Editing Alexander polynomial

{{short description|Knot invariant}}
In [[mathematics]], the '''Alexander polynomial''' is a [[knot invariant]] which assigns a [[polynomial]] with integer coefficients to each knot type.  [[James Waddell Alexander II]] discovered this, the first [[knot polynomial]], in 1923.  In 1969, [[John Horton Conway|John Conway]] showed a version of this polynomial, now called the '''Alexander–Conway polynomial''', could be computed using a [[skein relation]], although its significance was not realized until the discovery of the [[Jones polynomial]] in 1984.  Soon after Conway's reworking of the Alexander polynomial, it was realized that a similar skein relation was exhibited in Alexander's paper on his polynomial.{{efn|Alexander describes his skein relation toward the end of his paper under the heading "miscellaneous theorems", which is possibly why it got lost. [[Joan Birman]] mentions in her paper that Mark Kidwell brought her attention to Alexander's relation in 1970.{{sfn|Birman|1993}}}}

==Definition==
Let ''K'' be a [[Knot (mathematics)|knot]] in the [[3-sphere]].  Let ''X'' be the infinite [[cyclic cover|cyclic cover]] of the [[knot complement]] of ''K''.  This covering can be obtained by cutting the knot complement along a [[Seifert surface]] of ''K'' and gluing together infinitely many copies of the resulting manifold with boundary in a cyclic manner.  There is a [[deck transformation|covering transformation]] ''t'' acting on ''X''.  Consider the first homology (with integer coefficients) of ''X'', denoted <math>H_1(X)</math>.  The transformation ''t'' acts on the homology and so we can consider <math>H_1(X)</math> a [[module (mathematics)|module]] over the ring of [[Laurent polynomial]]s <math>\mathbb{Z}[t, t^{-1}]</math>.  This is called the '''Alexander invariant''' or '''Alexander module'''.

The module is finitely presentable; a [[presentation matrix]] for this module is called the '''Alexander matrix'''.  If the number of generators, <math>r</math>, is less than or equal to the number of relations, <math>s</math> , then we consider the ideal generated by all <math>r \times r</math> minors of the matrix; this is the zeroth [[Fitting ideal]] or '''Alexander ideal''' and does not depend on choice of presentation matrix.  If <math>r > s</math>, set the ideal equal to 0.  If the Alexander ideal is [[principal ideal|principal]], take a generator; this is called an Alexander polynomial of the knot.  Since this is only unique up to multiplication by the Laurent monomial <math>\pm t^n</math>, one often fixes a particular unique form.  Alexander's choice of normalization is to make the polynomial have a positive [[constant term]].

Alexander proved that the Alexander ideal is nonzero and always principal.  Thus an Alexander polynomial always exists, and is clearly a knot invariant, denoted <math>\Delta_K(t)</math>.  It turns out that the Alexander polynomial of a knot is the same polynomial for the mirror image knot.  In other words, it cannot distinguish between a knot and its mirror image.

==Computing the polynomial==
The following procedure for computing the Alexander polynomial was given by J. W. Alexander in his paper.{{sfn|Alexander|1928}}

Take an [[oriented]] diagram  of the knot with <math>n</math> crossings; there are <math>n+2</math> regions of the knot diagram. To work out the Alexander polynomial, first one must create an [[incidence matrix]] of size <math>(n, n + 2)</math>.  The <math>n</math> rows correspond to the <math>n</math> crossings, and the <math>n+2</math> columns to the regions. The values for the matrix entries are either <math>0,1,-1,t,-t</math>.

Consider the entry corresponding to a particular region and crossing. If the region is not adjacent to the crossing, the entry is 0. If the region is adjacent to the crossing, the entry depends on its location. The following table gives the entry, determined by the location of the region at the crossing from the perspective of the incoming undercrossing line.

: on the left before undercrossing: <math>-t</math>
: on the right before undercrossing: <math>1</math> 
: on the left after undercrossing: <math>t</math>
: on the right after undercrossing: <math>-1</math>

Remove two columns corresponding to adjacent regions from the matrix, and work out the determinant of the new <math>n \times n</math> matrix.  Depending on the columns removed, the answer will differ by multiplication by <math>\pm t^n</math>, where the power of <math>n</math> is not necessarily the number of crossings in the knot.  To resolve this ambiguity, divide out the largest possible power of <math>t</math> and multiply by <math>-1</math> if necessary, so that the constant term is positive. This gives the Alexander polynomial.

The Alexander polynomial can also be computed from the [[Seifert matrix]].

