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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

##### MSC:
 57M25 Knots and links in the $$3$$-sphere (MSC2010)