Questions about these patterns?
Email mrwright at math dot uchicago dot edu!



Seifert surface of a trefoil (crochet):

Ch 15. Mark the current stitch (I often just make it looser); ch 15
more, give a full twist, and connect with ss to form a loop. You
should have a loop, but the stitches should gradually "twist" around;
at this point if you were to sc along one of the edges you'd end up
with an orientable strip with a full twist in it and with two edges.

Ch 15 more. With a half twist in the same direction as the full twist,
connect to the marked stitch (which should be exactly opposite the
previous joined stitch).

At this point you have the skeleton of the Seifert surface. If
everything was done right, there should be just a single edge; sc
along that edge until you're at the desired size. Skip one or two
stitches each time you go from one strip to the next.

If you got one of the twists wrong, it's not a big deal as long as you
notice during the first row: crochet a few of the stitches, then twist
the chain stitch fully around just after the last of the sc stitches
you just did, and continue. The ch. skeleton will have a twist in it
(between two consecutive sc stitches), but it won't be very noticeable
after you've done a row of sc on each side.

I like to go over the entire thing once more at the end in a different
colour, to emphasize the boundary.


Any Seifert surface at all (crochet):

Seifert's algorithm gives you several discs, and twisted strips
connecting them. It's pretty obvious what to do, then: crochet the
appropriate number of discs (these will grow later on, so you'll want
them to be smaller than they'll be in the final surface). In the last
disc, ch an appropriate number of stitches, connect to another disc
with an appropriate twist, and so forth until they're all connected;
if you're making the Seifert surface of a knot, you'll have only one
edge and so you can keep crocheting along that; if you're making the
surface of a link you'll need to do each component separately. Skip a
stitch where the strips connect to the discs.


Any Seifert surface at all (knit):

About the same as with crochet. Using double-pointed needles, make the
appropriate number of discs. One way to do this is to cast on, say,
five stitches (2 sts. on two of the needles; 1 on the other), with
five markers (initially between each pair of consecutive
stitches). Keep knitting around, increasing at each marker (such as by
knitting the front and back of the next stitch). If this is too
difficult to start, cast on more stitches (this will result in a
larger hole in the middle of the disc).

When the disc has reached the desired size, it's time to do the
strips. If you don't want any spacing between them you can purl as
many stitches as you want the strip to be wide, and continue working
the strip in stockinette stitch (holding the rest of the stitches with
a stitch holder).

If you do want spacing between the strips, cast off the number of
stitches for the gap, then knit as many stitches as the strip will be
wide, and so forth; then knit one of the strips as above.

Either way, once the strip gets to half of the desired length, hold
the stitches with a stitch holder and knit the rest of the strips,
again to half of their ultimate length.

You now have a bunch of discs, each with strips coming out of
them. Join them together using a Kitchener stitch. They should all be
joined "compatibly"; that is, there should never be a transition from
a "right" side to a "wrong" side (after all, part of the point of
Seifert surfaces is that they're orientable!)

If you want to emphasize the boundary in a different colour, you can
do it by crocheting or using your favourite applied I-cord.


One thing I don't like about this pattern is the tendency for the
stockinette stitch to curl. I found that I-cord along the edge is
stiff enough to mostly straighten it out, though. Another alternative
for dealing with curl is to change the stitch; the disadvantage is
that most stitches that don't curl look no different on the right and
wrong sides, making it less obvious that the surface is orientable.