From: Eli the Bearded <*@qz.little-neck.ny.us> Newsgroups: alt.ascii-art Subject: an intro to knots Date: 14 Aug 2001 00:30:56 GMT Organization: Some absurd concept Lines: 197 Message-ID: NNTP-Posting-Host: panix1.panix.com X-Trace: news.panix.com 997749056 12867 166.84.1.1 (14 Aug 2001 00:30:56 GMT) X-Complaints-To: abuse@panix.com NNTP-Posting-Date: 14 Aug 2001 00:30:56 GMT X-Files: Used for sharpening claws and teeth on your hawk and hacksaw X-Motto: "Erosion of rights never seems to reverse itself." -- kenny@panix X-US-Congress: Moronic Fucks. X-Attribution: EtB X-Usenet-II: Because it is time for October. Encrypted: double rot-13 X-Newsreader: Sony Playstation 5.0MIPS Path: chonsp.franklin.ch!pfaff.ethz.ch!news-zh.switch.ch!news-ge.switch.ch!news.maxwell.syr.edu!howland.erols.net!panix!news.panix.com!qz!not-for-mail Xref: chonsp.franklin.ch alt.ascii-art:20505 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 ------ ,mmm. ,nn. ,mmm. ,mmm. ,nn. ,mmm. ,mmm. ,nn. ,mmm. TMbm, NP`?N> TMbm, NP`?N> 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 References: X-Newsreader: Forte Free Agent 1.21/32.243 Lines: 40 Date: Tue, 14 Aug 2001 09:52:24 GMT NNTP-Posting-Host: 62.253.84.97 X-Complaints-To: abuse@ntlworld.com X-Trace: news2-win.server.ntlworld.com 997782632 62.253.84.97 (Tue, 14 Aug 2001 10:50:32 BST) NNTP-Posting-Date: Tue, 14 Aug 2001 10:50:32 BST Organization: ntlworld News Service Path: chonsp.franklin.ch!pfaff.ethz.ch!news-zh.switch.ch!news-ge.switch.ch!newsfeed00.sul.t-online.de!t-online.de!news-spur1.maxwell.syr.edu!news.maxwell.syr.edu!btnet-peer1!btnet-peer0!btnet!news5-gui.server.ntli.net!ntli.net!news2-win.server.ntlworld.com.POSTED!not-for-mail Xref: chonsp.franklin.ch alt.ascii-art:20499 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. > TMbm, NP`?N> TMbm, NP`?N> 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 > `"""' `~~' `"""' `"""' `~~' `"""' `"""' `~~' `"""' That last one's one hell of a knot... ;-) -- o |\ _[] L_ === Stuart Moore []\, stm