Narrow Results

In this paper, we give a definition of $\mathbb{Z}$-valued functions from the ambient isotopy classes of spherical/plane curves derived from chord diagrams, denoted by ${\sum }_i{\alpha }_i{x}_i$. Then, we introduce certain elements of the free $\mathbb{Z}$-module generated by the chord diagrams with at most $l$ chords, called relators of Type (I) ((SII), (WII), (SIII), or (WIII), respectively), and introduce another function ${\sum }_i{\alpha }_i{\tilde{x}}_i$ derived from ${\sum }_i{\alpha }_i{x}_i$. The main result (Theorem 1) shows that if ${\sum }_i{\alpha }_i{\tilde{x}}_i$ vanishes for the relators of Type (I) ((SII), (WII), (SIII), or (WIII), respectively), then ${\sum }_i{\alpha }_i{x}_i$ is invariant under the Reidemeister move of type RI (strong RII, weak RII, strong RIII, or weak RIII, respectively) that is defined in [N. Ito and Y. Takimura, $(1,2)$ and weak $(1,3)$ homotopies on knot projections, J. Knot Theory Ramifications22 (2013) 1350085 14 pp].

An axis of a link projection is a closed curve which lies symmetrically on each region of the link projection. In this paper we define axis systems of link projections and characterize axis systems of the standard projections of twist knots.

In this paper, we introduce a distance ${\tilde{d}}_{w3}$ on the equivalence classes of spherical curves under deformations of type RI and ambient isotopies. We obtain an inequality that estimate its lower bound (Theorem 1). In Theorem 2, we show that if for a pair of spherical curves $P$ and $P^{\prime }$, ${\tilde{d}}_{w3}([P],[P^{\prime }])=1$ and $P$ and $P^{\prime }$ satisfy a certain technical condition, then $P^{\prime }$ is obtained from $P$ by a single weak RIII only. In Theorem 3, we show that if $P$ and $P^{\prime }$ satisfy other conditions, then $P^{\prime }$ is ambient isotopic to a spherical curve that is obtained from $P$ by a sequence of a particular local deformations, which realizes ${\tilde{d}}_{w3}([P],[P^{\prime }])$.

We give a definition of an integer-valued function ${\sum }_i{\alpha }_i{x}_i^{\ast }$ derived from arrow diagrams for the ambient isotopy classes of oriented spherical curves. Then, we introduce certain elements of the free $\mathbb{Z}$-module generated by the arrow diagrams with at most $l$ arrows, called relators of Type ($\check{\mathrm{\text{I}}}$) (($\check{\mathrm{\text{SII}}}$), ($\check{\mathrm{\text{WII}}}$), ($\check{\mathrm{\text{SIII}}}$) or ($\check{\mathrm{\text{WIII}}}$), respectively), and introduce another function ${\sum }_i{\alpha }_i{\tilde{x}}_i^{\ast }$ to obtain ${\sum }_i{\alpha }_i{x}_i^{\ast }$. One of the main results shows that if ${\sum }_i{\alpha }_i{\tilde{x}}_i^{\ast }$ vanishes on finitely many relators of Type ($\check{\mathrm{\text{I}}}$) (($\check{\mathrm{\text{SII}}}$), ($\check{\mathrm{\text{WII}}}$), ($\check{\mathrm{\text{SIII}}}$) or ($\check{\mathrm{\text{WIII}}}$), respectively), then ${\sum }_i{\alpha }_i{x}_i^{\ast }$ is invariant under the deformation of type RI (strong RII, weak RII, strong RIII or weak RIII, respectively). The other main result is that we obtain new functions of arrow diagrams with up to six arrows explicitly. This computation is done with the aid of computers.