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Lee Mosher
Lee Mosher
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The train track $\tau$ has a dual bigon track $\tau^\perp$, described in (for example) Penner's book. The bigon track $\tau^\perp$ might not be maximal, but it can be enlarged in various ways by triangulating its complementary regions. Every transverse multiloop to $\tau$ is carried by a unique minimal enlargement of $\tau^\perp$. Furthermore, although the transverse measure on $\tau^\perp$ that you get on $\tau^\perp$ (or one of its enlargements) is not unique, the bigons can be used to precisely define linear relations, representing the operation of isotoping strands across the bigon. Altogether this gives a parameterization as you like.

The train track $\tau$ has a dual bigon track $\tau^\perp$, described in (for example) Penner's book. The bigon track $\tau^\perp$ might not be maximal, but it can be enlarged in various ways by triangulating its complementary regions. Every transverse multiloop to $\tau$ is carried by a unique minimal enlargement of $\tau^\perp$. Furthermore, although the transverse measure on $\tau^\perp$ that you get is not unique, the bigons can be used to precisely define linear relations, representing the operation of isotoping strands across the bigon. Altogether this gives a parameterization as you like.

The train track $\tau$ has a dual bigon track $\tau^\perp$, described in (for example) Penner's book. The bigon track $\tau^\perp$ might not be maximal, but it can be enlarged in various ways by triangulating its complementary regions. Every transverse multiloop to $\tau$ is carried by a unique minimal enlargement of $\tau^\perp$. Furthermore, although the transverse measure that you get on $\tau^\perp$ (or one of its enlargements) is not unique, the bigons can be used to precisely define linear relations, representing the operation of isotoping strands across the bigon. Altogether this gives a parameterization as you like.

Source Link
Lee Mosher
Lee Mosher
  • 15.4k
  • 2
  • 42
  • 81

The train track $\tau$ has a dual bigon track $\tau^\perp$, described in (for example) Penner's book. The bigon track $\tau^\perp$ might not be maximal, but it can be enlarged in various ways by triangulating its complementary regions. Every transverse multiloop to $\tau$ is carried by a unique minimal enlargement of $\tau^\perp$. Furthermore, although the transverse measure on $\tau^\perp$ that you get is not unique, the bigons can be used to precisely define linear relations, representing the operation of isotoping strands across the bigon. Altogether this gives a parameterization as you like.