (z), and \partial / \partial x_j (w^{-1}) = - w^{-1} \cdot \partial / \partial x_j (w), extended linearly to the group ring. For a presentation \langle x_1, \dots, x_n \mid r_1, \dots, r_m \rangle with m \geq n, compute the Jacobian matrix A whose (i,j)-entry is the image of \partial r_i / \partial x_j under the abelianization map \phi: \mathbb{Z}[F] \to \mathbb{Z}[t, t^{-1}], where F is the free group on the x_k and \phi(x_k) = t for all k (since the abelianization of the knot group is \mathbb{Z}, generated by the meridian). This matrix A is m \times n.^[13]^[14]^[1] The Alexander polynomial \Delta_K(t) is then the determinant of any (n-1) \times (n-1) minor of A, obtained by deleting one row and one column; all such minors yield determinants differing by units \pm t^k in the Laurent polynomial ring \mathbb{Z}[t, t^{-1}]. The overall process involves: (1) abelianizing the knot group to \mathbb{Z} via the Hurewicz homomorphism, reflecting the infinite cyclic covering space of the knot complement; (2) lifting to the chain complex of the covering, where the Alexander module's presentation arises from the Fox derivatives; and (3) computing the torsion of the first homology of the infinite cyclic cover, encoded by \Delta_K(t) as the order ideal generator. Normalization conventions fix \Delta_K(1) = 1 and make the polynomial symmetric, \Delta_K(t) = \Delta_K(t^{-1}), up to units.^[13]^[1] For the trefoil knot, consider its standard diagram with three arcs and Wirtinger generators x, y. The single essential relator is r = x y x y^{-1} x^{-1} y^{-1}. The Fox derivatives are \partial r / \partial x = 1 + x y - x y x y^{-1} x^{-1} and \partial r / \partial y = x - x y x y^{-1} - x y x y^{-1} x^{-1} y^{-1}. Abelianizing with \phi(x) = \phi(y) = t yields the $1 \times 2 Alexander matrix entries $1 + t^2 - t and t - t^2 - 1. The 1×1 minor obtained by deleting the y-column is $1 - t + t^2, which is the Alexander polynomial up to units.^[14]^[1]

Seifert matrix approach

The Seifert matrix approach provides a geometric method to compute the Alexander polynomial of an oriented knot in three-dimensional space by leveraging a bounding orientable surface known as a Seifert surface. According to Seifert's theorem from 1934, every oriented knot bounds such a surface, which can be explicitly constructed from a knot diagram using Seifert's algorithm. This algorithm begins by resolving each crossing in the diagram: at an overcrossing followed by an undercrossing in the orientation direction, the strands are smoothed to form disjoint, oriented circles called Seifert circles. These circles are then filled with disks, and at each original crossing, a rectangular band (or ribbon) is attached between the corresponding disks, twisted according to the crossing sign—right-handed for positive crossings and left-handed for negative—to ensure the boundary of the resulting surface is exactly the knot. This disk-band decomposition yields an embedded orientable surface of minimal genus for many knots, facilitating the extraction of topological invariants.^[15] Given a Seifert surface F for the knot, with genus g, a basis \{\alpha_1, \dots, \alpha_{2g}\} is chosen for the first homology group H_1(F; \mathbb{Z}), consisting of simple closed curves on the surface. The Seifert matrix V is the $2g \times 2g integer matrix defined by V_{ij} = \mathrm{lk}(\alpha_i, \hat{\alpha}_j), where \hat{\alpha}_j is the positive push-off of \alpha_j (a parallel curve displaced along the positive normal direction to F in \mathbb{R}^3 \setminus K), and \mathrm{lk} denotes the linking number in S^3. The linking number is computed as half the signed crossings between the two curves, ensuring V captures the intersection form on the surface relative to the ambient space. This matrix is well-defined up to S-equivalence (congruence by integer matrices P with P^T P = I), independent of the choice of surface or basis, as established by Seifert.^[15] The Alexander polynomial is then obtained from the Seifert matrix via the formula \Delta_K(t) = \det(V - t V^T), where V^T is the transpose of V; this determinant is a Laurent polynomial in t with integer coefficients. Different choices of Seifert surface or basis yield polynomials differing by multiplication by units in the Laurent polynomial ring, specifically \pm t^k for some integer k. Normalization conventions fix a unique representative by requiring \Delta_K(1) = 1 (which holds for knots since \det(V - V^T) = \pm 1, reflecting the unimodular nature of the intersection form) and symmetry \Delta_K(t) = \Delta_K(t^{-1}), up to the unit factor; orientation reversal transposes V to V^T, negating the polynomial up to units. These conventions ensure the polynomial is a well-defined knot invariant.^[15]^[16] For the figure-eight knot $4_1, a minimal Seifert surface of genus 1 is constructed via Seifert's algorithm from its standard four-crossing diagram, yielding two Seifert circles connected by two twisted bands. A basis for H_1(F; \mathbb{Z}) consists of two curves c_1 and c_2 along the cores of the bands. The corresponding Seifert matrix is

