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Im et al. [Signature, nullity and determinant of checkerboard colorable virtual links, J. Knot Theory Ramifications 19(8) (2010) 1093–1114] introduced how to define Goeritz matrices for checkerboard colorable virtual links. In this paper, we extend this for the Goeritz matrices of virtual knots. And we consider its signature and determinant and show they are invariants for virtual knots.

In this paper, we introduce new index polynomial invariants for virtual knots. They are defined by using invariants of flat virtual knots. And we give some properties of these invariants and examples.

We introduce a family of polynomial invariants for flat virtual knots which extend the polynomial introduced by Kauffman and Richter, and we give several properties and examples.

We introduce some polynomial invariants for flat virtual links which are similar to the Jones–Kauffman polynomial, the Miyazawa polynomial and the arrow polynomial for virtual link diagrams, and we give several properties and examples.

We investigate embeddings from the set of long flat virtual knot diagrams to the set of long virtual knot diagrams so that we can construct invariants for long flat virtual knots. Also, we give properties and examples of several invariants for long flat virtual knots via these embeddings and invariants for long virtual knots.

We give a quotient of the ring so that the Miyazawa polynomial is a non-trivial invariant of welded links. Furthermore we show that this is also an invariant under the other forbidden move , and so it is a fused isotopy invariant. Also, we give some quotient ring so that the index polynomial can be an invariant for welded links.

We introduce invariants of flat virtual links which are induced from Vassiliev invariants of degree one for virtual links. Also we give several properties of these invariants for flat virtual links and examples. In particular, if the value of some invariants of flat virtual knots F are non-zero, then F is non-invertible so that every virtual knot overlying F is non-invertible.

In this paper, we present the Goeritz matrix for checkerboard colorable virtual links or, equivalently, checkerboard colorable links in thickened surfaces Sg × [0, 1], which is an extension of the Goeritz matrix for classical knots and links in ℝ3. Using this, we show that the signature, nullity and determinant of classical oriented knots and links extend to those of checkerboard colorable oriented virtual links.

We introduce a polynomial invariant of virtual links that is non-trivial for many virtuals, but is trivial on classical links. Also this polynomial is sometimes useful to find the virtual crossing number of virtual knots. We give various properties of this polynomial and examples.

We introduce a polynomial invariant of long virtual knots that is non-trivial for many virtual knots, but is trivial on classical knots. Also this polynomial is sometimes useful for finding the virtual crossing number of long virtual knots. We give various properties of this polynomial and examples.

H. Dye defined the parity mapping for a virtual knot diagram, which is a map from the set of real crossings of the diagram to ℤ. The notion generalizes the parity which is studied extensively by V. Manturov. The mapping induces the ith writhe (i ∈ ℤ\{0}) which is an invariant of the representing virtual knot. She applied the parity mapping to introduce a grade to the Henrich S-invariant for a virtual knot, and showed that the invariants are Vassiliev invariants of degree one. Following it, we define the parity mappings for a virtual link diagram, and define the similar invariants as above for a virtual link by using the parity mappings. We show that some of the invariants are Vassiliev invariants of degree one. We also checked necessary conditions for invertibility and amphicheirality via the invariants.

We construct some polynomial invariants for virtual links by the recursive method, which are different from the index polynomial invariant defined in [Y. H. Im, K. Lee and S. Y. Lee, Index polynomial invariant of virtual links, J. Knot Theory Ramifications19(5) (2010) 709–725]. We show that these polynomials can distinguish whether virtual knots can be invertible or not although the index polynomial cannot distinguish the invertibility of virtual knots.

We introduce a polynomial invariant of flat virtual knots which is sometimes useful for determining whether given flat virtual knots are invertible or not, and for finding the virtual crossing number of flat virtual knots. Also we give several properties of the polynomial invariant for flat virtual knots and examples. In particular, we show that the conjecture given by Hrencecin and Kauffman (2003) [8] is true and so an infinite class of flat virtual knots {Un}(n=1,2,…) are mutually distinct.