Alexander polynomial

In knot theory, the Alexander polynomial is an invariant of a knot . The polynomial was discovered by the topologist James Alexander in 1928 and is the first vertex polynomial.

definition

For the node K in the 3- sphere , consider the infinite cyclic superposition X of the node complement. (This can be constructed by cutting open the node complement along a Seifert surface and gluing countably many copies of the manifold created in this way cyclically along the intersections.) The cover transformation group of this overlay is cyclic, be its producer. The - module 's Alexander module . The Alexander module is finally presented , every presentation matrix is ​​called the Alexander matrix. The Alexander ideal in is the ideal generated by the - minors of this matrix , where the number is the producers of the presentation. (One can show that the Alexander ideal is independent of the chosen presentation and depends only on the knot.) Alexander proved that the Alexander ideal is a main ideal . The Alexander polynomial is defined as the generator of the Alexander ideal (and is thus only clearly defined except for multiplication with ). ${\displaystyle t}$ ${\displaystyle \mathbb {Z} [t,t^{-1}]}$ ${\displaystyle H_{1}X}$ ${\displaystyle \mathbb {Z} [t,t^{-1}]}$ ${\displaystyle r\times r}$ ${\displaystyle r}$ ${\displaystyle \Delta _{K}(t)}$ ${\displaystyle \pm t^{n}}$

calculation

In 1969, John Horton Conway showed that the Alexander polynomial can be calculated using two rules: ${\displaystyle \Delta }$

• ${\displaystyle \Delta (O)=1}$

for each projection O of the trivial node, and

• ${\displaystyle \Delta (L_{+})-\Delta (L_{-})+(t^{-{\frac {1}{2}}}-t^{\frac {1}{2}})\Delta (L_{0})=0}$,

wherein , , and oriented link diagrams as differ within a small area in the picture below and are identical outside of this area. ${\displaystyle L_{+}}$ ${\displaystyle L_{-}}$ ${\displaystyle L_{0}}$

In particular, this provides a normalization of the Alexander polynomial, which is actually only up to multiplication with certain. Closely related to this normalization is the Alexander-Conway polynomial or Conway polynomial , which is determined by the relations ${\displaystyle \pm t^{n}}$

• ${\displaystyle \nabla (O)=1}$
• ${\displaystyle \nabla (L_{+})-\nabla (L_{-})=z\nabla (L_{0})}$

is defined and is related to the Alexander polynomial standardized as above via the equation . ${\displaystyle \Delta _{L}(t^{2})=\nabla _{L}(t-t^{-1})}$

The Alexander polynomial can also be through the Seifert matrix V determine: . ${\displaystyle \Delta _{K}(t)=\det(V-tV^{*})}$

The Alexander polynomial is symmetric in and . One has . ${\displaystyle t}$ ${\displaystyle t^{-1}}$ ${\displaystyle \Delta _{K}(1)=\pm 1}$

Examples and Applications

The degree of the Alexander polynomial gives a lower bound for , where is the gender of any Seifert surface. (This follows directly from the calculation of the Alexander polynomial using the Seifert matrix, because this is a matrix.) ${\displaystyle 2g}$ ${\displaystyle g}$ ${\displaystyle 2g\times 2g}$

Example: The Alexander polynomial of the trefoil knot is . The trefoil knot has a Seifert surface with gender 1 and the degree of its Alexander polynomial is 2. ${\displaystyle \Delta (t)=t-1+t^{-1}\,}$

If the nodal complement allows a grain over the circle with a Seifert surface as a fiber (one then speaks of a fibered node ), then the degree of the Alexander polynomial is exact . Furthermore, it must then be monic, i.e. H. the coefficients of the highest and lowest terms are 1 or −1. In fact, the Alexander polynomial in this case is precisely the characteristic polynomial for the effect of the monodromy of the fibers on the 1st homology of the Seifert surface. ${\displaystyle \Delta _{K}(t)}$ ${\displaystyle 2g}$ ${\displaystyle \Delta _{K}(t)}$

The Alexander polynomial of a disc knot is always of the form for an integer Laurent polynomial (Fox-Milnor theorem). ${\displaystyle \Delta _{K}(t)=P(t)P(t^{-1})}$ ${\displaystyle P}$

Generalizations

The HOMFLY polynomial , in addition to other node polynomials, also generalizes the Alexander polynomial: it applies ${\displaystyle P(a,z)}$

${\displaystyle P(1,z)=\Delta _{K}(z)}$.

The Alexander polynomial is the Euler characteristic of a big graduated homology theory.

See also

• Jones polynomial

Individual evidence

1. ↑ Knot theory (PDF; 360 kB) on Reuter.mit.edu (accessed on May 20, 2012)
2. ↑ About the Alexander polynomial ( memento of March 2, 2005 in the Internet Archive ) on Uni-Hannover.de (accessed on May 20, 2012)
3. ↑ Mikhail Khovanov: Link homology and categorification Proceedings of the ICM 2006 Madrid