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 and called the Alexander–Conway polynomial (also known as Conway polynomial or Conway–Alexander polynomial). Suppose we are given an oriented link diagram, where 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. Here are Conway's skein relations: (where O is any diagram of the unknot) The relationship to the standard Alexander polynomial is given by . Here must be properly normalized (by multiplication of ) to satisfy the skein relation . Note that this relation gives a Laurent polynomial in t1/2. See knot theory for an example computing the Conway polynomial of the trefoil. Relation to Floer homology Using pseudo-holomorphic curves, and 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, showed that knot Floer homology detects the genus. Similarly, while the Alexander polynomial gives an obstruction to a knot complement fibering over the circle, 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 References (accessible introduction utilizing a skein relation approach) (covers several different approaches, explains relations between different versions of the Alexander polynomial) (explains classical approach using the Alexander invariant; knot and link table with Alexander polynomials) External links – knot and link tables with computed Alexander and Conway polynomials Knot theory Diagram algebras Polynomials John Horton Conway Knot invariants