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I have been looking for a clear definition of the n-framing of a knot unsuccessfully. A framing here refers to a choice of homeomorphism between a solid torus neighborhood (a.k.a, tubular neighborhood) of a knot and $S^1 \times D^2$ .I vaguely understand the number (which is an integer ) has to see with the number of times a basis for the normal space to the knot turns when doing a full $2\pi$ turn about the knot. I also know there are framings defined by a vector field V , where $V(k)$, the image of the (nowhere-zero)vector field at a point $k$ in the knot $K$ determines the framing. I assume a 0-framing means we take a normal vector and somehow parallel-displace it along the knot (having chosen a Riemannian metric $g$ to define the displacement), so that it returns exactly to its initial condition.I know that the framing number is an isotopy invariant. But this is not enough for me to know how to assign an integer to a framing. Any ideas, refs, please?

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3 Answers 3

up vote 3 down vote accepted

In my understanding, framing of a knot is not something you can calculate from the knot, but rather an additional piece of information. It is defined as follows. On the boundary of a tubular neighborhood of the knot there is a unique nontrivial simple closed curve which bounds a disc in the tubular neighbourhood. We call such a curve the meridian. Then there are infinitely many curves on the boundary of the tubular neighbourhood, which intersect the meridian once. They are called longitudes. Now a choice of a longitude is the framing of our knot. It is naturally in bijection with integers: all longitudes differ by some number of twists around the meridian (the sign of the difference is pinned down by orientation) and there is a unique longitude that is homologically zero in the complement of the knot, which is mapped to zero in our correspondence between integers and longitudes

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nice, +1; a followup: I guess that the number of intersections (aka, the twists around the meridian)of the meridians depends on the homology class of the longitude? Both answers are nice, I don't know which to accept. –  user99680 Feb 8 at 23:37
    
oh yes, exactly, homology of the knot complement is generated by the meridian, so the number of twists (which one may think of as adding meridians) is exactly the value of your chosen longitude as seen in first homology of the knot complement. –  Fyodor Feb 9 at 21:34

Knots are made of string, but framed knots are made of ribbon. More formally, the framing is a choice of a nowhere zero section of the normal bundle of the knot, which is essentially what you have written. This is the tubular neighborhood theorem!

Note that the collection of possible framings of a knot is $\Bbb{Z}$ as a torsor, not as a group. This means that we can speak of the difference between two framings, but there is no absolute sense of the framing of a knot.

When we draw diagrams (also called projections) of a framed knot in the plane, we draw two parallel strands that sometimes twist around each other. The blackboard framing is well-defined on a diagram (but meaningless for the knot itself), where the parallel strand does not twist around the original strand. Having established a blackboard framing for a knot, and after a choice of convention about what constitutes a positive crossing vs. negative crossing, the framing is just the linking number of the two edges of the ribbon, considered as a two-component link.

If we only allow ourselves the Reidemeister II and III moves, then the equivalence relation on diagrams is sometimes called regular isotopy and it respects framing.

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+1 ; does the "torsor class" (meaning constant difference ) agree with homology, and/or isotopy, i.e., are any two framings in the same torsor class isotopic? Both answers are very good; I'm having trouble choosing which to accept. –  user99680 Feb 8 at 23:45

This is my best understanding hopefully will help illustrate, together with other responses:

1) We choose a normal vector field for the core of $T:=S^1 \times D^2$ . Then the framing number is the winding number of the normal vector field about the core.

2)Say we select l, a longitude , as a basis element for $H^1 (T;\mathbb Z)$. Then the framing number is the linking number of the longitude with the core.

3)Any two n-framings are isotopic.

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