zbMATH — the first resource for mathematics

Invariants of links of Conway type. (English) Zbl 0655.57002
Let A be an algebra with (possibly infinitely many) specified elements \(a_ 1,a_ 2,...,a_ n,..\). in which two binary operations \(|\) and * are defined. If A satisfies a certain set of axioms, for example, \(a_ n| a_{n+1}=a_ n\), \(a_ n*a_{n+1}=a_ n\), \((a| b)| (c| d)=(a| c)| (b| d)\), etc., then A is called a Conway algebra. To each Conway algebra, one can define an oriented link invariant, say \(w_ L\). \(w_ L\) satisfies the skein relations: (1) \(w_{T_ n}=a_ n\), where \(T_ n\) denotes a trivial n-component link, (2) \(w_{L_+}=w_{L_-}| w_{L_ 0}\), (3) \(W_{L_- }=w_{L_+}*w_{L_ 0}\) (Theorem 1.8). Many link invariants, including Jones polynomials and homfly polynomials can be interpreted as invariants \(w_ L\) by defining \(|\) and * appropriately. (Example 1.11). The linking number of 2-component link is also described as an invariant of this type (Example 1.14). A complete proof of Theorem 1.8 occupies most of the paper. The authors discuss a slight generalization of Theorem 1.8 in section 3.
Reviewer: K.Murasugi

57M25 Knots and links in the \(3\)-sphere (MSC2010)