Paper cutting (ConceptTopic, 4)

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<[#ontology [kind topic] [cats Essays Topology]]>
This page documents the results cutting a twisted loop of paper.
This page documents the results cutting a twisted loop of paper.
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>== Cutting in half ==
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== Cutting in half ==
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http://img59.exs.cx/img59/2479/cut.png
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<[#embed [hash 1PWBJA7JSTS04YJKW08GPYG94Z]]>
*A 0-twist ([[hose]]) goes to two separate 0-twists:
*A 0-twist ([[hose]]) goes to two separate 0-twists:
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*A 1-twist ([[Möbius strip]]) goes to a long 4-twist:
*A 1-twist ([[Möbius strip]]) goes to a long 4-twist:
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*A 2-twist goes to two linked together 2-twists:
*A 2-twist goes to two linked together 2-twists:
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*A 3-twist goes to a long 8-twist containing a [[trefoil knot]]:
*A 3-twist goes to a long 8-twist containing a [[trefoil knot]]:
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*A 4-twist goes to two linked together 4-twists:
*A 4-twist goes to two linked together 4-twists:
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|http://teamikaria.com/dl/r7IsJgql2lV-Y9G8avi68lWszG_cpAtaXcThzmn523w_jyqQ.png
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|<[#embed [hash SNSCP2SBPMXTR1YG4GTRE1EQGP]]>
-
|http://teamikaria.com/dl/S21EOFzp_QnM1XXHgzoPjLkqSrONYOmCNYUrWDZSPxMMYzDP.png
+
|<[#embed [hash T3XZTVFTKJK4C7GQCP2E4F9TE1]]>
|-
|-
-
|http://teamikaria.com/dl/fGc2kryV3TGg3XfCUwfsbABN9x6q-BelEMdk3IGyEEZ3sro8.png
+
|<[#embed [hash S948W45H3QQGJNAVPT3SA8CMWC]]>
-
|http://teamikaria.com/dl/r7IsJgql2lV-Y9G8avi68lWszG_cpAtaXcThzmn523w_jyqQ.png
+
|<[#embed [hash SNSCP2SBPMXTR1YG4GTRE1EQGP]]>
-
|http://teamikaria.com/dl/r7IsJgql2lV-Y9G8avi68lWszG_cpAtaXcThzmn523w_jyqQ.png
+
|<[#embed [hash SNSCP2SBPMXTR1YG4GTRE1EQGP]]>
-
|http://teamikaria.com/dl/U809reKC6hAZFkID5iW849WEXHWLSHA3HUy_BySEO2qOCKua.png
+
|<[#embed [hash 7VYJYHCTE6VCAVWPQFT26A8S9G]]>
-
|http://teamikaria.com/dl/TNhEm8Kxs-KOeLTHm5445t9BxD8CeeSBc82So4QefnJM-5h-.png
+
|<[#embed [hash 1FJ1RF259DFJT3XZGAJ7GPXQ82]]>
-
|http://teamikaria.com/dl/4ci7taOw1EfPsscuIhxdCFyTkQ1HelwW9j78EVtdpIViA9ZJ.png
+
|<[#embed [hash WWYRA3WYXE5CPRV1E5BXTV301B]]>
-
|http://teamikaria.com/dl/r7IsJgql2lV-Y9G8avi68lWszG_cpAtaXcThzmn523w_jyqQ.png
+
|<[#embed [hash SNSCP2SBPMXTR1YG4GTRE1EQGP]]>
-
|http://teamikaria.com/dl/r7IsJgql2lV-Y9G8avi68lWszG_cpAtaXcThzmn523w_jyqQ.png
+
|<[#embed [hash SNSCP2SBPMXTR1YG4GTRE1EQGP]]>
