Abstract: In this talk, I will introduce an embedding presentation of a diagram, which is proved to be a unique

presentation of a diagram. Let $\cal L$ be a set of all of diagrams, called also links in this paper. An algebraic

system $(\cal L, \sim)$ is constructed. In fact, a link in $R^3$ (or $S^3$) is the equivalent class $[L]$ where

$L$ is one of its embedding presentations. Based on $(\cal L, \sim)$, Reduction Crossing Algorithm is

proposed which is used to reduce the number of crossings in an embedding presentation by introducing

a main tool called a pass replacement. For an infinite set of unknots $\cal U$, each $K$ in $\cal U$ can

be transformed  into the trivial unknot in at most  $O(n^c)$ by applying  the algorithm where $c$ is a

constant, $K\in {\cal U}$ and $n=|V(K)|$. As special consequences, three unknots are unknotted,  

which are Goeritz's unknot, Thistlethwaite's unknot and Haken's unknot (image courtesy of Cameron

Gordon). Moreover, an infinite family of unknots $K_{G_{2k,2l}}\in {\cal U}$ are unknotted in

$O(n\log\log n)$ time. In addition, unique presentations of a virtual link, an oriented link and oriented virtual link are introduced respectively.