Narrow Results

The zeroth coefficient polynomial of the skein (HOMFLYPT) knot polynomial called the Γ-polynomial is studied from a viewpoint of regular homotopy of knot diagrams. In particular, an elementary existence proof of the knot invariance of the Γ-polynomial is given. After observing that there are three types for 2-string tangle diagrams, the Γ-polynomial is generalized to a polynomial invariant of a 2-string tangle. As an application, we have a new proof of the assertion that Kinoshita's θ-curve is not equivalent to the trivial θ-curve.

We prove a folklore theorem of Thurston, which provides necessary and sufficient conditions for primality of a certain class of theta-curves. Namely, a theta-curve in the 3-sphere with an unknotted constituent knot $\kappa$ is prime, if and only if lifting the third arc of the theta-curve to the double branched cover over $\kappa$ produces a prime knot. We apply this result to Kinoshita’s theta-curve.

The height of a knotoid is a measure of how far it is from being a knot. Here, we define the positive and negative parts of the height, and we prove that they determine the unsigned height. Some polynomial invariants provide lower bounds for the signed heights. We also study a set of sequences associated to a knotoid.

We define invariants of higher dimensional theta-curves, and give a characterization of a trivial curve in the case of the classical dimension. We introduce a ribbon presentation of a knot and equivalence in ribbon presentations. Then a ribbon presentation induces a theta-curve, and invariants of theta-curves give those of ribbon presentations. Using the invariants, we can distinguish ribbon presentations of a knot.