Knots And Polynomial Invariants

Knots theory is one important branch of mathematics.From the beginning of20th cen-trury, remains to be a dynamic branch of mathematics. In the early20th centrury, American mathmatician Alexander discovered a knot invariant.Later on, in the1960s, when studying Alexander’s ivariants,an English mathematician named Conway put forward skein relation this is the first polynomial invariant in history.In1984, a mathematician from New Zealand Jones discovered a new polynomial. It astonished the mathematical world. Knots theory became one focus of mathematicans. In1990, American physicist E. Witten reinterpreted Jones’s work from the viepoint of physics.Then mathematicians became aware of the intimite relationship between knot theory and mathematical physics.Later on, lots of works generalizing Jones’work came into existence. HOMFLPT polynomial invariant and Kaufman invariant are the well-known ones.In chapter l,we review basics of knots theory.In chapter2,we will talk about some well-known polynomial invariants:Alexander polyno-mial, Conway polynomial, Jones polynomial and Kaufman’s bracket.In chapter3,we introduce a new knot invariant inspired by Kaufman’s construction. At the end, we talk about the significance of this kont invariant and the future work. invariant for knots.