Identifying Knots within a List - Revision history

Drorbn at 19:04, 20 October 2013

2013-10-20T19:04:46Z

← Older revision		Revision as of 15:04, 20 October 2013
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	{{Startup Note}}		{{Startup Note}}

	<!--$$Import["http://katlas.org/~~wiki~~/IdentifyWithin.m&action=raw"];$$-->		<!--$$Import["http://katlas.org/w/index.php?title=IdentifyWithin.m&action=raw"];$$-->
	<!--Robot Land, no human edits to "END"-->		<!--Robot Land, no human edits to "END"-->
	{{In\|		{{In\|
	n = 2 \|		n = 2 \|
	in = <nowiki>Import["http://katlas.org/~~wiki~~/IdentifyWithin.m&action=raw"];</nowiki>}}		in = <nowiki>Import["http://katlas.org/w/index.php?title=IdentifyWithin.m&action=raw"];</nowiki>}}
	<!--END-->		<!--END-->

	<!--$$Import["http://katlas.org/~~wiki~~/SubLink.m&action=raw"];$$-->		<!--$$Import["http://katlas.org/w/index.php?title=SubLink.m&action=raw"];$$-->
	<!--Robot Land, no human edits to "END"-->		<!--Robot Land, no human edits to "END"-->
	{{In\|		{{In\|
	n = 3 \|		n = 3 \|
	in = <nowiki>Import["http://katlas.org/~~wiki~~/SubLink.m&action=raw"];</nowiki>}}		in = <nowiki>Import["http://katlas.org/w/index.php?title=SubLink.m&action=raw"];</nowiki>}}
	<!--END-->		<!--END-->

IvaH at 01:18, 19 November 2007

2007-11-19T01:18:11Z

← Older revision		Revision as of 21:18, 18 November 2007
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	<!--END-->		<!--END-->

	<!--$$Import["http://katlas.org/wiki/SubLink.m&action=raw"];$$-->~~<!--END-->~~		<!--$$Import["http://katlas.org/wiki/SubLink.m&action=raw"];$$-->
			<!--Robot Land, no human edits to "END"-->
			{{In\|
			n = 3 \|
			in = <nowiki>Import["http://katlas.org/wiki/SubLink.m&action=raw"];</nowiki>}}
			<!--END-->

	<!--$$IdentifyWithin[SubLink[Link["L11n150"], 2], AllKnots[]]$$-->		<!--$$IdentifyWithin[SubLink[Link["L11n150"], 2], AllKnots[]]$$-->

IvaH at 01:17, 19 November 2007

2007-11-19T01:17:37Z

← Older revision		Revision as of 21:17, 18 November 2007
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	<!--END-->		<!--END-->

			<!--$$Import["http://katlas.org/wiki/SubLink.m&action=raw"];$$--><!--END-->
	<!--$$SubLink[pd_PD, js_List] := Module[
	{k, t0, t, t1, t2, S, P},
	t0 = Flatten[List @@@ Skeleton[pd][[js]]];
	t = pd /. x_X :> Select[x, MemberQ[t0, #] &];
	t = DeleteCases[t, X[]];
	k = 1;
	While[
	k <= Length[t],
	If[ Length[t[[k]]] < 4,
	t = Delete[t, k] /. (Rule @@ t[[k]]), ++k];
	];
	t1 = List @@ Union @@ t;
	t2 = Thread[(t1) -> Range[Length[t1]]];
	S = t /. t2;
	P = If[S != PD[] && Length[S] >= 3, S, PD[Knot[0, 1]], S]
	];
	SubLink[pd_PD, j_] := SubLink[pd, {j}];
	SubLink[L_, js_] := SubLink[PD[L], js];$$-->
	<!--Robot Land, no human edits to "END"-->
	{{In\|
	n = 3 \|
	in = <nowiki>SubLink[pd_PD, js_List] := Module[
	{k, t0, t, t1, t2, S, P},
	t0 = Flatten[List @@@ Skeleton[pd][[js]]];
	t = pd /. x_X :> Select[x, MemberQ[t0, #] &];
	t = DeleteCases[t, X[]];
	k = 1;
	While[
	k <= Length[t],
	If[ Length[t[[k]]] < 4,
	t = Delete[t, k] /. (Rule @@ t[[k]]), ++k];
	];
	t1 = List @@ Union @@ t;
	t2 = Thread[(t1) -> Range[Length[t1]]];
	S = t /. t2;
	P = If[S != PD[] && Length[S] >= 3, S, PD[Knot[0, 1]], S]
	];
	SubLink[pd_PD, j_] := SubLink[pd, {j}];
	SubLink[L_, js_] := SubLink[PD[L], js];</nowiki>}}
	<!--END-->

