## Virtual Knots

Virtual knots are a generalization of ordinary knots. These knots have an additional type of crossing, namely "virtual" or "sterile" one.

These virtual crossings satisfy their own Reidemeister moves, similar to the ordinary ones:

The way to calculate knot invariants for virtual knots is provided by Hypercube approach. The corresponding hypercube dimensions are listed in the table below.   HOMFLY polynomials for some virtual knots

The pictures were taken from the paper by L.Kauffman

 Correlator Genus $\Gamma^{\mathcal{L}}$ $\mathcal{L}$ $d_{\bar S}^{\mathcal{L}}(q=1)$ $d_{\bar S}^{\mathcal{L}}$ < Tr I> 0 Unknot $N$ $[N]$ < Tr $M^2$> 0 Virtual Hopf $N-N^2$ $-[N][N-1]$ < Tr $M^4$> 0 Composite of two virtual Hopfs $N(N-1)^2$ $[N][N-1]^2$ 1 Virtual trefoil (2.1) $-2N(N-1)$ $-[2][N][N-1]$ < Tr $M^6$> 0 Composite of three virtual Hopfs $-N(N-1)^3$ $-[N][N-1]^3$ 0 Different composite of three virtual Hopfs $-N(N-1)^3$ $-[N][N-1]^3$ 1 Composite of virtual Hopf and virtual 2.1 $2N(N-1)^2$ $[2][N][N-1]^2$ 1 Virtual link T[4,2} $-4N(N-1)$ $-[2]^2[N][N-1]$ 1 $N(N-1)(N-3)$ $[N][N-1]\Big([N-2]-1\Big)$ < Tr $M$ Tr $M$> 0 Twisted unknot $N(N-1)$ $[N][N-1]$ < Tr $M$ Tr $M^3$> 0 Virtual Hopf with a twist $-N(N-1)^2$ $-[N][N-1]^2$ < Tr $M^2$ Tr $M^2$> 0 Two virtual Hopfs $N^2(N-1)^2$ $[N]^2[N-1]^2$ 1 Hopf link $2N(N-1)$ $[2][N][N-1]$ < Tr $M$ Tr $M^5$> 0 Composite of two virtual Hopfs and twisted unknot $N(N-1)^3$ $[N][N-1]^3$ 0 Different composite of two virtual Hopfs and twisted unknot $N(N-1)^3$ $[N][N-1]^3$ 1 Composite of virtual 2.1 and twisted unknot $-2N(N-1)^2$ $-[2][N][N-1]^2$ < Tr $M^2$ Tr $M^4$> 0 Virtual Hopf and composite of two virtual Hopfs $- N^2(N-1)^3$ $-[N]^2[N-1]^3$ 0 Virtual Hopf and virtual 2.1 $2N^2(N-1)^2$ $[2][N]^2[N-1]^2$ 1 Composite of Hopf and virtual Hopf $-2N(N-1)^2$ $-[2][N][N-1]^2$ 1 Virtual 3.1, 3.2, 3.3, 3.4 $-N(N-1)(N-3)$ $-[N][N-1]\Big([N-2]-1\Big)$ < Tr $M^3$ Tr $M^3$> 0 Twisted composite of two virtual Hopfs $N(N-1)^3$ $[N][N-1]^3$ 0 Trefoil or virtual 3.5, 3.7 $4N(N-1)$ $[2]^2[N][N-1]$ 1 Virtual Borromean Rings $-N(N-1)(N-3)$ $-[N][N-1]\Big([N-2]-1\Big)$

• Original papers