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Subsections

Real world examples

In the previous section, I showed simple examples. This section deals with complete examples and shows how to put everything together.

The Reidemeister moves

Every introduction in Knot theory notes the three Reidemeister moves, here you can see how to typeset them with XY-pic. It is also the first example made up of many knot pieces carefully arranged.

First Reidemeister Move 

\xygraph{
$$ \xygraph{
     !{0;/r1.0pc/:} 
     [u] 
     !{\vover}
     !{\vcap-}
     [ul]!{\xcaph@(0)}
     [r]!{\xcaph@(0)}
    } \, \leftrightarrow \,
    \xygraph{
     !{0;/r1.0pc/:} [uu]
     !{\xcaph[3]@(0)}
  } $$
}

$\displaystyle \xygraph{ !{0;/r1.0pc/:} [u] !{\vover} !{\vcap-} [ul]!{\xcaph@(... ...ph@(0)} } \leftrightarrow \xygraph{ !{0;/r1.0pc/:} [uu] !{\xcaph[3]@(0)} }$

Second Reidemeister Move 

$$ \xygraph{
     !{0;/r1.0pc/:} 
     [u(0.8)]
     !{\xcaph@(0)}
     !{\vover}
     !{\vunder-}
     [l]!{\xcaph@(0)}
     [r]!{\xcaph@(0)} 
     [uul]!{\xcaph@(0)}           
   } \, \leftrightarrow \,
   \xygraph{
     !{0;/r1.0pc/:}             
     [u(0.8)]!{\huncross[2]}
   } $$
}

$\displaystyle \xygraph{ !{0;/r1.0pc/:} [u(0.8)] !{\xcaph@(0)} !{\vover} !{\vun... ...0)} } \leftrightarrow \xygraph{ !{0;/r1.0pc/:} [u(0.8)]!{\huncross[2]} }$

Third Reidemeister Move 

$$ \xygraph{
     !{0;/r1.0pc/:} 
     [u(0.7)]
     !{\xoverh[3]}
     [ull][ul(0.5)]!{\sbendv@(0)}
     [rrrr]!{\sbendh@(0)}
     [llllll][d(1.25)]!{\xcaph[-6]@(0)}           
   } \, \leftrightarrow \,
   \xygraph{
     !{0;/r1.0pc/:}             
     [uu]
     !{\xoverh[3]}
     [lldddd][ld(0.5)]!{\sbendh@(0)}
     [rrrr]!{\sbendv@(0)}
     [llllll][u(1.25)]!{\xcaph[-6]@(0)}
   } $$
}

$\displaystyle \xygraph{ !{0;/r1.0pc/:} [u(0.7)] !{\xoverh[3]} [ull][ul(0.5)]!{... ...d(0.5)]!{\sbendh@(0)} [rrrr]!{\sbendv@(0)} [llllll][u(1.25)]!{\xcaph[-6]@(0)} }$

Trefoil is made up with polygons

This section is based on the example given in [RM99]. It uses the polygon feature to arrange the knot pieces between the edges of stacked polygons. To do something similar you have to understand XY-pic's polygon feature described in [RM99], too. I have added orientation and labels, describing the writhe of the trefoil. 

\xygraph{
  !{0;/r2.0pc/:}
  !P3"a"{~>{}}
  !P9"b"{~:{(1.3288,0):}~>{}}
  !P3"c"{~:{(2.5,0):}~>{}}
  !{\vunder~{"b2"}{"b1"}{"a1"}{"a3"}<{-1}}
  !{\vcap~{"c1"}{"c1"}{"b4"}{"b2"}=<}
  !{\vunder~{"b5"}{"b4"}{"a2"}{"a1"}<{-1}}
  !{\vcap~{"c2"}{"c2"}{"b7"}{"b5"}=<}
  !{\vunder~{"b8"}{"b7"}{"a3"}{"a2"}<{-1}}
  !{\vcap~{"c3"}{"c3"}{"b1"}{"b8"}=<}
}
\xygraph{ !{0;/r2.0pc/:} !P3''a''{ >{}}!P9''b''{ :{(1.3288,0):} >{}}!P3''c''{ ... ...'}{''b7''}{''a3''}{''a2''}<{-1}} !{\vcap {''c3''}{''c3''}{''b1''}{''b8''}=<} }


