Identifying Knots within a List: Difference between revisions
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{{Manual TOC Sidebar}} |
{{Manual TOC Sidebar}} |
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<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>. |
<code>IdentifyWithin[L,H]</code>, whose code is available [[IdentifyWithin.m|here]], 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>. |
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<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: |
<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: |
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{{Startup Note}} |
{{Startup Note}} |
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<!--$$ |
<!--$$Import["http://katlas.org/wiki/IdentifyWithin.m&action=raw"];$$--><!--END--> |
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Invariants -> {Jones[#][q] &, HOMFLYPT[#][a, z] &, |
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Kauffman[#][a, z] &}, |
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ConnectedSum -> "True"}; |
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IdentifyWithin[L_, H_List, opts___Rule] := |
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Module[ |
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{div, j = 1, l, i = 1, u, mu, t, mt, out = {}, out1 = {}, nk, mnk, |
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mnk1, p, mp, m, p1, |
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invariants = (Invariants /. {opts} /. Options[IdentifyWithin]), |
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connectedsum = (ConnectedSum /. {opts} /. |
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Options[IdentifyWithin])}, |
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NormalizeP[poly_] := Module[{t1, i1}, |
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(For[i1 = 1 ; t1 := FactorList[poly], |
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i1 <= Length[Variables[poly]], i1++, |
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t1 = |
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DeleteCases[t1, {Variables[poly][[i1]], _Integer} | {1, 1}]]; |
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Times @@ Power @@@ t1 )]; |
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l := Length[invariants]; |
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u[0] = mu[0] = H; |
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While[i <= l && ! Length[out] === 1, |
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t[i] = invariants[[i]][L]; |
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mt[i] = invariants[[i]][Mirror[L]]; |
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u[i] = Select[u[i - 1], t[i] == invariants[[i]][#] &]; |
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mu[i] = Select[mu[i - 1], mt[i] == invariants[[i]][#] &]; |
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out = Flatten[{u[i], Mirror /@ mu[i]}]; |
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i++]; |
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Which[ |
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Length[out] >= 2, DeleteCases[out, Mirror[Knot[0, 1]]], |
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Length[out] == 1, |
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out = If[u[i - 1] != {}, u[i - 1], Mirror /@ mu[i - 1]], |
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connectedsum === "True", i = 1; nk[0] = mnk[0] = H; |
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While[Length[out1] != 1 && i <= l, |
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p[i] = NormalizeP[t[i]]; |
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mp[i] = NormalizeP[mt[i]]; |
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nk[i] = |
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Select[nk[i - 1], (p1 = NormalizeP[invariants[[i]][#]]; z = 3; |
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PolynomialRemainder[p[i], p1, Variables[p[i]][[1]]] === |
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0 ) &]; |
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mnk[i] = |
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Select[mnk[i - 1], (p1 = NormalizeP[invariants[[i]][#]]; z = 3; |
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PolynomialRemainder[mp[i], p1, Variables[p[i]][[1]]] === |
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0 ) &]; |
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Clear[z]; |
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mnk1[i] = Mirror /@ mnk[i]; |
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div = Flatten[{nk[i], mnk1[i]}]; |
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div = DeleteCases[div, Knot[0, 1] | Mirror[Knot[0, 1]]]; |
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If[div == {}, out1 = {}, |
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For[m = 1; |
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W[0] = |
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CS[0] = Select[ |
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Flatten /@ Flatten[Outer[List, div, div, 1], 1], OrderedQ], |
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Length[W[m - 1][[1]]] < 4, m++, |
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W[m] = Select[ |
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Flatten /@ Flatten[Outer[List, div, W[m - 1], 1], 1], |
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OrderedQ]; |
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CS[m] = Flatten[{CS[m - 1], W[m]}, 1]; |
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]; |
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out1 = |
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Select[CS[m - 1], |
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Expand[Times @@ invariants[[i]] /@ #] == t[i] &]; |
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]; |
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i++]; |
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If[out1 == {}, {}, ConnectedSum @@@ out1], True, {} |
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] |
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];$$--> |
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<!--Robot Land, no human edits to "END"--> |
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{{In| |
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n = 3 | |