After the work of J. W. Alexander, [[Ralph Fox]] considered a copresentation of the knot group <math>\pi_1(S^3\backslash K)</math>, and introduced non-commutative differential calculus, which also permits one to compute <math>\Delta_K(t)</math>.{{sfn|Fox|1961}}{{efn|Detailed exposition of this approach about higher Alexander polynomials can be found in {{harvp|Crowell|Fox|1963}}.}}

==Basic properties of the polynomial==

The Alexander polynomial is symmetric: <math>\Delta_K(t^{-1}) = \Delta_K(t)</math> for all knots K.

: From the point of view of the definition, this is an expression of the [[Poincaré duality|Poincaré Duality isomorphism]] <math> \overline{H_1 X} \simeq \mathrm{Hom}_{\mathbb Z[t,t^{-1}]}(H_1 X, G) </math> where <math>G</math> is the quotient of the field of fractions of <math>\mathbb Z[t,t^{-1}]</math> by <math>\mathbb Z[t,t^{-1}]</math>, considered as a <math>\mathbb Z[t,t^{-1}]</math>-module, and where <math>\overline{H_1 X}</math> is the conjugate <math>\mathbb Z[t,t^{-1}]</math>-module to <math>H_1 X</math> ie: as an abelian group it is identical to <math>H_1 X</math> but the covering transformation <math>t</math> acts by <math>t^{-1}</math>.

Furthermore, the Alexander polynomial evaluates to a unit on 1: <math>\Delta_K(1)=\pm 1</math>.

: From the point of view of the definition, this is an expression of the fact that the knot complement is a homology circle, generated by the covering transformation <math>t</math>.  More generally if <math>M</math> is a 3-manifold such that <math>rank(H_1 M) = 1</math> it has an Alexander polynomial <math>\Delta_M(t)</math> defined as the order ideal of its infinite-cyclic covering space.  In this case <math>\Delta_M(1)</math> is, up to sign, equal to the order of the torsion subgroup of <math>H_1 M</math>.

Every integral Laurent polynomial which is both symmetric and evaluates to a unit at 1 is the Alexander polynomial of a knot.<ref>{{harvnb|Kawauchi|2012|loc=Theorem 11.5.3, p. 150}}. Kawauchi credits this result to Kondo, H. (1979), "Knots of unknotting number 1 and their Alexander polynomials", ''Osaka J. Math.'' 16: 551-559, and to Sakai, T. (1977), "A remark on the Alexander polynomials of knots", ''Math. Sem. Notes Kobe Univ.'' 5: 451~456.</ref>

==Geometric significance of the polynomial==
Since the Alexander ideal is principal, <math>\Delta_K(t)=1</math> [[if and only if]] the commutator subgroup of the knot group is [[perfect group|perfect]] (i.e. equal to its own [[commutator subgroup]]).

For a [[topologically slice]] knot, the Alexander polynomial satisfies the Fox–Milnor condition <math>\Delta_K(t) = f(t)f(t^{-1})</math> where <math>f(t)</math> is some other integral Laurent polynomial.

Twice the [[Seifert surface|knot genus]] is bounded below by the degree of the Alexander polynomial.

[[Michael Freedman]] proved that a knot in the 3-sphere is [[topologically slice]]; i.e., bounds a "locally-flat" topological disc in the 4-ball, if the Alexander polynomial of the knot is trivial.{{sfn|Freedman|Quinn|1990}}

Kauffman describes the first construction of the Alexander polynomial via state sums derived from physical models. A survey of these topics and other connections with physics are given in.{{sfn|Kauffman|1983}}{{sfn|Kauffman|2012}}

There are other relations with surfaces and smooth 4-dimensional topology. For example, under certain assumptions, there is a way of modifying a smooth [[4-manifold]] by performing a [[surgery theory|surgery]] that consists of removing a neighborhood of a two-dimensional torus and replacing it with a knot complement crossed with ''S''<sup>1</sup>. The result is a smooth 4-manifold homeomorphic to the original, though now the [[Seiberg–Witten invariant]] has been modified by multiplication with the Alexander polynomial of the knot.{{sfn|Fintushel|Stern|1998}}

Knots with symmetries are known to have restricted Alexander polynomials.{{sfn|Kawauchi|2012|loc=symmetry section}} Nonetheless, the Alexander polynomial can fail to detect some symmetries, such as strong invertibility.

If the [[knot complement]] fibers over the circle, then the Alexander polynomial of the knot is known to be ''monic'' (the coefficients of the highest and lowest order terms are equal to <math>\pm 1</math>). In fact, if <math>S \to C_K \to S^1</math> is a fiber bundle where <math>C_K</math> is the knot complement, let <math>g : S \to S</math> represent the [[monodromy]], then <math>\Delta_K(t) = {\rm Det}(tI-g_*)</math> where <math>g_*\colon H_1 S \to H_1 S</math> is the induced map on homology.

