A Presentation Of (1,2)-knots Via The Mapping Class Group Of The 4th-punctured Torus

(g, n) - knots is a very active branch of the topology in 3 - manifold. (1,1)-knot is a very special type of (g, n) -knot. Its structure is simple and topological properties are easy to express. Some conclusions have been given recently. (1,2) - knots which is closed to (1, n) - knots is more complex than (1,1) - knots , so it is meaningful to explore (1,2) - knots.A(g, n) - knots is also called g genus n -bridge knots. Similarly A (1,1) - knot is called 1 genus 1 - bridge knots. In generally, a (1,1) - knot can be obtained by gluing along the boundarty two torus with a trivial arc properly embedded. There is no relation with the attching manners for the (1,1) - knots.In corresponding to this, A (1,2) - knot is called 1 genus 2 - bridge knot. (1,2) - knots can be obtained by gluing along the boundarty two torus with two trivial arcs properly embedded. But we can obtain different through attching manners for the (1,2) -knots. We may get a (1,2) - link which is a (1,2) - knot or two (1,1) - knots.This chapter introduced an algebraic representation for (1,2) - knots. we prove that every (1, 2) - knot in a lens space can be represented by the composition of an element of a certain rank two free subgroup of MCG₄(T)with a standard element only depending on the ambient space.