	<!--$$IdentifyWithin[SubLink[Link["L11n150"], 2], AllKnots[]]$$-->		<!--$$IdentifyWithin[SubLink[Link["L11n150"], 2], AllKnots[]]$$-->

IvaH at 01:16, 19 November 2007

2007-11-19T01:16:22Z

← Older revision		Revision as of 21:16, 18 November 2007
Line 6:		Line 6:
	{{Startup Note}}		{{Startup Note}}

	<!--$$Import["http://katlas.org/wiki/IdentifyWithin.m&action=raw"];$$~~--><!--END~~-->		<!--$$Import["http://katlas.org/wiki/IdentifyWithin.m&action=raw"];$$-->
			<!--Robot Land, no human edits to "END"-->
			{{In\|
			n = 2 \|
			in = <nowiki>Import["http://katlas.org/wiki/IdentifyWithin.m&action=raw"];</nowiki>}}
			<!--END-->

	<!--$$SubLink[pd_PD, js_List] := Module[		<!--$$SubLink[pd_PD, js_List] := Module[
Line 25:		Line 30:
	];		];
	SubLink[pd_PD, j_] := SubLink[pd, {j}];		SubLink[pd_PD, j_] := SubLink[pd, {j}];
	SubLink[L_, js_] := SubLink[PD[L], js];$$-->~~<!--END-->~~		SubLink[L_, js_] := SubLink[PD[L], js];$$-->
			<!--Robot Land, no human edits to "END"-->
			{{In\|
			n = 3 \|
			in = <nowiki>SubLink[pd_PD, js_List] := Module[
			{k, t0, t, t1, t2, S, P},
			t0 = Flatten[List @@@ Skeleton[pd][[js]]];
			t = pd /. x_X :> Select[x, MemberQ[t0, #] &];
			t = DeleteCases[t, X[]];
			k = 1;
			While[
			k <= Length[t],
			If[ Length[t[[k]]] < 4,
			t = Delete[t, k] /. (Rule @@ t[[k]]), ++k];
			];
			t1 = List @@ Union @@ t;
			t2 = Thread[(t1) -> Range[Length[t1]]];
			S = t /. t2;
			P = If[S != PD[] && Length[S] >= 3, S, PD[Knot[0, 1]], S]
			];
			SubLink[pd_PD, j_] := SubLink[pd, {j}];
			SubLink[L_, js_] := SubLink[PD[L], js];</nowiki>}}
			<!--END-->

	<!--$$IdentifyWithin[SubLink[Link["L11n150"], 2], AllKnots[]]$$-->		<!--$$IdentifyWithin[SubLink[Link["L11n150"], 2], AllKnots[]]$$-->
	<!--Robot Land, no human edits to "END"-->		<!--Robot Land, no human edits to "END"-->
	{{InOut\|		{{InOut\|
	n = 2 \|		n = 4 \|
	in = <nowiki>IdentifyWithin[SubLink[Link["L11n150"], 2], AllKnots[]]</nowiki> \|		in = <nowiki>IdentifyWithin[SubLink[Link["L11n150"], 2], AllKnots[]]</nowiki> \|
	out= <nowiki>{Knot[5, 2]}</nowiki>}}		out= <nowiki>{Knot[5, 2]}</nowiki>}}

IvaH at 01:16, 19 November 2007

2007-11-19T01:16:03Z

← Older revision		Revision as of 21:16, 18 November 2007
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	];		];
	SubLink[pd_PD, j_] := SubLink[pd, {j}];		SubLink[pd_PD, j_] := SubLink[pd, {j}];
	SubLink[L_, js_] := SubLink[PD[L], js];$$-->		SubLink[L_, js_] := SubLink[PD[L], js];$$--><!--END-->


	<!--$$IdentifyWithin[SubLink[Link["L11n150"], 2], AllKnots[]]$$-->		<!--$$IdentifyWithin[SubLink[Link["L11n150"], 2], AllKnots[]]$$-->
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	out= <nowiki>{Knot[5, 2]}</nowiki>}}		out= <nowiki>{Knot[5, 2]}</nowiki>}}
	<!--END-->		<!--END-->


	{{Knot Image Pair\|L11n150\|gif\|5_2\|gif}}		{{Knot Image Pair\|L11n150\|gif\|5_2\|gif}}


	Unfortunately, the program does not provide absolute identification when all the used invariants cannot distinguish between two or more different knots. In that case, a list of possible candidates for <math>L</math> is returned.		Unfortunately, the program does not provide absolute identification when all the used invariants cannot distinguish between two or more different knots. In that case, a list of possible candidates for <math>L</math> is returned.

