# How can i change 8_19 to (3,4)-torus knot K(3,4)?

In the knot table, it is well-known that 8_19 is (3,4)-torus knot. But, it is not clear to me. How can i change 8_19 to (3,4)-torus knot K(3,4)? Moreover, it is well-known that braids of two equivalent knots are related by Markov moves. How can i change a braid presentation of 8_19 to a braid presentation of K(3,4) by using Markov moves?

• You might want to provide a link to some place where we can see what $8_{19}$ is. Mar 31 '15 at 1:54
• I added a picture. It looks like a torus knot to me. Perhaps you are using a different projection, seogman? Mar 31 '15 at 2:13

Place a point $0$ in the central triangle of the figure. Orient $K$ so it winds $+3$ times about $0$. Draw a straight ray $R$ going directly down from $0$. Rotate $R$ around $0$ in the anti-clockwise direction. Record the crossings as $R$ passes them. The orientation allows us to record each crossing as $\sigma_1^{\pm 1}$ or $\sigma_2^{\pm 1}$. (In fact, in this example all crossings are positive.)
Write out the word you get. A single Markov III move (in the braid group, $\sigma_2 \sigma_1 \sigma_2 = \sigma_1 \sigma_2 \sigma_1$) transforms your word into $(\sigma_1 \sigma_2)^4$: the closure of this word is the $(3,4)$ torus knot.
Added later: Different knot tables seem to show different projections of $8_{19}$. The figure that Ryan added to the original question is the $8_{19}$ knot as shown in what the Knot Atlas webpage calls "The Rolfsen Knot Table", but Rolfsen's book actually has a different projection where it takes two moves of the sort I described to get to the standard projection of a torus knot.