mathematics, the Alexander polynomial is a knot invariant which assigns a polynomial with integer coefficients to each knot type. James Waddell Alexander II discovered...

knot. The first knot polynomial, the Alexander polynomial, was introduced by James Waddell Alexander II in 1923. Other knot polynomials were not found until...

the Alexander polynomial and the Jones polynomial, both of which can be obtained by appropriate substitutions from HOMFLY. The HOMFLY polynomial is also...

corresponding to this projection map. Much like in the construction of the Alexander polynomial, consider H1(Cn) as a module over the group-ring of covering transformations...

knot polynomials, such as the Conway, Alexander, and Jones polynomials, the relevant skein relations are sufficient to calculate the polynomial recursively...

Touchard polynomials Wilkinson's polynomial Wilson polynomials Zernike polynomials Pseudo-Zernike polynomials Alexander polynomial HOMFLY polynomial Jones...

In the mathematical field of knot theory, the Jones polynomial is a knot polynomial discovered by Vaughan Jones in 1984. Specifically, it is an invariant...

(t)=c_{0}+c_{1}t+\cdots +c_{n}t^{n}+\cdots +c_{0}t^{2n}} be the Alexander polynomial of the knot. Then the Arf invariant is the residue of c n − 1 + c...

polynomial, and the Kauffman polynomial. A variant of the Alexander polynomial, the Alexander–Conway polynomial, is a polynomial in the variable z with integer...

\left(V-tV^{*}\right),} which is a polynomial of degree at most 2g in the indeterminate t . {\displaystyle t.} The Alexander polynomial is independent of the choice...

homology is a homology theory whose Euler characteristic is the Alexander polynomial of the knot. It has been proven effective in deducing new results...

amphichiral. Its Alexander polynomial is Δ ( t ) = − 2 t + 5 − 2 t − 1 , {\displaystyle \Delta (t)=-2t+5-2t^{-1},\,} its Conway polynomial is ∇ ( z ) = 1...

to note that its signature is not zero. Another proof is that its Alexander polynomial does not satisfy the Fox-Milnor condition. The trefoil is a fibered...

shares the same Jones polynomial. Both knots also have the curious property of having the same Alexander polynomial and Conway polynomial as the unknot. The...

used to classify and distinguish knots and links. For instance, the Alexander polynomial associates certain numbers with any given knot; these numbers are...

1971. Alexander's Chimney in Rocky Mountain National Park is named after him. Alexander horned sphere Alexander polynomial Alexander cochain Alexander–Spanier...

Examples of knots with nonmonic Alexander polynomials abound, for example the twist knots have Alexander polynomials q t − ( 2 q + 1 ) + q t − 1 {\displaystyle...

(mathematics) Chern–Simons form Topological quantum field theory Alexander polynomial Jones polynomial 2+1D topological gravity Skyrmion ∞-Chern–Simons theory...

knot, the granny knot is not a ribbon knot or a slice knot. The Alexander polynomial of the granny knot is Δ ( t ) = ( t − 1 + t − 1 ) 2 , {\displaystyle...

Dehn, J. W. Alexander, and Kurt Reidemeister, investigated knots. Out of this sprang the Reidemeister moves and the Alexander polynomial.: 15–45  Dehn...

of quandle cohomology. The Alexander quandles are also important, since they can be used to compute the Alexander polynomial of a knot. Let A {\displaystyle...

cochain complex. It may be regarded as a categorification of the Jones polynomial. It was developed in the late 1990s by Mikhail Khovanov. To any link diagram...

genus of a knot; a theorem of Michael Freedman says that if the Alexander polynomial of K is 1, then the topologically locally flat slice genus of K is...

_{1}^{2}\sigma _{2}^{2}\sigma _{1}^{-1}\sigma _{2}^{-2}.\,} Its Alexander polynomial is Δ ( t ) = t 3 / 2 − 3 t 1 / 2 + 3 t − 1 / 2 − t − 3 / 2 , {\displaystyle...

the smallest possible crossing number for a composite knot. The Alexander polynomial of the square knot is Δ ( t ) = ( t − 1 + t − 1 ) 2 , {\displaystyle...

subdividing its segments before it can be simplified. The Alexander–Conway polynomial and Jones polynomial of the unknot are trivial: Δ ( t ) = 1 , ∇ ( z ) =...

{\displaystyle ({\begin{smallmatrix}2&1\\1&1\end{smallmatrix}})} . The Alexander polynomial of the figure-eight knot is Δ ( t ) = − t + 3 − t − 1 ,   {\displaystyle...

Crossing number Linking number Skein relation Knot polynomials Alexander polynomial Jones polynomial Knot group Writhe Quandle Seifert surface Braids Braid...

the Alexander polynomial. In the odd case, the Alexander polynomial of the Lissajous knot must be a perfect square. In the even case, the Alexander polynomial...

Alexander matrix is a presentation matrix for the Alexander invariant of a knot. The determinant of an Alexander matrix is the Alexander polynomial for...