==Relations to satellite operations==

If a knot <math>K</math> is a [[satellite knot]] with pattern knot <math>K'</math> (there exists an embedding <math>f : S^1 \times D^2 \to S^3</math> such that <math>K=f(K')</math>, where <math>S^1 \times D^2 \subset S^3</math> is an unknotted solid torus containing <math>K'</math>), then <math>\Delta_K(t) = \Delta_{f(S^1 \times \{0\})}(t^a) \Delta_{K'}(t)</math>, where <math>a \in \mathbb Z</math> is the integer that represents <math>K' \subset S^1 \times D^2</math> in <math>H_1(S^1\times D^2) = \mathbb Z</math>.

Examples: For a connect-sum <math>\Delta_{K_1 \# K_2}(t) = \Delta_{K_1}(t) \Delta_{K_2}(t)</math>. If <math>K</math> is an untwisted [[Satellite knot|Whitehead double]], then <math>\Delta_K(t)=\pm 1</math>.

==Alexander–Conway polynomial==
Alexander proved the Alexander polynomial satisfies a skein relation.  [[John Horton Conway|John Conway]] later rediscovered this in a different form and showed that the skein relation together with a choice of value on the unknot was enough to determine the polynomial.  Conway's version is a polynomial in ''z'' with integer coefficients, denoted <math>\nabla(z)</math> and called the '''Alexander–Conway polynomial''' (also known as '''Conway polynomial''' or '''Conway–Alexander polynomial''').

Suppose we are given an oriented link diagram, where <math>L_+, L_-, L_0</math> are link diagrams resulting from crossing and smoothing changes on a local region of a specified crossing of the diagram, as indicated in the figure.
[[Image:Skein (HOMFLY).svg|200px|center]]

Here are Conway's skein relations:

* <math>\nabla(O) = 1</math> (where O is any diagram of the unknot)
* <math>\nabla(L_+) - \nabla(L_-) = z \nabla(L_0)</math>

The relationship to the standard Alexander polynomial is given by <math>\Delta_L(t^2) = \nabla_L(t - t^{-1})</math>.  Here <math>\Delta_L</math> must be properly normalized (by multiplication of <math>\pm t^{n/2}</math>) to satisfy the skein relation <math>\Delta(L_+) - \Delta(L_-) = (t^{1/2} - t^{-1/2}) \Delta(L_0)</math>.  Note that this relation gives a Laurent polynomial in ''t<sup>1/2</sup>''.

See [[knot theory]] for an example computing the Conway polynomial of the trefoil.

==Relation to Floer homology==

Using pseudo-holomorphic curves, Ozsváth-Szabó{{sfn|Ozsváth|Szabó|2004}} and Rasmussen{{sfn|Rasmussen|2003}} associated a bigraded abelian group, called knot Floer homology, to each isotopy class of knots. The graded [[Euler characteristic]] of knot Floer homology is the Alexander polynomial. While the Alexander polynomial gives a lower bound on the genus of a knot, {{sfn|Ozsváth|Szabó|2004b}} showed that knot Floer homology detects the genus. Similarly, while the Alexander polynomial gives an obstruction to a knot complement fibering over the circle, {{sfn|Ni|2007}} showed that knot Floer homology completely determines when a knot complement fibers over the circle. The knot Floer homology groups are part of the Heegaard Floer homology family of invariants; see [[Floer homology]] for further discussion.