IvaH at 00:38, 19 November 2007

2007-11-19T00:38:22Z

Show changes

IvaH at 15:41, 8 November 2007

2007-11-08T15:41:42Z

← Older revision		Revision as of 11:41, 8 November 2007
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	];</nowiki>}}		];</nowiki>}}
	<!--END-->		<!--END-->

			<!--$$SubLink[pd_PD, js_List] := Module[
			{k, t0, t, t1, t2, S, P},
			t0 = Flatten[List @@@ Skeleton[pd][[js]]];
			t = pd /. x_X :> Select[x, MemberQ[t0, #] &];
			t = DeleteCases[t, X[]];
			k = 1;
			While[
			k <= Length[t],
			If[ Length[t[[k]]] < 4,
			t = Delete[t, k] /. (Rule @@ t[[k]]), ++k];
			];
			t1 = List @@ Union @@ t;
			t2 = Thread[(t1) -> Range[Length[t1]]];
			S = t /. t2;
			P = If[S != PD[] && Length[S] >= 3, S, PD[Knot[0, 1]], S]
			];
			SubLink[pd_PD, j_] := SubLink[pd, {j}];
			SubLink[L_, js_] := SubLink[PD[L], js];$$-->

IvaH at 15:23, 8 November 2007

2007-11-08T15:23:20Z

@@ Line 1: / Line 1: @@
 {{Manual TOC Sidebar}}
-IdentifyWithin[L,H] returns those elements from the list of knots H, whose invariant matches that of the knot L. It can also recognize mirrors and connected sums of the knots in the list. Its options include turning off (on) the search for connected sums with ConnectedSum->False (True) and choosing the invariants to be used in identification by selecting, for example, Invariants->{Jones[#][q]&, HOMFLYPT[#][a,z]&}.
+<code>IdentifyWithin[L,H]</code> returns those elements from the list of knots <math>H</math>, whose invariant matches that of the knot <math>L</math>. It can also recognize mirrors and connected sums of the knots in the list. Its options include turning off (on) the search for connected sums with <code>ConnectedSum->False (True)</code> and choosing the invariants to be used in identification by selecting, for example, <code>Invariants->{Jones[#][q]&, HOMFLYPT[#][a,z]&}</code>.
-IdentifyWithin can be used together with [[Prime Links with a Non-Prime Component|SubLink]] to determine the components of a link. For the second component of link [[L11n150]], for instance, we get:
+<code>IdentifyWithin</code> can be used together with [[Prime Links with a Non-Prime Component|<code>SubLink</code>]] to determine the components of a link. For the second component of link [[L11n150]], for instance, we get:
 {{Startup Note}}
 <!--$$Options[IdentifyWithin] = {
@@ Line 170: / Line 156: @@
    ];</nowiki>}}
 <!--END-->

IvaH at 13:25, 25 October 2007

2007-10-25T13:25:57Z

← Older revision		Revision as of 09:25, 25 October 2007
Line 1:		Line 1:
	{{Manual TOC Sidebar}}		{{Manual TOC Sidebar}}

	==IdentifyWithin==

	IdentifyWithin[L,H] returns those elements from the list of knots H, whose invariant matches that of the knot L. It can also recognize mirrors and connected sums of the knots in the list. Its options include turning off (on) the search for connected sums with ConnectedSum->False (True) and choosing the invariants to be used in identification by selecting, for example, Invariants->{Jones[#][q]&, HOMFLYPT[#][a,z]&}.		IdentifyWithin[L,H] returns those elements from the list of knots H, whose invariant matches that of the knot L. It can also recognize mirrors and connected sums of the knots in the list. Its options include turning off (on) the search for connected sums with ConnectedSum->False (True) and choosing the invariants to be used in identification by selecting, for example, Invariants->{Jones[#][q]&, HOMFLYPT[#][a,z]&}.

IvaH: New page: {{Manual TOC Sidebar}} ==IdentifyWithin== IdentifyWithin[L,H] returns those elements from the list of knots H, whose invariant matches that of the knot L. It can also recognize mirrors a...

