L4a1
From Knot Atlas
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![]() (Knotscape image) | See the full Thistlethwaite Link Table (up to 11 crossings).
Visit L4a1's page at Knotilus. Visit L4a1's page at the original Knot Atlas. |
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L4a1 is |
A Kolam with two cycles[1] | Hearst Castle tile [2] | Mosaic seen at Kibbutz Lahav [3] | ||||
[edit] Link Presentations
[edit Notes on L4a1's Link Presentations] Why such an ugly Braid Representative?
| Planar diagram presentation | X6172 X8354 X2536 X4718 |
| Gauss code | {1, -3, 2, -4}, {3, -1, 4, -2} |
| A Braid Representative | | ||||
| A Morse Link Presentation |
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[edit] Polynomial invariants
| Multivariable Alexander Polynomial (in u, v, w, ...) | (db)
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| Jones polynomial | (db)
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| Signature | -1 (db) |
| HOMFLY-PT polynomial | a5z−1−a3z−a3z−1−az (db) |
| Kauffman polynomial | a5z3−3a5z + a5z−1 + a4z2−a4 + a3z3−2a3z + a3z−1 + a2z2 + az (db) |
[edit] Khovanov Homology
| The coefficients of the monomials trqj are shown, along with their alternating sums χ (fixed j, alternation over r). The squares with yellow highlighting are those on the "critical diagonals", where j−2r = s + 1 or j−2r = s−1, where s = -1 is the signature of L4a1. Nonzero entries off the critical diagonals (if any exist) are highlighted in red. | <table border=1> <tr align=center> <td width=22.2222%><table cellpadding=0 cellspacing=0> <tr><td>\</td><td> </td><td>r</td></tr> <tr><td> </td><td> \ </td><td> </td></tr> <tr><td>j</td><td> </td><td>\</td></tr> </table></td> <td width=11.1111%>-4</td><td width=11.1111%>-3</td><td width=11.1111%>-2</td><td width=11.1111%>-1</td><td width=11.1111%>0</td><td width=22.2222%>χ</td></tr> <tr align=center><td>0</td><td> </td><td> </td><td> </td><td> </td><td bgcolor=yellow>1</td><td>1</td></tr> <tr align=center><td>-2</td><td> </td><td> </td><td> </td><td bgcolor=yellow>1</td><td bgcolor=yellow>1</td><td>0</td></tr> <tr align=center><td>-4</td><td> </td><td> </td><td bgcolor=yellow> </td><td bgcolor=yellow> </td><td> </td><td>0</td></tr> <tr align=center><td>-6</td><td> </td><td bgcolor=yellow> </td><td bgcolor=yellow>1</td><td> </td><td> </td><td>1</td></tr> <tr align=center><td>-8</td><td bgcolor=yellow>1</td><td bgcolor=yellow> </td><td> </td><td> </td><td> </td><td>1</td></tr> <tr align=center><td>-10</td><td bgcolor=yellow>1</td><td> </td><td> </td><td> </td><td> </td><td>1</td></tr> </table> |
| Integral Khovanov Homology
(db, data source) |
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[edit] Computer Talk
Much of the above data can be recomputed by Mathematica using the packageKnotTheory`. See A Sample KnotTheory` Session.[edit] Modifying This Page
| Read me first: Modifying Knot Pages
See/edit the Link Page master template (intermediate). See/edit the Link_Splice_Base (expert). Back to the top. |
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in the Rolfsen table of links. It frequently occurs in late Roman mosaics and some medieval decorations. In this context, it is called the "Solomon's knot" (sigillum Salomonis) or "guilloche knot". It is also the "Kramo-bone" symbol (meaning "one being bad makes all appear to be bad") of the Adinkra symbol system. Link 
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