In this paper, we define the delta derivative in the integer group ring and we show that the delta derivative is well defined on the free groups. We also define a polynomial invariant of oriented knot and link by carrying the delta derivative to the link group. Since the delta derivative is a generalization of the free derivative, this polynomial invariant called the delta polynomial is a generalization of the Alexander polynomial. In addition, we present a new polynomial called the difference polynomial of oriented knot and link, which is similar to the Alexander polynomial and is a special case of the delta polynomial.
ALTINTAŞ, İSMET and TAŞKÖPRÜ, KEMAL
"A generalization of the Alexander polynomial as an application of the delta derivative,"
Turkish Journal of Mathematics: Vol. 42:
2, Article 8.
Available at: https://journals.tubitak.gov.tr/math/vol42/iss2/8