Virtual Knots

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

Virtual and ordinary crossings

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

Virtual Reidemeister moves

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 Diagram
Unknot
$N$
$[N]$
< Tr $M^2$>
0
Virtual Hopf Diagram
Virtual Hopf
$N-N^2$
$-[N][N-1]$
< Tr $M^4$>
0
Two Virtual Hopfs Diagram
Composite of two virtual Hopfs
$N(N-1)^2$
$[N][N-1]^2$
1
Virtual trefoil Diagram
Virtual trefoil (2.1)
$-2N(N-1)$
$-[2][N][N-1]$
< Tr $M^6$>
0
Three Virtual Hopfs Diagram
Composite of three virtual Hopfs
$-N(N-1)^3$
$-[N][N-1]^3$
0
Three Virtual Hopfs Diagram
Different composite of three virtual Hopfs
$-N(N-1)^3$
$-[N][N-1]^3$
1
Virtual Hopf and Virtual Trefoil Diagram
Composite of virtual Hopf and virtual 2.1
$2N(N-1)^2$
$[2][N][N-1]^2$
1
Virtual T[4,2} Diagram
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
Unknot Diagram
Twisted unknot
$N(N-1)$
$[N][N-1]$
< Tr $M$ Tr $M^3$>
0
Virtual Hopf Diagram
Virtual Hopf with a twist
$-N(N-1)^2$
$-[N][N-1]^2$
< Tr $M^2$ Tr $M^2$>
0
Two Virtual Hopfs Diagram
Two virtual Hopfs
$N^2(N-1)^2$
$[N]^2[N-1]^2$
1
Hopf Diagram
Hopf link
$2N(N-1)$
$[2][N][N-1]$
< Tr $M$ Tr $M^5$>
0
Two Virtual Hopfs and Unknot Diagram
Composite of two virtual Hopfs and twisted unknot
$N(N-1)^3$
$[N][N-1]^3$
0
Two Virtual Hopfs and Unknot Diagram
Different composite of two virtual Hopfs and twisted unknot
$N(N-1)^3$
$[N][N-1]^3$
1
Virtual trefoil and Unknot Diagram
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 Diagram
Virtual Hopf and composite of two virtual Hopfs
$- N^2(N-1)^3$
$-[N]^2[N-1]^3$
0
Virtual Hopf Diagram
Virtual Hopf and virtual 2.1
$2N^2(N-1)^2$
$[2][N]^2[N-1]^2$
1
Hopf and Virtual Hopf Diagram
Composite of Hopf and virtual Hopf
$-2N(N-1)^2$
$-[2][N][N-1]^2$
1
Virtual 3.1
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
Two Virtual Hopfs Diagram
Twisted composite of two virtual Hopfs
$N(N-1)^3$
$[N][N-1]^3$
0
Trefoil Diagram
Trefoil or virtual 3.5, 3.7
$4N(N-1)$
$[2]^2[N][N-1]$
1
Virtual Borromean Rings Diagram
Virtual Borromean Rings
$-N(N-1)(N-3)$
$-[N][N-1]\Big([N-2]-1\Big)$