By Khan R.A.

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**Additional info for A bivariate extension of Bleimann-Butzer-Hahn operator**

**Example text**

The twisted Alexander polynomial for a knot K ⊂ S 3 was first introduced by Xiao–Song Lin in 1990 (cf. [Lin01]). Whereas Lin’s original definition used ‘regular Seifert surfaces’ of knots, later extensions to links and 3-manifolds either generalized the Reidemeister–Milnor–Turaev torsion (cf. [Wa94, Ki96, KL99a, FK06]) or generalized the homological definition of the Alexander polynomial (cf. [JW93, KL99a, Ch03, FK06, HKL10]). In most cases the setup for twisted invariants is as follows: Let N be a 3-manifold with empty or toroidal boundary, ψ : π1 (N ) → F an epimorphism onto a free abelian group F and γ : π1 (N ) → GL(k, R) a representation with R a domain.

2. As pointed out in Sect. 5, the twisted Alexander polynomials N,i can be computed from a presentation of the fundamental group, whereas the computation of τ (N, γ ⊗ ψ) requires in general an understanding of the CW-structure of N (cf. Sect. 2). In particular the equality of Proposition 2 (8) is often a faster method for computing τ (N, γ ⊗ ψ) (at the price of a higher indeterminacy). 3. The twisted Alexander polynomial is only defined for representations over a Noetherian UFD, whereas the twisted Reidemeister torsion is defined for a finite dimensional representation over any commutative ring.

We observe that under virtualization, this knot is equivalent to a classical knot. 42 K. Bhandari et al. Fig. 14 Z-equivalence Fig. 46 Fig. 72 Fig. 107 have arrow polynomial one and are equivalent to the unknot via a sequence of classical and virtual Reidemeister moves and the Z-equivalence (shown in Fig. 14). The knots in Figs. 18 have arrow polynomial one. In the paper [FKM05], the authors (Fenn, Kauffman, and Manturov) made the following conjecture: Conjecture 2 Let K be a virtual knot. If the bracket polynomial of K, K = 1 then K is Z-equivalent to the unknot.

### A bivariate extension of Bleimann-Butzer-Hahn operator by Khan R.A.

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