V = \begin{pmatrix} -1 & 0 \\ 1 & 1 \end{pmatrix},

with entries computed from the linking numbers of the curves and their positive push-offs (e.g., V_{11} = -1 from one self-linking, V_{12} = 0 from no intersection in push-off). Then,

V - t V^T = \begin{pmatrix} t-1 & -t \\ 1 & 1-t \end{pmatrix},

and

\det(V - t V^T) = (t-1)(1-t) + t = -t^2 + 3t - 1.

Normalizing by multiplying by t^{-1} gives \Delta_{4_1}(t) = -t^{-1} + 3 - t, satisfying \Delta(1) = 1 and the symmetry condition.^[16]^[17]

Properties

Algebraic characteristics

The Alexander polynomial \Delta_K(t) of a knot K resides in the Laurent polynomial ring \Lambda = \mathbb{Z}[t, t^{-1}] and exhibits a fundamental symmetry property: \Delta_K(t) = \pm t^k \Delta_K(t^{-1}) for some integer k, which stems from the involution t \mapsto t^{-1} acting on the Alexander module of the knot complement.^[18] This symmetry ensures that the coefficients of \Delta_K(t) are palindromic when the polynomial is normalized appropriately.^[19] Conventionally, the Alexander polynomial is normalized to be monic (leading coefficient 1) with integer coefficients, and for knots, it satisfies \Delta_K(1) = 1, reflecting the fact that the knot complement has trivial homology in degree 1.^[18] Additionally, the degree of \Delta_K(t) is even under this normalization, as the constant term and leading coefficient must align due to the symmetry.^[19] The degree of the Alexander polynomial provides bounds related to the knot's Seifert genus g(K): specifically, \deg \Delta_K(t) \leq 2g(K), with equality achieved when a minimal genus Seifert surface yields the presentation matrix.^[20] For alternating knots, this equality holds using the canonical Seifert surface derived from a reduced alternating diagram, highlighting the polynomial's role in measuring knot complexity.^[21] As an invariant of the Alexander module H_1(\tilde{X}_K; \mathbb{Z})—the first homology of the infinite cyclic cover of the knot complement—the polynomial is determined up to units \pm t^k in \Lambda by the \Lambda-module structure: isomorphic Alexander modules yield identical Alexander polynomials.^[19] For links with \mu components, the Alexander polynomial generalizes to a multivariable form \Delta_L(t_1, \dots, t_\mu) \in \mathbb{Z}[t_1^{\pm 1}, \dots, t_\mu^{\pm 1}], satisfying analogous symmetry \Delta_L(t_1, \dots, t_\mu) = \pm t_1^{k_1} \cdots t_\mu^{k_\mu} \Delta_L(t_1^{-1}, \dots, t_\mu^{-1}) for integers k_i, with the total degree connected to the pairwise linking numbers among components.^[9]^[22]