-
|http://teamikaria.com/dl/ET_Axau-K9zuB2hAzCqUlj29JC1BD4ybV0pc4oJKQI04bnA5.png
+
|<[#embed [hash W228RY1J3K4BPRG0GKWVWM6QPT]]>
-
|http://teamikaria.com/dl/r7IsJgql2lV-Y9G8avi68lWszG_cpAtaXcThzmn523w_jyqQ.png
+
|<[#embed [hash SNSCP2SBPMXTR1YG4GTRE1EQGP]]>
-
|http://teamikaria.com/dl/r7IsJgql2lV-Y9G8avi68lWszG_cpAtaXcThzmn523w_jyqQ.png
+
|<[#embed [hash SNSCP2SBPMXTR1YG4GTRE1EQGP]]>
-
|http://teamikaria.com/dl/U809reKC6hAZFkID5iW849WEXHWLSHA3HUy_BySEO2qOCKua.png
+
|<[#embed [hash 7VYJYHCTE6VCAVWPQFT26A8S9G]]>
|}
|}
*A 5-twist goes to a long 12-twist containing a knot with five crossings:
*A 5-twist goes to a long 12-twist containing a knot with five crossings:
-
{|cellpadding=&quot;0&quot; cellspacing=&quot;0&quot;
+
{|cellpadding="0" cellspacing="0"
-
|http://teamikaria.com/dl/qjNL1AqhdfJF-QuxIA34_ywBleTcHIRDzoffL-mE8PIEobT8.png
+
|<[#embed [hash 456DGFJD4RGHCGVR4AA96SV9FF]]>
-
|http://teamikaria.com/dl/r7IsJgql2lV-Y9G8avi68lWszG_cpAtaXcThzmn523w_jyqQ.png
+
|<[#embed [hash SNSCP2SBPMXTR1YG4GTRE1EQGP]]>
-
|http://teamikaria.com/dl/r7IsJgql2lV-Y9G8avi68lWszG_cpAtaXcThzmn523w_jyqQ.png
+
|<[#embed [hash SNSCP2SBPMXTR1YG4GTRE1EQGP]]>
-
|http://teamikaria.com/dl/r7IsJgql2lV-Y9G8avi68lWszG_cpAtaXcThzmn523w_jyqQ.png
+
|<[#embed [hash SNSCP2SBPMXTR1YG4GTRE1EQGP]]>
-
|http://teamikaria.com/dl/S21EOFzp_QnM1XXHgzoPjLkqSrONYOmCNYUrWDZSPxMMYzDP.png
+
|<[#embed [hash T3XZTVFTKJK4C7GQCP2E4F9TE1]]>
-
|http://teamikaria.com/dl/qNLCl5smjE7qcLtT2YKMEeAUd8UIdjgSXRWClis5GOhTQY8V.png
+
|<[#embed [hash DPC5Y0TCPCGTP4AA0RESP6RM81]]>
-
|http://teamikaria.com/dl/qjNL1AqhdfJF-QuxIA34_ywBleTcHIRDzoffL-mE8PIEobT8.png
+
|<[#embed [hash 456DGFJD4RGHCGVR4AA96SV9FF]]>
-
|http://teamikaria.com/dl/r7IsJgql2lV-Y9G8avi68lWszG_cpAtaXcThzmn523w_jyqQ.png
+
|<[#embed [hash SNSCP2SBPMXTR1YG4GTRE1EQGP]]>
-
|http://teamikaria.com/dl/r7IsJgql2lV-Y9G8avi68lWszG_cpAtaXcThzmn523w_jyqQ.png
+
|<[#embed [hash SNSCP2SBPMXTR1YG4GTRE1EQGP]]>
-
|http://teamikaria.com/dl/r7IsJgql2lV-Y9G8avi68lWszG_cpAtaXcThzmn523w_jyqQ.png
+
|<[#embed [hash SNSCP2SBPMXTR1YG4GTRE1EQGP]]>
-
|http://teamikaria.com/dl/r7IsJgql2lV-Y9G8avi68lWszG_cpAtaXcThzmn523w_jyqQ.png
+
|<[#embed [hash SNSCP2SBPMXTR1YG4GTRE1EQGP]]>
-
|http://teamikaria.com/dl/r7IsJgql2lV-Y9G8avi68lWszG_cpAtaXcThzmn523w_jyqQ.png
+
|<[#embed [hash SNSCP2SBPMXTR1YG4GTRE1EQGP]]>
-
|http://teamikaria.com/dl/r7IsJgql2lV-Y9G8avi68lWszG_cpAtaXcThzmn523w_jyqQ.png
+
|<[#embed [hash SNSCP2SBPMXTR1YG4GTRE1EQGP]]>
|?
|?
|-
|-
-
|http://teamikaria.com/dl/fGc2kryV3TGg3XfCUwfsbABN9x6q-BelEMdk3IGyEEZ3sro8.png
+
|<[#embed [hash S948W45H3QQGJNAVPT3SA8CMWC]]>
-
|http://teamikaria.com/dl/r7IsJgql2lV-Y9G8avi68lWszG_cpAtaXcThzmn523w_jyqQ.png
+
|<[#embed [hash SNSCP2SBPMXTR1YG4GTRE1EQGP]]>
-
