% 12-4-1.tex
\documentclass{article}
\usepackage{amsmath,amssymb}
\usepackage[all,poly,knot]{xy}
\begin{document}
\xymatrix{ A & *\txt{B}    & C\\ {\bullet}  & {\sum_{k=1}^nk^3}  & {\Box} }

\xymatrix{
      A \ar@{=>}[d]\ar@{=}[r]\ar@{-}[dr]   &
      B\ar@{.}[r]\ar@{.>}[d]\ar"2,1" & C\ar@{~}[d]\ar@{:>}[dl]\ar@{:}[1,-2]\\
      {\bullet} \ar@{~>}[r] & {\sum_{k=1}^nk^3}  & {\Box} \ar@{-->}[l]\ar@{==>}[ul]
   }

\xymatrix{AAAAAAAAA \ar[r]^{up}_{dn} &B\ar[d]^a_b\\
AAAAAAAAA \ar[u]^a_b &B\ar[l]^-{up}_-{dn}\ar[ul]^1_2}

\xymatrix{ABCD \ar@{-}[r]|{xy} &B\ar@{}[d]|{xy}\\
ABCD \ar[u]|{xy}&B\ar@{-}[l]|-{xy}\ar[ul]|{xy}}

\xymatrix{ABCD \ar@{-}[r]|\hole &B\ar[d]|{\hole}\\
ABCD \ar[u]|\hole&B\ar@{-}[l]|-\hole\ar[ul]|\hole}

\xymatrix{A \ar [r] \ar @{} [dr] |=  & B \ar[d]\\
B\ar[u] \ar @{} [r] | \sim  & C}

\xymatrix{
1 \ar[rr] ^-{1000000x} \ar[dr] _(.2){2000x} |!{[d];[rr]}\hole
&& 1000000 \\ 1000 \ar[r] _{2x}\ar[urr] _>>>>(0.5){x^2}& 2000 }

\xymatrix{
A\ar[r]^-*+[o][F]{x} & *+[F=]{\sum_{i=n}^m {i^2}}
  && *+<10pt>[F-,]{D}\ar[ll]_>>><(0.4)*[F-:<2pt>]{f(x)} }

\xymatrix@1{
      A \ar @{|*\composite{{+}*{\times}}|} [rr]
      ^*+\txt{High\\label}
      && B \ar @{/*\composite{{\texttt{Y}}*{-}}/} [rr]
      && C  }

\xymatrix@1{ x \ar@/^/[r]*{\circ} & {\Box} }

\xymatrix@1{ x \ar@/^/[r]{yz} & {\Box} }

\xymatrix{ U \ar@/_7mm/[ddr]_y \ar@/^/[drr]^x
\ar@{.>}[dr]|-{(x,y)} \\
& X \times_Z Y \ar[d]^q \ar[r]_p
& X \ar[d]_f \\
& Y \ar[r]^g & Z }

\xymatrix{A\ar@(u,ul)[r]\ar@(dr,d)[r] &B}\qquad\qquad
\xymatrix@1{x \ar@(ul,dl)[]|{id} \ar@/^/[rr]|f && f(x)\ar@/^/[ll]|{f^{-1}} }

\xymatrix@1{A\ar@{{*}~>|}[r] & B\ar@_{<-)}[r] & \;C\ar@{_{(}->}[r] & D\ar@3{||-->>}[rr] &&E}

\xymatrix{A \ar @/^/ @{^{<}-_{>}} [r] \ar @/_/ @{{*}{x}{*}} [r] & B }

\xymatrix{ A \ar[d]  \\ B & C \ar@<1ex>[ul] \ar@<1ex>[ul];[] }

\xymatrix@1{ A \ar@/^/[r] \ar@/^/@<-1.5mm>[r] & B }

\xymatrix@1{ x \ar@/^/[r]+LU*{\circ} \ar@/_/[r]+<4mm,-1mm>*{*} & {\Box} }

\xymatrix{
       {\circ}
       \ar@{|-->} '[dr] ^a
                  '[rr]+D*{\bullet} ^b
                  [drrr] ^c
               & {\circ} & {\circ} & {\circ} \\
       {\circ} & {\circ} & {\circ} & {\circ} }

\xymatrix@!0{
        & \lambda\omega \ar@{-}[rr]\ar@{-}'[d][dd]
          & & \lambda C \ar@{-}[dd]
       \\
        \lambda2 \ar@{-}[ur]\ar@{-}[rr]\ar@{-}[dd]
          & & \lambda P2 \ar@{-}[ur]\ar@{-}[dd]
       \\
        & \lambda\underline\omega \ar@{-}'[r][rr]
          & & \lambda P\underline\omega
       \\
        \lambda{\to} \ar@{-}[rr]\ar@{-}[ur]
          & & \lambda P \ar@{-}[ur]
       }

\xymatrix{
  {\circ} \ar `r [d] ^a
              `[rr] ^b
              `/4pt[rr] ^c
              `[rrr] ^d
              `_dl [drrr]^e
                 [drrr]^f
          & {\circ} & {\circ} & {\circ} \\
  {\circ} & {\circ} & {\circ} & {\circ} }

\xymatrix{
  {\circ} \ar `r [d] ^a
              `[rr] ^b
              `/4pt[rr] ^c
              `[rrr] ^d  '[rrr] _{ee}
              `_dl [drrr]^e
                 [drrr]^f
          & {\circ} & {\circ} & {\circ} \\
  {\circ} & {\circ} & {\circ} & {\circ} }

\xymatrix{
  {\circ} \ar@{=>} `r [d] ^a
              `[rr] ^b
              `/4pt[rr] ^c
              `[rrr] ^d  '[rrr] _{ee}
              `_dl [drrr]^e
                 [drrr]^f
          & {\circ} & {\circ} & {\circ} \\
  {\circ} & {\circ} & {\circ} & {\circ} }

\xymatrix@1{A \ar `d[r]  [r] \ar@<+2pt> `d[r] `[r] [r] & B }

$$\vcenter{\SelectTips{cm}{}
\objectmargin={1pt}
\xygraph{ !{0;(.77,-.77):0}
 !~:{@{-}|@{>}}
 w (:[r(.6)]{x_1}
   ,:[d]z:[r]y:[u(.6)]{x_2}:"x_1":"z"
    :@( {"w";"z"}, {"y";"z"})"z":"x_2") }}\qquad\qquad\qquad
\vcenter{\xygraph{!{0;/r1.5pc/:}
 !P3"a"{~>{}}!P9"b"{~:{(1.3288,0):}~>{}}
 !P3"c"{~:{(2.5,0):}~>{}}
 !{\vover~{"b2"}{"b1"}{"a1"}{"a3"}}
 !{"b4";"b2"**\crv{"c1"}}
 !{\vover~{"b5"}{"b4"}{"a2"}{"a1"}}
 !{"b7";"b5"**\crv{"c2"}}
 !{\vover~{"b8"}{"b7"}{"a3"}{"a2"}}
 !{"b1";"b8"**\crv{"c3"}}}}
$$
\end{document}