At first three polygons are drawn, a triangle with the edges $ a1, \ldots, a3$, a polygon with 9 edges $ b1, \ldots, b9$ and at last a triangle with edges $ c1, \ldots, c3$. Then every knot piece is placed between 4 edges. In the next example the polygons are made visible and the numbering of the edges is shown: 


\xygraph{
  !{0;/r2.0pc/:}
  !P3"a"{~*{\xypolynode}>{}}
  !P9"b"{~:{(1.3288,0):}~*{\xypolynode}>{}}
  !P3"c"{~:{(2.5,0):}~*{\xypolynode}>{}}
}
\xygraph{ !{0;/r2.0pc/:} !P3''a''{ *{\xypolynode}>{}} !P9''b''{ :{(1.3288,0):} *{\xypolynode}>{}} !P3''c''{ :{(2.5,0):} *{\xypolynode}>{}} }


The figure-8 knot

Like the trefoil, the figure-8 knot is drawn using stacked polygons. The arrangement of the polygons is tricky and I cannot give you any general advice. 


 
\xygraph{ 
  !{0;/r2.0pc/:}
  !P9"e"{~:{(5,0):}~>{}}[u]
  !P5"d"{~:{(1.41421,0):}~>{}}[dd]
  !P4"a"{~:{(1.41421,0):}~>{}}[ddl]
  !P8"b"{~={45}~>{}} [rr]
  !P8"c"{~={45}~>{}} [ddl][u(0.1)]
  !P3"f"{~>{}}
  !{\xoverh~{"a2"}{"a1"}{"a3"}{"a4"}}
  !{\vover~{"b6"}{"b4"}{"f1"}{"b2"}}
  !{\vunder~{"c6"}{"c8"}{"f1"}{"c2"}}
  !{\vover~{"d3"}{"d1"}{"a2"}{"a1"}}
  !{\xcapv~{"c6"}{"f2"}{"b6"}{"f3"}}
  !{\hcap~{"d3"}{"e5"}{"b4"}{"e5"}}
  !{\hcap~{"d1"}{"e1"}{"c8"}{"e1"}}
}
\xygraph{ !{0;/r2.0pc/:} !P9''e''{ :{(5,0):} >{}}[u] !P5''d''{ :{(1.41421,0):... ... {''d3''}{''e5''}{''b4''}{''e5''}} !{\hcap {''d1''}{''e1''}{''c8''}{''e1''}} }


Here is the arrangement of the polygons: 



\xygraph{                                                        
  !{0;/r2.0pc/:}                                                 
  !P9"e"{~:{(5,0):}~*{
   \xypolynode}~>{-}}[u]                      
  !P5"d"{~:{(1.41421,0):}~*{
   \xypolynode}~>{-}}[dd]               
  !P4"a"{~:{(1.41421,0):}~*{
   \xypolynode}~>{-}}[ddl]              
  !P8"b"{~={45}~*{
   \xypolynode}~>{-}}[rr]                           
  !P8"c"{~={45}~*{
   \xypolynode}~>{-}}                             
  [ddl][u(0.1)]                                                  
  !P3"f"{~*{
   \xypolynode}~>{-}}                                   
}
\xygraph{ !{0;/r2.0pc/:} !P9''e''{ :{(5,0):} *{\xypolynode} >{-}}[u] !P5''d''... ...''{ ={45} *{\xypolynode} >{-}} [ddl][u(0.1)] !P3''f''{ *{\xypolynode} >{-}} }


next up previous contents
Next: Troubleshooting Up: Drawing knot pictures using Previous: Basic training   Contents
Lars Fischer 2003-06-15