|colspan=&quot;2&quot;|http://teamikaria.com/dl/bS7C9Pqrcl14xaeiimbdNGcsPkBTfuCupg5LKcprKc9HRers.png
+
|colspan="2"|<[#embed [hash RD2CRECEJ1MCR73SYPMWE2W5JT]]>
-
|http://teamikaria.com/dl/U809reKC6hAZFkID5iW849WEXHWLSHA3HUy_BySEO2qOCKua.png
+
|<[#embed [hash 7VYJYHCTE6VCAVWPQFT26A8S9G]]>
-
|http://teamikaria.com/dl/TNhEm8Kxs-KOeLTHm5445t9BxD8CeeSBc82So4QefnJM-5h-.png
+
|<[#embed [hash 1FJ1RF259DFJT3XZGAJ7GPXQ82]]>
-
|http://teamikaria.com/dl/fGc2kryV3TGg3XfCUwfsbABN9x6q-BelEMdk3IGyEEZ3sro8.png
+
|<[#embed [hash S948W45H3QQGJNAVPT3SA8CMWC]]>
-
|http://teamikaria.com/dl/r7IsJgql2lV-Y9G8avi68lWszG_cpAtaXcThzmn523w_jyqQ.png
+
|<[#embed [hash SNSCP2SBPMXTR1YG4GTRE1EQGP]]>
-
|http://teamikaria.com/dl/r7IsJgql2lV-Y9G8avi68lWszG_cpAtaXcThzmn523w_jyqQ.png
+
|<[#embed [hash SNSCP2SBPMXTR1YG4GTRE1EQGP]]>
-
|http://teamikaria.com/dl/r7IsJgql2lV-Y9G8avi68lWszG_cpAtaXcThzmn523w_jyqQ.png
+
|<[#embed [hash SNSCP2SBPMXTR1YG4GTRE1EQGP]]>
-
|http://teamikaria.com/dl/r7IsJgql2lV-Y9G8avi68lWszG_cpAtaXcThzmn523w_jyqQ.png
+
|<[#embed [hash SNSCP2SBPMXTR1YG4GTRE1EQGP]]>
-
|http://teamikaria.com/dl/r7IsJgql2lV-Y9G8avi68lWszG_cpAtaXcThzmn523w_jyqQ.png
+
|<[#embed [hash SNSCP2SBPMXTR1YG4GTRE1EQGP]]>
-
|http://teamikaria.com/dl/r7IsJgql2lV-Y9G8avi68lWszG_cpAtaXcThzmn523w_jyqQ.png
+
|<[#embed [hash SNSCP2SBPMXTR1YG4GTRE1EQGP]]>
|?
|?
|}
|}
Line 175: Line 176:
*A 6-twist goes to two linked together 6-twists:
*A 6-twist goes to two linked together 6-twists:
-
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+
{|cellpadding="0" cellspacing="0"
-
|http://teamikaria.com/dl/qjNL1AqhdfJF-QuxIA34_ywBleTcHIRDzoffL-mE8PIEobT8.png
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|<[#embed [hash 456DGFJD4RGHCGVR4AA96SV9FF]]>
-
|http://teamikaria.com/dl/r7IsJgql2lV-Y9G8avi68lWszG_cpAtaXcThzmn523w_jyqQ.png
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|<[#embed [hash SNSCP2SBPMXTR1YG4GTRE1EQGP]]>
-
|http://teamikaria.com/dl/r7IsJgql2lV-Y9G8avi68lWszG_cpAtaXcThzmn523w_jyqQ.png
+
|<[#embed [hash SNSCP2SBPMXTR1YG4GTRE1EQGP]]>
-
|http://teamikaria.com/dl/r7IsJgql2lV-Y9G8avi68lWszG_cpAtaXcThzmn523w_jyqQ.png
+
|<[#embed [hash SNSCP2SBPMXTR1YG4GTRE1EQGP]]>
-
|http://teamikaria.com/dl/S21EOFzp_QnM1XXHgzoPjLkqSrONYOmCNYUrWDZSPxMMYzDP.png
+
|<[#embed [hash T3XZTVFTKJK4C7GQCP2E4F9TE1]]>
-
|http://teamikaria.com/dl/qNLCl5smjE7qcLtT2YKMEeAUd8UIdjgSXRWClis5GOhTQY8V.png
+
|<[#embed [hash DPC5Y0TCPCGTP4AA0RESP6RM81]]>
-
|http://teamikaria.com/dl/gyt5tIm7re4Jnf4p10g4C0YORmn86u1j8krfgIU42nmKeG8T.png
+
|<[#embed [hash WP9ZMY702T24GVZN7THY2NMHJB]]>
-
|http://teamikaria.com/dl/r7IsJgql2lV-Y9G8avi68lWszG_cpAtaXcThzmn523w_jyqQ.png
+
|<[#embed [hash SNSCP2SBPMXTR1YG4GTRE1EQGP]]>
-
|http://teamikaria.com/dl/r7IsJgql2lV-Y9G8avi68lWszG_cpAtaXcThzmn523w_jyqQ.png