Multiplicativity under knot operations

The Alexander polynomial exhibits multiplicativity under the connected sum of knots. Specifically, if K_1 and K_2 are knots in S^3, then the Alexander polynomial of their connected sum K = K_1 \# K_2 satisfies

\Delta_K(t) = \Delta_{K_1}(t) \Delta_{K_2}(t),

up to units in the Laurent polynomial ring \mathbb{Z}[t, t^{-1}]. This property arises from the direct sum decomposition of the homology groups of the infinite cyclic covers of the knot complements.^[23]^[24] Under the mirror image operation, the Alexander polynomial transforms in a specific way. For a knot K, the polynomial of its mirror mK is given by \Delta_{mK}(t) = \Delta_K(t^{-1}), again up to units. This relation implies that the Alexander polynomial cannot distinguish a knot from its mirror image, as the substitution t \mapsto t^{-1} yields an equivalent Laurent polynomial under normalization conventions.^[24] The Alexander polynomial remains invariant under certain mutation operations on knots. In particular, it is unchanged by genus 2 mutations, which preserve the structure of the knot complement relevant to the polynomial's definition via Seifert matrices or Fox calculus. Similarly, for knot diagrams, the polynomial is invariant under flype operations, as these are part of the equivalence relations that define ambient isotopy. This invariance holds in cases where the mutations or flypes do not alter the underlying Alexander module.^[25]^[26] For cable constructions, the Alexander polynomial admits an explicit formula. The (p, q)-cable knot C_{p,q}(K) of a knot K, where p and q are coprime integers with p > 0, has Alexander polynomial

\Delta_{C_{p,q}(K)}(t) = \Delta_K(t^p) \Delta_{T_{p,q}}(t),

where T_{p,q} denotes the (p, q)-torus knot and \Delta_{T_{p,q}}(t) = \frac{(t^{pq} - 1)(t - 1)}{(t^p - 1)(t^q - 1)}. This cabling formula reflects the satellite nature of the construction, combining the companion knot's polynomial with that of the pattern torus knot.^[27] These operational properties highlight the Alexander polynomial's utility, yet it is not a complete invariant. Counterexamples include the Kinoshita-Terasaka knot and the Conway knot, which are distinct 11-crossing knots related by mutation but share the identical trivial Alexander polynomial \Delta(t) = 1. Such mutants demonstrate that the polynomial fails to detect all topological differences arising from local changes in knot diagrams.

