Narrow Results

We introduce a local deformation called the virtualized $\mathrm{\Delta }$-move for virtual knots and links. We prove that the virtualized $\mathrm{\Delta }$-move is an unknotting operation for virtual knots. Furthermore we give a necessary and sufficient condition for two virtual links to be related by a finite sequence of virtualized $\mathrm{\Delta }$-moves.

We introduce a parity of classical crossings of virtual link diagrams which extends the Gaussian parity of virtual knot diagrams and the odd writhe of virtual links that extends that of virtual knots introduced by Kauffman [A self-linking invariants of virtual knots, Fund. Math.184 (2004) 135–158]. Also, we introduce a multi-variable polynomial invariant for virtual links by using the parity of classical crossings, which refines the index polynomial introduced in [Index polynomial invariants of virtual links, J. Knot Theory Ramifications19(5) (2010) 709–725]. As consequences, we give some properties of our invariant, and raise some 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 give a new interpretation of the Alexander polynomial ${\mathrm{\Delta }}_0$ for virtual knots due to Sawollek [On Alexander–Conway polynomials for virtual knots and Links, preprint (2001), arXiv:math/9912173] and Silver and Williams [Polynomial invariants of virtual links, J. Knot Theory Ramifications12 (2003) 987–1000], and use it to show that, for any virtual knot, ${\mathrm{\Delta }}_0$ determines the writhe polynomial of Cheng and Gao [A polynomial invariant of virtual links, J. Knot Theory Ramifications22(12) (2013), Article ID: 1341002, 33pp.] (equivalently, Kauffman’s affine index polynomial [An affine index polynomial invariant of virtual knots, J. Knot Theory Ramifications22(4) (2013), Article ID: 1340007, 30pp.]). We also use it to define a second-order writhe polynomial, and give some applications.

In this paper, we give two new criteria of detecting the checkerboard colorability of virtual links by using the odd writhe and the arrow polynomial of virtual links, respectively. As a result, we prove that 6 virtual knots are not checkerboard colorable, leaving only one virtual knot whose checkerboard colorability is unknown among all virtual knots up to four classical crossings.

In this paper, we study the invariants of twisted knots derived from the invariants of virtual knots such as the arc shift number and the odd writhe. We provide a class of twisted knots such that the arc shift number of every member is 1.