**Definition 2.1** A *contact structure* ξξ on R3R3 assigns to each point p∈R3p∈R3 a plane ξp⊂R3ξp⊂R3. The *standard contact structure* assigns to the point p=(x,y,z)p=(x,y,z) the plane

ξ
p
=
Span
{
e
→
2
,
e
→
1
+
y
e
→
3
}
,
ξ
p
=
Span
{
e
→
2
,
e
→
1
+
y
e
→
3
}
,

(1)where we orient R3R3 via the Right Hand Rule.

**Definition 2.2** A *knot* in R3R3 is the image of an embedding Φ:S1→R3Φ:S1→R3. A *Legendrian knot* in R3,ξR3,ξ is a knot such that

Φ
'
(
θ
)
∈
ξ
Φ
(
θ
)
Φ
'
(
θ
)
∈
ξ
Φ
(
θ
)

(2)for all θθ.

Observe that if Φ(θ)=x(θ),y(θ),z(θ)Φ(θ)=x(θ),y(θ),z(θ) parametrizes a knot LL then for LL to be Legendrian in the standard contact structure, Φ'(θ)Φ'(θ) must be orthogonal to e→2×(e→1+ye→3)=ye→1-e→3. In particular, y(θ)x'(θ)-z'(θ)=0 for all θ .e→2×(e→1+ye→3)=ye→1-e→3. In particular, y(θ)x'(θ)-z'(θ)=0 for all θ .

Whenever we draw a knot in R3R3 we are actually drawing a projection of the knot in some plane with labeled crossings. We use the *front* and *Lagrangian* projections to draw Legendrian knots.

**Definition 2.3** Let Π:R3→R2Π:R3→R2 such that Π(x,y,z)=(x,z)Π(x,y,z)=(x,z). The *front projection of LL*, denoted Π(L)Π(L), is the image of LL under ΠΠ. If ΦΦ above parametrizes LL then ΦΠ(θ):=x(θ),z(θ)ΦΠ(θ):=x(θ),z(θ) parametrizes Π(L)Π(L).

We see from the identity y(θ)x'(θ)-z'(θ)=0y(θ)x'(θ)-z'(θ)=0 that the front projection of a Legendrian knot cannot have vertical tangencies, and at each crossing the slope of the over-crossing segment must be less than the slope of the under-crossing segment.

**Definition 2.4** Let π:R3→R2π:R3→R2 such that π(x,y,z)=(x,y)π(x,y,z)=(x,y). The *Lagrangian projection of LL* is the image of LL under ππ and is denoted π(L)π(L). As with the front projection, π(L)π(L) is parametrized by Φπ(θ):=x(θ),y(θ)Φπ(θ):=x(θ),y(θ).