Interpretations

Geometric and topological meaning

The Alexander polynomial \Delta_K(t) encodes significant topological information about a knot K \subset S^3 through the homology of its covering spaces. Specifically, the knot complement S^3 \setminus K admits an infinite cyclic cover \widetilde{M_K} corresponding to the kernel of the abelianization map \pi_1(S^3 \setminus K) \to [\mathbb{Z}](/page/Z), where the deck transformations are generated by a meridian. The first homology group H_1(\widetilde{M_K}; \mathbb{Z}) forms a torsion module over the ring \Lambda = \mathbb{Z}[t, t^{-1}], and \Delta_K(t) is a generator of the annihilator ideal of this module, up to units in \Lambda. This representation captures the torsion structure intrinsic to the knot's topology, distinguishing it from free parts of the homology. Similarly, for the n-fold cyclic branched cover \Sigma_n(K) of S^3 branched along K, the order of the torsion subgroup of H_1(\Sigma_n(K); \mathbb{Z}) equals \left| \prod_{j=0}^{n-1} \Delta_K([\omega](/page/Omega)^j) \right|, where \omega is a primitive nth root of unity; in particular, for the double branched cover (n=2), |H_1(\Sigma_2(K); \mathbb{Z})| = |\Delta_K(-1)|.^[28] A key geometric consequence arises in relation to the Seifert genus g(K), the minimal genus of an orientable surface in S^3 bounded by K. The degree of \Delta_K(t) provides a lower bound: g(K) \geq \frac{1}{2} \deg \Delta_K(t), reflecting the minimal dimension required for a Seifert matrix to produce a polynomial of that degree.^[29] The evaluation |\Delta_K(-1)|, known as the knot determinant, further ties into this via the double branched cover, where large values indicate substantial torsion that constrains minimal surface complexities in certain knot classes, such as alternating knots where it often aligns closely with $2g(K) + 1. S-equivalence offers another topological lens: two knots are S-equivalent if their Seifert matrices are related by integer congruence and stabilizations (adding trivial bands), and this equivalence preserves \Delta_K(t).^[30] The Blanchfield duality links this to the Alexander module, endowing H_1(\widetilde{M_K}; \mathbb{Q}(t)/\mathbb{Z}[t, t^{-1}]) with a nondegenerate Hermitian form over \mathbb{Q}(t), ensuring the module's self-duality and tying S-equivalence classes to algebraic concordance invariants.^[31] Geometrically, \Delta_K(t) admits an interpretation as the Reidemeister torsion of the chain complex of the infinite cyclic cover \widetilde{M_K}, specifically a regularized determinant \tau(\widetilde{M_K}) = \Delta_K(t) up to sign and units, measuring the "size" of the acyclic complex after tensoring with the Novikov ring.^[31] This torsion invariant, originally developed by Reidemeister in the 1930s and refined by Fox and Milnor, highlights the polynomial's role in quantifying deviations from acyclicity in the knot complement's topology. Finally, evaluations of \Delta_K(t) at roots of unity connect to cyclotomic fields: for a primitive nth root \zeta, |\Delta_K(\zeta)| divides the order of H_1(\Sigma_n(K); \mathbb{Z}), mirroring how Dedekind zeta values at roots of unity relate to class numbers in cyclotomic extensions. In arithmetic topology, this analogy portrays knots as "primes" in the "3-manifold world," with \Delta_K(t) akin to an Iwasawa polynomial governing growth in infinite towers of covers, akin to \mathbb{Z}_p-extensions of number fields.^[32]

Connections to modern homology theories

The Alexander polynomial of a knot K in S^3 serves as the graded Euler characteristic of the knot Floer homology groups \widehat{\HFK}(K), a bigraded invariant introduced by Ozsváth and Szabó in their development of Heegaard Floer homology. Specifically, if \widehat{\HFK}_{i,j}(K) denotes the bigraded groups with Maslov grading i and Alexander grading j, then

\Delta_K(t) = \sum_{i,j} (-1)^i t^j \rank \widehat{\HFK}_{i,j}(K),

where \Delta_K(t) is the symmetrized Alexander polynomial. This relation positions the Alexander polynomial as the decategorification of \widehat{\HFK}(K), with the full homology providing a refinement that detects additional topological features, such as concordance obstructions beyond those from \Delta_K(t). In the Ozsváth-Szabó framework, the knot Floer complex \CFK^-(K) further refines this connection, where the Alexander polynomial equals the Euler characteristic \sum_i (-1)^i t^{\gr(i)} \dim \HFK^i(K, t) in the associated graded theory, with \gr denoting the Alexander grading. This categorification has been instrumental in modern low-dimensional topology, enabling computations of Seiberg-Witten invariants and Dehn surgery effects through the surgery exact triangle. Connections also extend to Khovanov homology, a categorification of the Jones polynomial, though it does not directly lift the Alexander polynomial; instead, related constructions in Khovanov-Rozansky homology yield Euler characteristics that specialize to quantum invariants encompassing the Alexander in certain limits. For instance, the sl(n) Khovanov-Rozansky theories provide a framework where the Poincaré polynomial in quantum and homological gradings relates to the Alexander polynomial via specialization at q = t and a = 1 in the HOMFLY-PT polynomial. In gauge-theoretic approaches, instanton Floer homology for knots, developed by Kronheimer and Mrowka, recovers the Alexander polynomial from the torsion structure of the homology groups \KHI(K), mirroring the Heegaard Floer relation and providing evidence for the conjectural equivalence between these theories. Recent post-2000 advancements, particularly through bordered Heegaard Floer homology, refine these links via pairing theorems and surgery formulas; for example, the bordered invariants categorify the Alexander polynomial's behavior under satellite operations and Dehn fillings, as shown in the link surgery formula.

