Original Usenet thread from alt.ascii-art, started 14 Aug 2001.
an intro to knots
alt.ascii-art
· 2 messages · 14 Aug 2001 - 14 Aug 2001
I've been working on a ascii-art comic about a trefoil named
Knoty Kned. It inspired me to write up this basic introduction
to knot theory with ascii-art diagrams. While knot theory is
a form of math, no math is needed to understand what is here.
An Introduction to Knot Diagrams
All text and diagrams by Eli the Bearded
In knot diagrams all the string exists at one of two levels,
top (visible) or bottom (invisible). The bottom level is only
used when strings need to cross. The string gets stretched and
or rearranged so that it never happens that more than two levels
are needed.
String on one level: String on two levels:
_ _
\\__ __ \\__ __ _
\ \_/ \ \ \_/ \ / \
\ \ \ / |
| ___/ | ___/ ._/
| / | /
/ | / |
/ \ / \
\___/ \___/
In knot diagrams the strings are always presumed to be loops without
ends. If the strings had ends, then nothing would hold the knots
together. (In real life friction holds knots together, but friction
doesn't count for much in a picture.) Sometimes just sections of
knots are shown, and the viewer must remember that the knots continue
outside of the picture.
The Reidemeister Moves for Knot Diagram Rearrangement
Move 1(a): Untwisting a free loop
__ _ _______
\ / \ \
... / | => ... |
___/ ._/ _______/
Move 1(b): Twisting a free loop
_______ __ _
\ \ / \
... | => ... / |
_______/ ___/ ._/
Move 2(a): Removing a double overlap
_____ _____ _____ _____
\ / \ /
\ ' | |
. \ | |
... | | ... => ... | | ...
` / | |
/ . | |
_____/ \_____ _____/ \_____
Move 2(a): Adding a double overlap
_____ _____ _____ _____
\ / \ /
| | \ '
| | . \
... | | ... => ... | | ...
| | ` /
| | / .
_____/ \_____ _____/ \_____
Move 3: Crossing an overlap
_____ _____ _____ _____
____ \ _ / ____ \ /
\ ' \ '
... \ ... => ... \ ...
/ \ ___ /._\ _____
____/ \_____ ____/ \_____
There is no (a) and (b) version of this one since rotating
the image would produce the two different versions.
Some Simple Knots In Diagrams
Single String Examples
The "Unknot"
___
/ \
| |
\___/
The "Trefoil" (also an "overhand knot")
__ __
/ ` / \
/ / /
| / \
| / / \
\____/ |
.____/
The "Figure 8 Knot"
_____
/ `
.- | ---------.
/ \ , \
| \ / |
| \ / |
| / |
| / \ |
| / \ |
| / | |
\ ' / |
`---------' /
._________/
Multipe String Examples
Two Unknots in a Link
____ ____
/ \ / \
/ \ \
/ , \ \
\ \ ' /
\ \ /
\____, \_____/
The "Borromean Rings" (no two are joined, but the three are)
______
/ \
/ \
| __ _____
/ | \ / \
/ \ \ | \
/ / ` / \
| | ----' |
\ | / /
\ / /
\_____/ .______/
The "Whitehead Link"
______
/ `
.- | ---------.
/ \ , \
| \ / |
| \ / |
| / |
| / \ |
| / \ |
| / | |
\ ' | /
`--------- / -'
._____/
Elijah
------
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"'.ch,"YF .m,"YP` 'N" "'.ch,"YF .m,"YP` 'N" "'.ch,"YF .m,"YP` 'N"
,dNT" "'.dN' "T'.YT ,dNT" "'.dN' "T'.YT ,dNT" "'.dN' "T'.YT
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N" `TY. Y: `TYo "'.dT, dN" `TY. Y: `TYo "'.dT, dN" `TY. Y: `TYo "'.d
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,dY' ,dNF . ` ,dY' ,dNF . ` ,dY' ,dN
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. `Y" .d' Yo ,dNP .o. `?h. `Y" .d' Yo ,dNP .o. `?h. `Y" .d' Yo ,dNP .