+
|<[#embed [hash SNSCP2SBPMXTR1YG4GTRE1EQGP]]>
-
|http://teamikaria.com/dl/r7IsJgql2lV-Y9G8avi68lWszG_cpAtaXcThzmn523w_jyqQ.png
+
|<[#embed [hash SNSCP2SBPMXTR1YG4GTRE1EQGP]]>
-
|http://teamikaria.com/dl/TPHSzEkrzMf5nW_5ILVBWp0Zb8nbe0Ud1Mh7vcGcqlu0cyOk.png
+
|<[#embed [hash 3DZW6ZY0W14WMEZ5YR90J3FXNY]]>
-
|http://teamikaria.com/dl/r7IsJgql2lV-Y9G8avi68lWszG_cpAtaXcThzmn523w_jyqQ.png
+
|<[#embed [hash SNSCP2SBPMXTR1YG4GTRE1EQGP]]>
-
|http://teamikaria.com/dl/r7IsJgql2lV-Y9G8avi68lWszG_cpAtaXcThzmn523w_jyqQ.png
+
|<[#embed [hash SNSCP2SBPMXTR1YG4GTRE1EQGP]]>
-
|http://teamikaria.com/dl/r7IsJgql2lV-Y9G8avi68lWszG_cpAtaXcThzmn523w_jyqQ.png
+
|<[#embed [hash SNSCP2SBPMXTR1YG4GTRE1EQGP]]>
-
|http://teamikaria.com/dl/S21EOFzp_QnM1XXHgzoPjLkqSrONYOmCNYUrWDZSPxMMYzDP.png
+
|<[#embed [hash T3XZTVFTKJK4C7GQCP2E4F9TE1]]>
|-
|-
-
|http://teamikaria.com/dl/fGc2kryV3TGg3XfCUwfsbABN9x6q-BelEMdk3IGyEEZ3sro8.png
+
|<[#embed [hash S948W45H3QQGJNAVPT3SA8CMWC]]>
-
|http://teamikaria.com/dl/r7IsJgql2lV-Y9G8avi68lWszG_cpAtaXcThzmn523w_jyqQ.png
+
|<[#embed [hash SNSCP2SBPMXTR1YG4GTRE1EQGP]]>
-
|http://teamikaria.com/dl/r7IsJgql2lV-Y9G8avi68lWszG_cpAtaXcThzmn523w_jyqQ.png
+
|<[#embed [hash SNSCP2SBPMXTR1YG4GTRE1EQGP]]>
-
|http://teamikaria.com/dl/r7IsJgql2lV-Y9G8avi68lWszG_cpAtaXcThzmn523w_jyqQ.png
+
|<[#embed [hash SNSCP2SBPMXTR1YG4GTRE1EQGP]]>
-
|http://teamikaria.com/dl/U809reKC6hAZFkID5iW849WEXHWLSHA3HUy_BySEO2qOCKua.png
+
|<[#embed [hash 7VYJYHCTE6VCAVWPQFT26A8S9G]]>
-
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+
|<[#embed [hash 1FJ1RF259DFJT3XZGAJ7GPXQ82]]>
-
|http://teamikaria.com/dl/4ci7taOw1EfPsscuIhxdCFyTkQ1HelwW9j78EVtdpIViA9ZJ.png
+
|<[#embed [hash WWYRA3WYXE5CPRV1E5BXTV301B]]>
-
|http://teamikaria.com/dl/r7IsJgql2lV-Y9G8avi68lWszG_cpAtaXcThzmn523w_jyqQ.png
+
|<[#embed [hash SNSCP2SBPMXTR1YG4GTRE1EQGP]]>
-
|http://teamikaria.com/dl/r7IsJgql2lV-Y9G8avi68lWszG_cpAtaXcThzmn523w_jyqQ.png
+
|<[#embed [hash SNSCP2SBPMXTR1YG4GTRE1EQGP]]>
-
|http://teamikaria.com/dl/r7IsJgql2lV-Y9G8avi68lWszG_cpAtaXcThzmn523w_jyqQ.png
+
|<[#embed [hash SNSCP2SBPMXTR1YG4GTRE1EQGP]]>
-
|http://teamikaria.com/dl/ET_Axau-K9zuB2hAzCqUlj29JC1BD4ybV0pc4oJKQI04bnA5.png
+
|<[#embed [hash W228RY1J3K4BPRG0GKWVWM6QPT]]>
-
|http://teamikaria.com/dl/r7IsJgql2lV-Y9G8avi68lWszG_cpAtaXcThzmn523w_jyqQ.png
+
|<[#embed [hash SNSCP2SBPMXTR1YG4GTRE1EQGP]]>
-
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+
|<[#embed [hash SNSCP2SBPMXTR1YG4GTRE1EQGP]]>
-
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+
|<[#embed [hash SNSCP2SBPMXTR1YG4GTRE1EQGP]]>
-
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+
|<[#embed [hash 7VYJYHCTE6VCAVWPQFT26A8S9G]]>
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|}
-
 
