skein

《Symplectic Geometry》 · 第 174 页 · Section 7.7；后文的分离链环为零结论有误，明示构造下 \(z[O_2]=0\) 不推出 \([O_2]=0\)

We shall consider the set of all formal linear combinations of ambient isotopy classes of links with coefficients taken from a ring \(R\). We shall take \(R\) to be the ring of polynomials in one variable, \(z\), with integer coefficients: \(R=\mathbb Z[z]\). […] We can manipulate expressions of this type by applying the Conway skein relation as a rewriting rule. The resulting algebraic structure is called a linear skein and is denoted by \(\mathcal L(\mathbb R^3)\). Note that these expressions are purely formal in character — the plus operator used to form the expression does not correspond to any operation on links. An expression cannot be evaluated, it can only be replaced by an equivalent one using the skein relation.

《Symplectic Geometry》 · 第 151、162、163–164 页 · Definition 7.4.1 的有向输入与关系 (3)；Section 7.3 的局部相同条件；Figure 6.8

The Conway polynomial of an oriented link \(L\), denoted variously by \(\nabla(L)\) or \(\nabla_L(z)\) depending on the required emphasis, is defined by the three following axioms. […] 3. Skein relation: \(\nabla(L_+)-\nabla(L_-)=z\nabla(L_0)\) where \(L_+\), \(L_-\) and \(L_0\) have diagrams \(D_+\), \(D_-\) and \(D_0\) which differ as shown in Figure 6.8. […] Figure 6.8 shows small parts of three link diagrams which are identical outside a small neighbourhood. The broken circle indicates the boundary of the neighbourhood, and the links differ inside as shown.