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Kevin Walker
Kevin Walker
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I think you are confused about what "framing" means in this context, based on your clarification of the question in the comments.

A framing of a graph inside a 3-manifold $M$ is, by definition, a choice of (isotopy class of) ribbon whose core is the original graph. If the graph is a single loop and if $M$ is an integer homology sphere, then we can identity framings with integers (the linking number of a boundary component of the ribbon with its core). Otherwise we can't. So saying that the "standard ribbons in the standard handlebody has framing 0" is incorrect -- you can't describe a framing in a handlebody with an integer (such as 0).

If we glue two handlebodies together to form a closed 3-manifold $M$, and if each handlebody contains a ribbon graph, then image under gluing of the union of these two ribbon graphs is a ribbon graph in $M$, i.e. a framed graph in $M$.

I think you are confused about what "framing" means in this context, based on your clarification of the question in the comments.

A framing of a graph inside a 3-manifold $M$ is, by definition, a choice of (isotopy class of) ribbon whose core is the original graph. If the graph is a single loop and if $M$ is an integer homology sphere, then we can identity framings with integers. Otherwise we can't. So saying that the "standard ribbons in the standard handlebody has framing 0" is incorrect -- you can't describe a framing in a handlebody with an integer (such as 0).

If we glue two handlebodies together to form a closed 3-manifold $M$, and if each handlebody contains a ribbon graph, then image under gluing of the union of these two ribbon graphs is a ribbon graph in $M$, i.e. a framed graph in $M$.

I think you are confused about what "framing" means in this context, based on your clarification of the question in the comments.

A framing of a graph inside a 3-manifold $M$ is, by definition, a choice of (isotopy class of) ribbon whose core is the original graph. If the graph is a single loop and if $M$ is an integer homology sphere, then we can identity framings with integers (the linking number of a boundary component of the ribbon with its core). Otherwise we can't. So saying that the "standard ribbons in the standard handlebody has framing 0" is incorrect -- you can't describe a framing in a handlebody with an integer (such as 0).

If we glue two handlebodies together to form a closed 3-manifold $M$, and if each handlebody contains a ribbon graph, then image under gluing of the union of these two ribbon graphs is a ribbon graph in $M$, i.e. a framed graph in $M$.

Source Link
Kevin Walker
Kevin Walker
  • 12.8k
  • 2
  • 42
  • 91

I think you are confused about what "framing" means in this context, based on your clarification of the question in the comments.

A framing of a graph inside a 3-manifold $M$ is, by definition, a choice of (isotopy class of) ribbon whose core is the original graph. If the graph is a single loop and if $M$ is an integer homology sphere, then we can identity framings with integers. Otherwise we can't. So saying that the "standard ribbons in the standard handlebody has framing 0" is incorrect -- you can't describe a framing in a handlebody with an integer (such as 0).

If we glue two handlebodies together to form a closed 3-manifold $M$, and if each handlebody contains a ribbon graph, then image under gluing of the union of these two ribbon graphs is a ribbon graph in $M$, i.e. a framed graph in $M$.