From Knot Atlas
[edit] Link Presentations
[edit Notes on L10n75's Link Presentations]
| Planar diagram presentation
| X6172 X5,14,6,15 X3849 X2,16,3,15 X16,7,17,8 X9,18,10,19 X17,1,18,4 X19,12,20,5 X11,20,12,13 X13,10,14,11
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| Gauss code
| {1, -4, -3, 7}, {-2, -1, 5, 3, -6, 10, -9, 8}, {-10, 2, 4, -5, -7, 6, -8, 9}
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[edit] Polynomial invariants
| Multivariable Alexander Polynomial (in u, v, w, ...)
| (db)
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| Jones polynomial
| 2q−1−3q−2 + 5q−3−4q−4 + 5q−5−4q−6 + 3q−7−q−8 + q−9 (db)
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| Signature
| -2 (db)
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| HOMFLY-PT polynomial
| z2a8 + a8z−2 + 2a8−z4a6−3z2a6−2a6z−2−5a6−z4a4−z2a4 + a4z−2 + a4 + 2z2a2 + 2a2 (db)
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| Kauffman polynomial
| z6a10−5z4a10 + 7z2a10−2a10 + z7a9−3z5a9 + z3a9 + za9 + z8a8−3z6a8 + 3z4a8−4z2a8−a8z−2 + 3a8 + 3z7a7−9z5a7 + 9z3a7−8za7 + 2a7z−1 + z8a6−2z6a6 + 6z4a6−14z2a6−2a6z−2 + 9a6 + 2z7a5−5z5a5 + 9z3a5−8za5 + 2a5z−1 + 2z6a4−2z4a4−a4z−2 + 3a4 + z5a3 + z3a3 + za3 + 3z2a2−2a2 (db)
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| 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 = -2 is the signature of L10n75. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.
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| <table border=1>
<tr align=center>
<td width=15.3846%><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=7.69231%>-8</td><td width=7.69231%>-7</td><td width=7.69231%>-6</td><td width=7.69231%>-5</td><td width=7.69231%>-4</td><td width=7.69231%>-3</td><td width=7.69231%>-2</td><td width=7.69231%>-1</td><td width=7.69231%>0</td><td width=15.3846%>χ</td></tr>
<tr align=center><td>-1</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td bgcolor=yellow>2</td><td>2</td></tr>
<tr align=center><td>-3</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td bgcolor=yellow>3</td><td bgcolor=yellow>2</td><td>-1</td></tr>
<tr align=center><td>-5</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td bgcolor=yellow>2</td><td bgcolor=yellow> </td><td> </td><td>2</td></tr>
<tr align=center><td>-7</td><td> </td><td> </td><td> </td><td> </td><td> </td><td bgcolor=yellow>2</td><td bgcolor=yellow>3</td><td> </td><td> </td><td>1</td></tr>
<tr align=center><td>-9</td><td> </td><td> </td><td> </td><td> </td><td bgcolor=yellow>3</td><td bgcolor=yellow>2</td><td> </td><td> </td><td> </td><td>1</td></tr>
<tr align=center><td>-11</td><td> </td><td> </td><td> </td><td bgcolor=yellow>2</td><td bgcolor=yellow>3</td><td> </td><td> </td><td> </td><td> </td><td>1</td></tr>
<tr align=center><td>-13</td><td> </td><td> </td><td bgcolor=yellow>1</td><td bgcolor=yellow>2</td><td> </td><td> </td><td> </td><td> </td><td> </td><td>-1</td></tr>
<tr align=center><td>-15</td><td> </td><td bgcolor=yellow> </td><td bgcolor=yellow>2</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td>2</td></tr>
<tr align=center><td>-17</td><td bgcolor=yellow>1</td><td bgcolor=yellow>1</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td>0</td></tr>
<tr align=center><td>-19</td><td bgcolor=yellow>1</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td>1</td></tr>
</table>
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