==Notes==
{{Notelist}}

==References==
{{Reflist|20em}}

==Sources==
*{{Cite book |first=Colin C. |last=Adams |author-link=Colin Adams (mathematician)|title=The Knot Book: An elementary introduction to the mathematical theory of knots |orig-year=1994 |publisher=American Mathematical Society |year=2004 |isbn=978-0-8218-3678-1 }} (accessible introduction utilizing a skein relation approach)
*{{Cite journal |first=J. W. |last=Alexander |author-link=James Waddell Alexander II |year=1928 |title=Topological Invariants of Knots and Links |journal=[[Transactions of the American Mathematical Society]] |volume=30 |issue=2 |pages=275–306 |doi=10.1090/S0002-9947-1928-1501429-1  |jstor=1989123 |doi-access=free |url=https://www.ams.org/journals/tran/1928-030-02/S0002-9947-1928-1501429-1/S0002-9947-1928-1501429-1.pdf}}
*{{cite journal |last=Birman |first=Joan |date=1993 |title=New points of view in knot theory |journal=Bull. Amer. Math. Soc. |series=N.S. |volume=28 |issue=2 |pages=253–287|doi=10.1090/S0273-0979-1993-00389-6 |arxiv=math/9304209 }}
*{{Cite book |first1=Richard |last1=Crowell |first2=Ralph |last2=Fox |author-link2=Ralph Fox| title=Introduction to Knot Theory |publisher=Ginn and Co. after 1977 Springer Verlag |year=1963 }}
*{{Cite journal |last1=Fintushel |first1=Ronald |author1-link=Ronald Fintushel|last2=Stern |first2=Ronald J.|author2-link=Ronald J. Stern |date=October 1998 |title=Knots, links, and 4-manifolds |journal=[[Inventiones Mathematicae]] |volume=134 |issue=2 |pages=363–400 |doi=10.1007/s002220050268 |issn=0020-9910 |arxiv=dg-ga/9612014 |bibcode=1998InMat.134..363F |mr=1650308 |s2cid=3752148}}
*{{Cite book |first=Ralph |last=Fox |author-link=Ralph Fox |chapter=A quick trip through knot theory |title=Proceedings of the University of Georgia Topology Institute |editor-first=M.K. |editor-last=Fort |publisher=Prentice-Hall |location=Englewood Cliffs. N. J. |year=1961 |pages=120–167 |oclc=73203715}}
*{{Cite book |author-link=Michael H. Freedman |first1=Michael H. |last1=Freedman |author-link2=Frank Quinn (mathematician) |first2=Frank |last2=Quinn |title=Topology of 4-manifolds |series=Princeton Mathematical Series |volume=39 |publisher=Princeton University Press |year=1990 |isbn=978-0-691-08577-7 |url-access=registration |url=https://archive.org/details/topologyof4manif0000free }}
*{{Cite book |first=Louis |last=Kauffman |author-link=Louis Kauffman| title=Formal Knot Theory |url=https://books.google.com/books?id=WzdbcxGdXTMC |date=2006 |orig-year=1983 |publisher=Courier |isbn=978-0-486-45052-0|ref={{sfnref|Kauffman1983}}}}
*{{Cite book |first=Louis |last=Kauffman |author-link=Louis Kauffman| title=Knots and Physics |publisher=World Scientific Publishing Company |year=2012 |edition=4th |isbn=978-981-4383-00-4}}
*{{Cite book |first=Akio |last=Kawauchi |title=A Survey of Knot Theory |url=https://books.google.com/books?id=RkEBCAAAQBAJ |date=2012 |publisher=Birkhäuser |isbn=978-3-0348-9227-8 |orig-year=1996}} (covers several different approaches, explains relations between different versions of the Alexander polynomial)
*{{Cite journal |first1=Yi |last1=Ni |title=Knot Floer homology detects fibred knots |journal=Inventiones Mathematicae |series=Invent. Math. |volume=170 |year=2007 |issue=3|pages=577–608 |arxiv=math/0607156 |doi=10.1007/s00222-007-0075-9|bibcode=2007InMat.170..577N |s2cid=119159648 }}
*{{Cite journal |first1=Peter |last1=Ozsváth |author-link1=Peter Ozsváth| first2=Zoltán |last2=Szabó |author-link2=Zoltán Szabó (mathematician)| year=2004  |title=Holomorphic disks and knot invariants |journal=[[Advances in Mathematics]] |volume=186 |issue=1 |pages=58–116 |arxiv=math/0209056 |bibcode=2002math......9056O  |doi=10.1016/j.aim.2003.05.001 |doi-access=free |s2cid=11246611 }}
*{{Cite journal |first1=Peter |last1=Ozsváth |author-link1=Peter Ozsváth| first2=Zoltán |last2=Szabó |author-link2=Zoltán Szabó (mathematician)| title=Holomorphic disks and genus bounds |journal=[[Geometry and Topology]] |volume=8 |issue=2004 |year=2004b |pages=311–334 |arxiv=math/0311496  |doi=10.2140/gt.2004.8.311 |s2cid=11374897 }}
*{{Cite thesis |first=Jacob |last=Rasmussen |title=Floer homology and knot complements |publisher=Harvard University |year=2003 |arxiv=math/0306378 |bibcode=2003math......6378R |pages=6378 }}
*{{Cite book |first=Dale |last=Rolfsen |author-link=Dale Rolfsen| title=Knots and Links |edition=2nd |publisher=Publish or Perish |year=1990 |isbn=978-0-914098-16-4 }} (explains classical approach using the Alexander invariant; knot and link table with Alexander polynomials)

==External links==
* {{springer|title=Alexander invariants|id=p/a011300}}
* {{Knot Atlas|Main_Page|Main Page|The_Alexander-Conway_Polynomial|The Alexander-Conway Polynomial}}&nbsp;– knot and link tables with computed Alexander and Conway polynomials

{{Knot theory}}

[[Category:Knot theory]]
[[Category:Diagram algebras]]
[[Category:Polynomials]]
[[Category:John Horton Conway]]
[[Category:Knot invariants]]