From Knot Atlas
[edit] Link Presentations
[edit Notes on L10n62's Link Presentations]
| Planar diagram presentation
| X12,1,13,2 X16,8,17,7 X5,1,6,10 X3746 X9,5,10,4 X18,14,19,13 X20,16,11,15 X14,20,15,19 X2,11,3,12 X8,18,9,17
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| Gauss code
| {1, -9, -4, 5, -3, 4, 2, -10, -5, 3}, {9, -1, 6, -8, 7, -2, 10, -6, 8, -7}
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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
| 5 (db)
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| HOMFLY-PT polynomial
| −za−9−a−9z−1 + z5a−7 + 5z3a−7 + 8za−7 + 5a−7z−1−z7a−5−6z5a−5−13z3a−5−15za−5−8a−5z−1 + z5a−3 + 5z3a−3 + 8za−3 + 4a−3z−1 (db)
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| Kauffman polynomial
| −z8a−4−z8a−6−z7a−3−5z7a−5−4z7a−7 + 3z6a−4−2z6a−6−5z6a−8 + 6z5a−3 + 24z5a−5 + 16z5a−7−2z5a−9 + 3z4a−4 + 23z4a−6 + 20z4a−8−13z3a−3−35z3a−5−18z3a−7 + 4z3a−9−13z2a−4−33z2a−6−23z2a−8−3z2a−10 + 12za−3 + 23za−5 + 11za−7−za−9−za−11 + 8a−4 + 14a−6 + 9a−8 + 2a−10−4a−3z−1−8a−5z−1−5a−7z−1−a−9z−1 (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 = 5 is the signature of L10n62. 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%>-2</td><td width=7.69231%>-1</td><td width=7.69231%>0</td><td width=7.69231%>1</td><td width=7.69231%>2</td><td width=7.69231%>3</td><td width=7.69231%>4</td><td width=7.69231%>5</td><td width=7.69231%>6</td><td width=15.3846%>χ</td></tr>
<tr align=center><td>18</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td bgcolor=yellow>1</td><td>1</td></tr>
<tr align=center><td>16</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td bgcolor=yellow>2</td><td bgcolor=yellow> </td><td>-2</td></tr>
<tr align=center><td>14</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td bgcolor=yellow>2</td><td bgcolor=yellow>2</td><td> </td><td>0</td></tr>
<tr align=center><td>12</td><td> </td><td> </td><td> </td><td> </td><td> </td><td bgcolor=yellow>3</td><td bgcolor=yellow>2</td><td> </td><td> </td><td>-1</td></tr>
<tr align=center><td>10</td><td> </td><td> </td><td> </td><td> </td><td bgcolor=yellow>1</td><td bgcolor=yellow>2</td><td bgcolor=red>1</td><td> </td><td> </td><td>0</td></tr>
<tr align=center><td>8</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>6</td><td> </td><td> </td><td bgcolor=yellow>2</td><td bgcolor=yellow>1</td><td> </td><td> </td><td> </td><td> </td><td> </td><td>1</td></tr>
<tr align=center><td>4</td><td> </td><td bgcolor=yellow>1</td><td bgcolor=yellow>4</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td>3</td></tr>
<tr align=center><td>2</td><td bgcolor=yellow> </td><td bgcolor=yellow> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td>0</td></tr>
<tr align=center><td>0</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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