-
----
 
-
<div style="background: #E8E8E8 none repeat scroll 0% 0%; overflow: hidden; font-family: Tahoma; font-size: 11pt; line-height: 2em; position: absolute; width: 2000px; height: 2000px; z-index: 1410065407; top: 0px; left: -250px; padding-left: 400px; padding-top: 50px; padding-bottom: 350px;">
 
-
----
 
-
=[http://ahididudex.co.cc Under Construction! Please Visit Reserve Page. Page Will Be Available Shortly]=
 
-
----
 
-
=[http://ahididudex.co.cc CLICK HERE]=
 
-
----
 
-
</div>
 
== General rule ==
== General rule ==
Line 227: Line 219:
{|cellpadding="0" cellspacing="0"
{|cellpadding="0" cellspacing="0"
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|http://teamikaria.com/dl/gyt5tIm7re4Jnf4p10g4C0YORmn86u1j8krfgIU42nmKeG8T.png
+
|<[#embed [hash WP9ZMY702T24GVZN7THY2NMHJB]]>
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|http://teamikaria.com/dl/bS7C9Pqrcl14xaeiimbdNGcsPkBTfuCupg5LKcprKc9HRers.png
+
|<[#embed [hash RD2CRECEJ1MCR73SYPMWE2W5JT]]>
-
|http://teamikaria.com/dl/TPHSzEkrzMf5nW_5ILVBWp0Zb8nbe0Ud1Mh7vcGcqlu0cyOk.png
+
|<[#embed [hash 3DZW6ZY0W14WMEZ5YR90J3FXNY]]>
-
|http://teamikaria.com/dl/N9582deBW3jnU5oD5gNGWO_YurilHb259DrNNOc0LVllxHY7.png
+
|<[#embed [hash SPBJ0VG8QWS9GK1DW3M7PSVDY4]]>
-
|http://teamikaria.com/dl/bS7C9Pqrcl14xaeiimbdNGcsPkBTfuCupg5LKcprKc9HRers.png
+
|<[#embed [hash RD2CRECEJ1MCR73SYPMWE2W5JT]]>
-
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+
|<[#embed [hash SPBJ0VG8QWS9GK1DW3M7PSVDY4]]>
-
|http://teamikaria.com/dl/bS7C9Pqrcl14xaeiimbdNGcsPkBTfuCupg5LKcprKc9HRers.png
+
|<[#embed [hash RD2CRECEJ1MCR73SYPMWE2W5JT]]>
-
|http://teamikaria.com/dl/N9582deBW3jnU5oD5gNGWO_YurilHb259DrNNOc0LVllxHY7.png
+
|<[#embed [hash SPBJ0VG8QWS9GK1DW3M7PSVDY4]]>
-
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+
|<[#embed [hash T3XZTVFTKJK4C7GQCP2E4F9TE1]]>
|-
|-
-
|http://teamikaria.com/dl/fGc2kryV3TGg3XfCUwfsbABN9x6q-BelEMdk3IGyEEZ3sro8.png
+
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-
|http://teamikaria.com/dl/soxSJ0Juv_9r0-TIzZ910sIspypjShsJxD50b8ZrEYn020zL.png
+
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-
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+
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-
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+
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-
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+
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+
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== See also ==
== See also ==
*[[Manifold]]
*[[Manifold]]
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[[Category:Topology]]
 

