A Note on Knot Invariants In

We prove a relation between several easy-to-define numerical invariants of generic knots in H × S, and discuss its implication to contact geometry. 1. Definitions of Invariants The 2-dimensional hyperbolic space H2 is defined to be the upper half plane {(x, y) ∈ R2| y > 0} equipped with the metric ds2 = dx +dy y2 . STH2 is the unit tangent bundle of H2, i.e., the S1-bundle over H2 formed by tangent vectors of length 1. There is a natural orientation preserving diffeomorphism Φ : H2 × S1 → STH2 given by (1) Φ(x, y, θ) = y(cos θ ∂ ∂x + sin θ ∂ ∂y )|(x,y). Let R0 beR 3 with the z-axis removed, and (z, r, φ) the standard cylindrical coordinates of R3. Then there is a natural orientation preserving diffeomorphism Ψ : H2×S1 → R0 given by (2)    z = x, r = y 1 2 , φ = θ. In the rest of this paper, we will always identify these three manifolds by the diffeomorphisms Φ and Ψ. An embedding of S1 into H2 × S1 is a called knot. The standard orientation of S1 induces an orientation on every knot. Unless otherwise specified, all knots in this paper are oriented this way. Since H1(H 2 × S1) ∼= H1(S 1) ∼= Z, and the homology class of a S1-fiber is a generator of H1(H 2 × S1), we have that, for any knot K in H1(H 2 × S1), there is a unique integer h(K) satisfying [K] = h(K)[{pt} × S1] ∈ H1(H 2 × S1). Definition 1.1. h(K) is called the homology of the knot K. A knot in H2×S1 is said to be generic if it is nowhere tangent to the S1-fibers. Two generic knots are said to be generically isotopic if they are isotopic through generic knots. Let Pr : H2 × S1 → H2 be the projection onto the first factor. For any generic knot K, Pr(K) is an immersed curve in H2. For an immersed curve L in H2, the canonical lifting of L to H2 × S1 is defined to be L̃ = dL ds ∈ STH2 = H2 × S1, where ds is the arc length element of L. Definition 1.2. For a generic knot K in H2 × S1, define the rotation number of K to be r(K) = h(P̃ r(K)), where P̃ r(K) is the canonical lifting of Pr(K). 1