Extensions

Behavior under satellite constructions

Satellite knots are constructed by embedding a pattern knot P in a solid torus V and then mapping V homeomorphically onto a tubular neighborhood of a companion knot C in S^3, with the image of P yielding the satellite knot S.^[33] The winding number w of P around the core of V measures how many times P wraps around C.^[33] The Alexander polynomial of a satellite knot satisfies the formula

\Delta_S(t) = \Delta_P(t) \cdot \Delta_C(t^w),

where \Delta_P(t) is the Alexander polynomial of the pattern P viewed in the solid torus (normalized such that the unknot pattern yields \Delta_{\text{unknot}}(t^w) = 1), and \Delta_C(t) is that of the companion.^[33] This multiplicative structure arises from the decomposition of the infinite cyclic cover of the satellite complement into covers associated to P and C.^[33] Cables represent a special class of satellites where the pattern is a torus knot T(m,n) on the boundary torus of the solid torus, with winding number m. For the (m,n)-cable of K, the Alexander polynomial is

\Delta_{m,n}(K)(t) = \Delta_K(t^m) \cdot \Delta_{T(m,n)}(t),

where \Delta_{T(m,n)}(t) = \frac{(t^{mn}-1)(t-1)}{(t^m-1)(t^n-1)} (up to units t^k).^[34] This reflects the pattern's contribution from the torus knot polynomial, scaled by the companion's polynomial evaluated at higher powers.^[34] Whitehead doubles are satellites with a specific pattern in the solid torus that clasps around the core with winding number w=0. Untwisted Whitehead doubles of any knot K have trivial Alexander polynomial \Delta_W(t) = 1, independent of \Delta_K(t), since the pattern polynomial \Delta_P(t) = 1 and t^0 = 1.^[35] This triviality implies that untwisted Whitehead doubles are non-fibered for non-trivial companions, as fibered knots have monic Alexander polynomials of even degree matching twice the genus, but here the degree is 0 while the genus exceeds 0.^[35] Satellite constructions preserve topological concordance properties linked to the Alexander polynomial; for instance, satellites with trivial Alexander polynomial, such as untwisted Whitehead doubles, are topologically slice, meaning concordant to the unknot in the topological category.^[36] This follows from the general result that knots with \Delta_K(t) = 1 bound locally flat disks in the 4-ball topologically.^[36]

Alexander–Conway variant

The Alexander–Conway polynomial, introduced by John Horton Conway in 1969 and published in 1970, reparametrizes the original Alexander polynomial by substituting z = t^{1/2} - t^{-1/2}, yielding \nabla(z) = \Delta(t), a polynomial in z with integer coefficients and only non-negative powers.^[37] This form addresses limitations of the Laurent polynomial structure in the t-variable by providing a cleaner, ordinary polynomial representation that simplifies computations.^[38] A key advantage of the Alexander–Conway variant is its presentation via Conway coefficients, which enable an arc index formulation suitable for open arcs and link diagrams, facilitating recursive calculations without fractional issues inherent in the original Laurent form.^[38] The polynomial is computed using the skein relation \nabla(L_+) - \nabla(L_-) = z \nabla(L_0), where L_+, L_-, and L_0 denote link diagrams differing locally at a crossing (positive, negative, and smoothed, respectively), with the normalization \nabla(\text{[unknot](/page/Unknot)}) = 1.^[5] For links, the Alexander–Conway polynomial extends naturally to a multi-variable version \nabla(z_1, \dots, z_\mu), where \mu is the number of components, related to the multi-variable Alexander polynomial through symmetrization over the variables. This multi-variable form preserves the skein relation in each variable while accommodating oriented components. The Alexander–Conway variant proves especially useful in finite type theory, where its coefficients generate Vassiliev invariants; for instance, the coefficient of z^2 yields the simplest nontrivial finite type invariant of order 2.^[39]