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in = <nowiki>Options[IdentifyWithin] = { |
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Invariants -> {Jones[#][q] &, HOMFLYPT[#][a, z] &, |
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Kauffman[#][a, z] &}, |
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ConnectedSum -> "True"}; |
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IdentifyWithin[L_, H_List, opts___Rule] := |
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Module[ |
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{div, j = 1, l, i = 1, u, mu, t, mt, out = {}, out1 = {}, nk, mnk, |
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mnk1, p, mp, m, p1, |
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invariants = (Invariants /. {opts} /. Options[IdentifyWithin]), |
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connectedsum = (ConnectedSum /. {opts} /. |
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Options[IdentifyWithin])}, |
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NormalizeP[poly_] := Module[{t1, i1}, |
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(For[i1 = 1 ; t1 := FactorList[poly], |
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i1 <= Length[Variables[poly]], i1++, |
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t1 = |
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DeleteCases[t1, {Variables[poly][[i1]], _Integer} | {1, 1}]]; |
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Times @@ Power @@@ t1 )]; |
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l := Length[invariants]; |
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u[0] = mu[0] = H; |
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While[i <= l && ! Length[out] === 1, |
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t[i] = invariants[[i]][L]; |
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mt[i] = invariants[[i]][Mirror[L]]; |
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u[i] = Select[u[i - 1], t[i] == invariants[[i]][#] &]; |
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mu[i] = Select[mu[i - 1], mt[i] == invariants[[i]][#] &]; |
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out = Flatten[{u[i], Mirror /@ mu[i]}]; |
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i++]; |
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Which[ |
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Length[out] >= 2, DeleteCases[out, Mirror[Knot[0, 1]]], |
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Length[out] == 1, |
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out = If[u[i - 1] != {}, u[i - 1], Mirror /@ mu[i - 1]], |
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connectedsum === "True", i = 1; nk[0] = mnk[0] = H; |
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While[Length[out1] != 1 && i <= l, |
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p[i] = NormalizeP[t[i]]; |
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mp[i] = NormalizeP[mt[i]]; |
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nk[i] = |
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Select[nk[i - 1], (p1 = NormalizeP[invariants[[i]][#]]; z = 3; |
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PolynomialRemainder[p[i], p1, Variables[p[i]][[1]]] === |
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0 ) &]; |
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mnk[i] = |
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Select[mnk[i - 1], (p1 = NormalizeP[invariants[[i]][#]]; z = 3; |
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PolynomialRemainder[mp[i], p1, Variables[p[i]][[1]]] === |
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0 ) &]; |
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Clear[z]; |
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mnk1[i] = Mirror /@ mnk[i]; |
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div = Flatten[{nk[i], mnk1[i]}]; |
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div = DeleteCases[div, Knot[0, 1] | Mirror[Knot[0, 1]]]; |
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If[div == {}, out1 = {}, |
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For[m = 1; |
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W[0] = |
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CS[0] = Select[ |
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Flatten /@ Flatten[Outer[List, div, div, 1], 1], OrderedQ], |
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Length[W[m - 1][[1]]] < 4, m++, |
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W[m] = Select[ |
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Flatten /@ Flatten[Outer[List, div, W[m - 1], 1], 1], |
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OrderedQ]; |
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CS[m] = Flatten[{CS[m - 1], W[m]}, 1]; |
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]; |
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out1 = |
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Select[CS[m - 1], |
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Expand[Times @@ invariants[[i]] /@ #] == t[i] &]; |
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]; |
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i++]; |
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If[out1 == {}, {}, ConnectedSum @@@ out1], True, {} |
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] |
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];</nowiki>}} |
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<!--END--> |
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<!--$$SubLink[pd_PD, js_List] := Module[ |
<!--$$SubLink[pd_PD, js_List] := Module[ |
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Revision as of 20:38, 18 November 2007
IdentifyWithin[L,H], whose code is available here, returns those elements from the list of knots , whose invariant matches that of the knot . 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 SubLink to determine the components of a link. For the second component of link L11n150, for instance, we get:
(For In[1] see Setup)
In[2]:=
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IdentifyWithin[SubLink[Link["L11n150"], 2], AllKnots[]]
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Out[2]=
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{Knot[5, 2]}
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 L11n150 |
 5_2 |
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 is returned.