No ,dP' d, ,db' ,.`TF' ?No ,dP' d, ,db' ,.`TF' ?No ,dP' d, ,db' ,.`T
?Nb, '": `?N,'T' "N, ` ?Nb, '": `?N,'T' "N, ` ?Nb, '": `?N,'T' "N,
`?Nb, o.`?N, .d.`db `?Nb, o.`?N, .d.`db `?Nb, o.`?N, .d.`db
o.`"Y'.db `"'.db. ,eo o.`"Y'.db `"'.db. ,eo o.`"Y'.db `"'.db. ,eo
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<Nb,dN"'.dbdMP"'.Nb.iN> <Nb,dN"'.dbdMP"'.Nb.iN> <Nb,dN"'.dbdMP"'.Nb.iN
`"""' `~~' `"""' `"""' `~~' `"""' `"""' `~~' `"""'
On 14 Aug 2001 00:30:56 GMT, Eli the Bearded <*@qz.little-neck.ny.us> enlightened us with: > > I've been working on a ascii-art comic about a trefoil named > Knoty Kned. It inspired me to write up this basic introduction > to knot theory with ascii-art diagrams. While knot theory is > a form of math, no math is needed to understand what is here. > > > An Introduction to Knot Diagrams > > All text and diagrams by Eli the Bearded > ,mmm. ,nn. ,mmm. ,mmm. ,nn. ,mmm. ,mmm. ,nn. ,mmm. > <NF'?Nm, ?>TMbm, NP`?N> <NF'?Nm, ?>TMbm, NP`?N> <NF'?Nm, ?>TMbm, NP`?N > YT `"7Nb ' `YNh,' TY YT `"7Nb ' `YNh,' TY YT `"7Nb ' `YNh,' TY > "'.ch,"YF .m,"YP` 'N" "'.ch,"YF .m,"YP` 'N" "'.ch,"YF .m,"YP` 'N" > ,dNT" "'.dN' "T'.YT ,dNT" "'.dN' "T'.YT ,dNT" "'.dN' "T'.YT > dNT" .m: ,dN'.Q,`.cN" dNT" .m: ,dN'.Q,`.cN" dNT" .m: ,dN'.Q,`.cN" > N" `TY. Y: `TYo "'.dT, dN" `TY. Y: `TYo "'.dT, dN" `TY. Y: `TYo "'.d > ' ,do `Y, do `?Nb `M. `F' ,do `Y, do `?Nb `M. `F' ,do `Y, do `?Nb > oNT' `TbdT' `T' ,. "bo. oNT' `TbdT' `T' ,. "bo. oNT' `TbdT' `T' , > ,dY' ,dNF . ` ,dY' ,dNF . ` ,dY' ,dN > 'T ,b. ,dPYb. ,dP" 'bo. 'T ,b. ,dPYb. ,dP" 'bo. 'T ,b. ,dPYb. ,dP" > . `Y" .d' Yo ,dNP .o. `?h. `Y" .d' Yo ,dNP .o. `?h. `Y" .d' Yo ,dNP . > No ,dP' d, ,db' ,.`TF' ?No ,dP' d, ,db' ,.`TF' ?No ,dP' d, ,db' ,.`T > ?Nb, '": `?N,'T' "N, ` ?Nb, '": `?N,'T' "N, ` ?Nb, '": `?N,'T' "N, > `?Nb, o.`?N, .d.`db `?Nb, o.`?N, .d.`db `?Nb, o.`?N, .d.`db > o.`"Y'.db `"'.db. ,eo o.`"Y'.db `"'.db. ,eo o.`"Y'.db `"'.db. ,eo > .N' ndNP , .dNP' . lb .N' ndNP , .dNP' . lb .N' ndNP , .dNP' . lb > <Nb,dN"'.dbdMP"'.Nb.iN> <Nb,dN"'.dbdMP"'.Nb.iN> <Nb,dN"'.dbdMP"'.Nb.iN > `"""' `~~' `"""' `"""' `~~' `"""' `"""' `~~' `"""' That last one's one hell of a knot... ;-) -- o |\ _[] L_ === Stuart Moore []\, stm
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