References

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[25]
[PDF] square numbers and polynomial invariants of achiral knots
Also, the skein and Alexander polynomial are invariant under mutation, so they are well-defined on a mutation equivalence class.
[26]
[PDF] A Cable Knot and BPS-Series
Jan 13, 2023 · ... knot [13]. 3.2 The Alexander polynomial. The cabling formula for the Alexander polynomial of a knot K is [18]. ∆C(p,q)(K)(t)=∆K(tp)∆T(p,q) ...
[27]
Computing knot Floer homology in cyclic branched covers - MSP
Jul 25, 2008 · The order of H1.†m.K// is equal to. Qm1. jD0 БK .!j /, where БK is the Alexander polynomial of K, and ! is a primitive mth root of unity (Fox ...
[28]
[PDF] Knots, Polynomials, and Categorification - Jacob Rasmussen
These lectures give an introduction to knot polynomials and their cat- egorifications. Topics covered include the Jones and Alexander polynomials,. Khovanov ...
[29]
[PDF] S-EQUIVALENCE OF KNOTS 1. Introduction An oriented knot k is a ...
Every oriented knot is spanned by an oriented surface, a Seifert surface, and this gives rise to a matrix of linking numbers called a Seifert matrix. Any two ...
[30]
[PDF] Reidemeister torsion, peripheral complex, and Alexander ...
A different approach to the study of Alexander polynomials relies on the use of Reidemeister torsion. Milnor [Mi62, Mi66] showed that the Alexander polynomial ...
[31]
[PDF] Knots in Number Theory - Universiteit Leiden
We build the theory to compute the Alexander polynomial from the ground up, after which we will show that the same construction very much applies to the Iwasawa ...
[32]
[PDF] Abelian invariants of satellite knots - Paul Melvin
Since the Alexander polynomial of a knot K is just detAK(t) , we have. Corollary (Seifert [S]). As(t) = AE(t)Ac(t w). Remarks. (i). If w = 0 , then the theorem ...
[33]
[PDF] arXiv:0806.2172v2 [math.GT] 15 Jun 2008
Jun 15, 2008 · Our intuition comes from the fact that the Alexander polynomial of cable knots is determined by the formula. (1). ∆Kp,q (t)=∆Tp,q (t) · ∆K ...
[34]
Knot Floer homology of Whitehead doubles - MSP
Dec 17, 2007 · In particular, the Alexander polynomial of the 0–twisted Whitehead double of K is trivial. It is thus an interesting question to ask how, if ...
[35]
New topologically slice knots - Project Euclid
In the early 1980's Mike Freedman showed that all knots with trivial Alexander polynomial are topologically slice (with fundamental group Z ℤ ).
[36]
[PDF] John Horton Conway: The Man and His Knot Theory
May 27, 2022 · An Alexander polynomial of a link can be calculated from its Conway polynomial by substituting z = t1/2−t-1/2. Laurent polynomials are not ...<|control11|><|separator|>
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[PDF] Section 10.1. The Conway Polynomial of a Knot
Feb 6, 2021 · −1 and in this polynomial we have (t − 1 + t. −1. )|t=1 = 1 so this is the normalized version of the Alexander polynomial of the trefoil knot.
[38]
Vassiliev Invariant -- from Wolfram MathWorld
Vassiliev invariants, discovered around 1989, provided a radically new way of looking at knots. The notion of finite type (a.k.a. Vassiliev) knot invariants ...