This pattern copyright 1998, Elaine Fraser. 
Its use for commercial gain is not allowed.

A Klein Bottle (a 3-dimentional object with a single-surface)

1x25g Double Knitting
3mm hook (tension must be quite tight)

Finished height: approx. 20cm

Base:
Begin with 3ch, join to form ring.
8 dc in ring, join with sl st.
Work spirally, 1dc in next dc, 2dc infollowing dc until work
measures approximately 7cm in diameter.

Weave in the starting end - this will be inaccessible when the
bottle is finished.

Body:
Continue cylindrically, in rounds, using the same number of stitches
per row as no the last spiral. Note the number of stitches (S1).

Work until cylinder measures 4cm from base. Work a chain of
one third the number of stitches (S1/3) and sl st to the opposite
side of the cylinder, half-way around the circumference. 

Internal tube:
Continue in dc around one half of the cylinder, and work 1dc in each
of the chain stitches. Skip one stitch of cylinder body after working
over the chain, and before next set of stitches into chain. Do this 
three times. 
On next round, work 2dc into the stitch after, and before the central
chain - at the same point as the decreases were worked. Do this three 
times. At end of round, chain through the work, between the central
chain and the half-cylinder just worked, to arrive at the beginning
point of the chain. Note that the yarn will now be pulling under the
bar, but this will be corrected soon - there is no need to pull the
ball through the work.

Work 1dc in each space on the underside of the chain, and repeat the
half-cylinder decreases and increases as before. When this is finished,
work slip-stitches through the dcs, to join the two halves of the 
bar together, at the top. 

Chain through the work, evenly, along the line joining the bar to 
the cylinder body, making 10 even chain stitches from the top
of the bar, around the bottom and back to the top. At this point,
the yarn will be feeding normally again.

Bottle top:
Continue working the cylinder, as from base to bar, until bottle
height is 12cm. 
For next three rounds, work (2dc, skip 1), narrowing the cylinder to 
a bottle neck. Count the number of stitches remaining (S2).

Divide the number of stitches by four (S2/4) and work that many 
rounds (S3). 

Connector:
With the smooth end of the central bar facing (the side with the
10 even chains) mark the 'bottom' (South) of the bottle top in line 
with this, and mark the top and two mid-points like the four points 
of the compass.

        N
    W       E
        S

Make elbows:
On the next round, when the East point is reached, make a sl st, turn,
and work back to the West point. Make a sl st, turn and work back to the
East.
Work even for for (S3) rounds. 

Repeat this (S3) times in all, but on subsequent repeats, skip one stitch
at the South on the first row after the elbow.

Join to bar:
Decrease 1 stitch at South, every (S3) rounds, until 10 stitches remain.
Work even until connector comfortably reaches to the mouth of the 
central bar. End with sl st.

Ensuring connector is not overly twisted, evenly sl st open end to the 
10 chain around the mouth of the central bar. Cast off and weave in end.