A question regarding knots: SKETCH A PROOF OF THEOREM 1 A regular projection sat
ID: 3111732 • Letter: A
Question
A question regarding knots:
SKETCH A PROOF OF THEOREM 1
A regular projection satisfies the following conditions:
1) No line joining two vertices is parallel to the vertical axis.
2) No vertices span a plane containing a line parallel to the vertical axis.
3) There are no triple points in the projection
Need help with this, thanks
determined by an ardered espmat theT termin ined by the veencell with the perign hed b and q, the same for all iThmsy sin extra points in the defining eec r bon There are two theorems that make regular projectione especially useful. The first sta have a regular projection then there is an equivalent knot nearby that does have a regular projection. The second states that if a knot does have a regular projection then al nearby knots are equivalent and also have regular projes tions. The notion of nearby is made precise by m the distance between vertices point on the knot. Next perform a seqpetce dl eeetary oui that replace each p with & in the deeing or K. These moves are fot appl d to al ,nas do not bound intervals whose points. Finally each aossing poin ca te hoded tes that if a knot does not projesticns ois meig TERMINOLOGY A knot diagram consists facolet onofunate These arcs are The points in the diagran which compand to points in the projection are called cring pts, crossings. Above the aosang put te tes- called either egea o ara d ite igo Let K be a knot determined by the on THEOREM 1, every number t> there is a knot K' determined by an ordered set (g..) such that the distance from q' to pi is less than t for all the knot ones cald number of arcs is the same s the me d oans emu undercmoasing. Noice tat the other the derp sorude nyti i,K, is equivalent to K, and the projection of K' s regular. is equivalent to K, and the projection of K" isExplanation / Answer
First label each arc a1; a2; .... an where a1 connects p1 to p2, a2 connects p2
to p3, and so on, ending with an connecting pn to p1. Then, for any t > 0
construct an open ball at every vertex with radius t. If a point pi is moved to
any other point qi in the open ball around it the distance between pi and qi will
be less than t.
Fix p1 (that is let q1 = p1). Now, if a1 causes part of the projection to not be
regular it can be moved (via moving p2) so that the irregularities are gone, and
the knot is still equivalent. There is a single line running through p1 which is
parallel to the vertical axis. If this line runs through the open ball around p2
then remove it from the open ball. Also, there will be a nite number crossings
in the diagram. At each of these crossings create a plane that is parallel to the
vertical axis, and runs through the projection of p1 and the crossing point. If
any of these planes intersect the ball around p2 then remove that region from
the ball. We know the deleted ball will be nonempty because a ball with a nite
number of slices taken out will always be nonempty. Now if p2 is moved to any
point in the ball that we have created it will satisfy the conditions of making
the projection regular, and will be within distane t of its old position. However,
we may still need to limit further the places p2 can be moved to ensure that
K0
is equivalent to K. Do not let a1 be moved anywhere that would cause it
to cross another arc in the knot. We know we can do this from theorem two.
This still leaves innitely many places in the ball around p2 that we can move
p2. Pick one and label it q2.
Follow this same procedure for a2, only this time we need to conscider condition
2 (above) for a regular projection. If the plane that contains a1 and is parallel
to the vertical axis runs through the ball around p3, then remove that portion
of the ball also. Continue doing this with the rest of the vertices. When you
are nished there will be a new knot K0
that is equivalent to K (made up of
vertices q1; q2; ...; qn), has a regular projection, and every qi is within distance t of pi