A linking number definition of the affine index polynomial and applications (Q2872206)

Virtual knot theory has many easily defined, easily computed, yet very powerful invariants. The authors of this well-written paper give a new definition of one of these: the affine index polynomial. The new definition, based on virtual linking numbers, is called the wriggle polynomial. The authors show that the wriggle polynomial is equal to the affine index polynomial. Two applications of this result are given.NEWLINENEWLINEFirst the authors prove the Cosmetic Crossing Conjecture for virtual knots having only odd crossings: switching an odd crossing from positive to negative, or vice versa, of an odd virtual knot must also change the isotopy class of the knot. An interesting corollary is that the only crossings in a virtual knot that can admit non-trivial cosmetic crossing changes are those which are even and have trivial weight.NEWLINENEWLINEA second application is to detecting mutations of virtual knots. The authors give infinitely many examples where the wriggle polynomial detects mutations by positive rotation. In fact, there is an order two Vassiliev invariant of virtual knots that detects mutations. For classical knots, the smallest order finite-type invariant that detects mutations is eleven. It is furthermore shown that mutations by positive reflection can never be detected by the affine index polynomial.