2007-10-25T13:17:24Z

New page: {{Manual TOC Sidebar}} ==IdentifyWithin== IdentifyWithin[L,H] returns those elements from the list of knots H, whose invariant matches that of the knot L. It can also recognize mirrors a...

New page

{{Manual TOC Sidebar}}

==IdentifyWithin==

IdentifyWithin[L,H] returns those elements from the list of knots H, whose invariant matches that of the knot L. It can also recognize mirrors and connected sums of the knots in the list. Its options include turning off (on) the search for connected sums with ConnectedSum->False (True) and choosing the invariants to be used in identification by selecting, for example, Invariants->{Jones[#][q]&, HOMFLYPT[#][a,z]&}.
IdentifyWithin can be used together with [[Prime Links with a Non-Prime Component|SubLink]] to determine the components of a link. For the second component of link [[L11n150]], for instance, we get:

{{Startup Note}}



{{InOut|
n = 2 |
in = <nowiki>IdentifyWithin[SubLink[Link["L11n150"], 2], AllKnots[]]</nowiki> |
out= <nowiki>{Knot[5, 2]}</nowiki>}}


{{Knot Image Pair|L11n150|gif|5_2|gif}}

Unfortunately, the program does not provide absolute identification when all the used invariants cannot distinguish between two or more different knots. In that case, a list of possible candidates for L is returned.



{{In|
n = 3 |
in = <nowiki>Options[IdentifyWithin] = {
Invariants -> {Jones[#][q] &, HOMFLYPT[#][a, z] &,
Kauffman[#][a, z] &},
ConnectedSum -> "True"};
IdentifyWithin[L_, H_List, opts___Rule] :=

Module[
{div, j = 1, l, i = 1, u, mu, t, mt, out = {}, out1 = {}, nk, mnk,
mnk1, p, mp, m, p1,
invariants = (Invariants /. {opts} /. Options[IdentifyWithin]),
connectedsum = (ConnectedSum /. {opts} /.
Options[IdentifyWithin])},

NormalizeP[poly_] := Module[{t1, i1},
(For[i1 = 1 ; t1 := FactorList[poly],
i1 <= Length[Variables[poly]], i1++,
t1 =
DeleteCases[t1, {Variables[poly][[i1]], _Integer} | {1, 1}]];
Times @@ Power @@@ t1 )];

l := Length[invariants];
u[0] = mu[0] = H;
While[i <= l && ! Length[out] === 1,
t[i] = invariants[[i]][L];
mt[i] = invariants[[i]][Mirror[L]];
u[i] = Select[u[i - 1], t[i] == invariants[[i]][#] &];
mu[i] = Select[mu[i - 1], mt[i] == invariants[[i]][#] &];
out = Flatten[{u[i], Mirror /@ mu[i]}];
i++];

Which[
Length[out] >= 2, DeleteCases[out, Mirror[Knot[0, 1]]],
Length[out] == 1,
out = If[u[i - 1] != {}, u[i - 1], Mirror /@ mu[i - 1]],
connectedsum === "True", i = 1; nk[0] = mnk[0] = H;

While[Length[out1] != 1 && i <= l,
p[i] = NormalizeP[t[i]];
mp[i] = NormalizeP[mt[i]];
nk[i] =
Select[nk[i - 1], (p1 = NormalizeP[invariants[[i]][#]]; z = 3;
PolynomialRemainder[p[i], p1, Variables[p[i]][[1]]] ===
0 ) &];
mnk[i] =
Select[mnk[i - 1], (p1 = NormalizeP[invariants[[i]][#]]; z = 3;
PolynomialRemainder[mp[i], p1, Variables[p[i]][[1]]] ===
0 ) &];

Clear[z];

mnk1[i] = Mirror /@ mnk[i];
div = Flatten[{nk[i], mnk1[i]}];
div = DeleteCases[div, Knot[0, 1] | Mirror[Knot[0, 1]]];

If[div == {}, out1 = {},
For[m = 1;
W[0] =
CS[0] = Select[
Flatten /@ Flatten[Outer[List, div, div, 1], 1], OrderedQ],
Length[W[m - 1][[1]]] < 4, m++,
W[m] = Select[
Flatten /@ Flatten[Outer[List, div, W[m - 1], 1], 1],
OrderedQ];
CS[m] = Flatten[{CS[m - 1], W[m]}, 1];
];
out1 =
Select[CS[m - 1],
Expand[Times @@ invariants[[i]] /@ #] == t[i] &];
];
i++];
If[out1 == {}, {}, ConnectedSum @@@ out1], True, {}
]
];</nowiki>}}