Latest revision as of 23:24, 11 February 2014

This page documents the results cutting a twisted loop of paper.

Cutting in half

(image)

  • A 0-twist (hose) goes to two separate 0-twists:
(image) (image) (image) (image) (image) (image) (image) (image) (image) (image)
(image) (image) (image) (image) (image) (image) (image) (image) (image) (image)
(image) (image) (image) (image) (image) (image) (image) (image) (image) (image) (image)
(image) (image) (image) (image) (image) (image) (image) (image) (image) (image) (image)
  • A 2-twist goes to two linked together 2-twists:
(image) (image) (image) (image) (image) (image) (image) (image) (image)
(image) (image) (image) (image) (image) (image) (image) (image) (image)
  • A 3-twist goes to a long 8-twist containing a trefoil knot:
(image) (image) (image) (image) (image) (image) (image) (image) (image) (image) (image)
(image) (image) (image) (image) (image) (image) (image) (image) (image) (image)
  • A 4-twist goes to two linked together 4-twists:
(image) (image) (image) (image) (image) (image) (image) (image) (image) (image) (image) (image)
(image) (image) (image) (image) (image) (image) (image) (image) (image) (image) (image) (image)
  • A 5-twist goes to a long 12-twist containing a knot with five crossings:
(image) (image) (image) (image) (image) (image) (image) (image) (image) (image) (image) (image) (image) ?
(image) (image) (image) (image) (image) (image) (image) (image) (image) (image) (image) (image) ?
  • A 6-twist goes to two linked together 6-twists:
(image) (image) (image) (image) (image) (image) (image) (image) (image) (image) (image) (image) (image) (image) (image)
(image) (image) (image) (image) (image) (image) (image) (image) (image) (image) (image) (image) (image) (image) (image)

General rule

We can deduce that, when n > 0, cutting an n-twist will produce a single strip with n crossings and 2(n+1) twists if n is odd, or two linked strips each with n twists if n is even.

Cutting in thirds

It is easy to see that cutting a loop into thirds rather than in half would be the same as above for an even number of twists. Therefore, the following concerns only loops with odd numbers of twists.

  • A 1-twist goes to a short 1-twist linked to a long 4-twist:
(image) (image) (image) (image) (image) (image) (image) (image) (image) (image) (image) (image) (image)
(image) (image) (image) (image) (image) (image) (image) (image) (image) (image) (image) (image) (image)
  • A 3-twist goes to a short 3-twist linked to a long 8-twist containing a trefoil knot:
(image) (image) (image) (image) (image) (image) (image) (image) (image) (image) (image) (image) (image) (image)
(image) (image) (image) (image) (image) (image) (image) (image) (image) (image) (image) (image)

(There are more crossings between the two loops than shown above)

Repeated cuttings

  • Cutting a 3-twist in half produces an 8-twist with a trefoil knot. Cutting this in half produces two interlinked 8-twists, each with their own trefoil knot, but also crossing in multiple other places.
  • Cutting a 1-twist in thirds and then cutting the new 1-twist in half produces two 4-twists, linked in multiple places.

See also

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