Self-dual Maps III: projective links

Abstract

In this paper, we present necessary and sufficient combinatorial conditions for a link to be projective, that is, a link in $RP^3$. This characterization is closely related to the notions of antipodally self-dual and antipodally symmetric maps. We also discuss the notion of symmetric cycle, an interesting issue arising in projective links leading us to an easy condition to prevent a projective link to be alternating.

Figures (14)

• Figure 1: (From left to right) A diagram of the Trefoil, its shadow with a 2-colored faces (vertices on white crossed circles), corresponding Black graph (bold edges and black circles) and White graph (dotted edges and white circles).
• Figure 2: (Left) Left-over-right rule from black point of view. (Right) Right-over-left rule from white point of view.
• Figure 3: (From left to right) diagram $D$ of the Hopf link, denoted by $2_1$, signed Black graph $(B_D,S_E)$ and signed White graph $(W_D,-S_E)$.
• Figure 4: Nonintersecting curves inside $\mathbb{B}^3$ (induced by a link in $\mathbb{R}\mathbb{P}^3$) and its projective diagram in $\mathbb{R}\mathbb{P}^2$.
• Figure 5: (Left) A $2$-antipodally symmetric map where the antipodal mapping is color-preserving. (Right) A $2$-antipodally symmetric map where the antipodal mapping is color-reversing.

Theorems & Definitions (11)

• Remark 1
• Remark 2
• Proposition 1
• Remark 3
• Lemma 1
• proof
• Theorem 1
• proof
• Corollary 1
• proof