The OU number and Reidemeister moves of type III for link diagrams

Abstract

We introduce the non-self OU sequence and the OU number for link diagrams. Using these, we give a lower bound for the number of necessary Reidemeister moves of type III between two diagrams of the same link.

Figures (9)

• Figure 1: Reidemeister moves.
• Figure 2: An RI I I$\alpha$ move.
• Figure 3: A 2-component link diagram $D=K_1 \cup K_2$ with non-self OU sequences $f(K_1)=O^2 U^2$ and $f(K_2)=OUOU$. Note that self crossings are not counted.
• Figure 4: A diagram $D=K_1 \cup K_2 \cup K_3$ with $f(K_1)=O^4$, $f(K_2)=OUOU$ and $f(K_3)=U^4$.
• Figure 5: The upper, middle, and lower segments $s_1, s_2$, and $s_3$. Middle segments are thickened.

Theorems & Definitions (55)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Definition 1
• Proposition 1
• proof
• Lemma 1
• proof
• Definition 2