From Knot Atlas
[edit] Link Presentations
[edit Notes on L11n334's Link Presentations]
| Planar diagram presentation
| X6172 X11,16,12,17 X8493 X2,18,3,17 X5,14,6,15 X18,7,19,8 X15,12,16,5 X13,20,14,21 X9,13,10,22 X21,11,22,10 X4,19,1,20
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| Gauss code
| {1, -4, 3, -11}, {-5, -1, 6, -3, -9, 10, -2, 7}, {-8, 5, -7, 2, 4, -6, 11, 8, -10, 9}
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[edit] Polynomial invariants
| Multivariable Alexander Polynomial (in u, v, w, ...)
| 0 (db)
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| Jones polynomial
| q3−2q2 + 2q−1 + 2q−1 + q−2 + 2q−4−2q−5 + 2q−6−q−7 (db)
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| Signature
| -1 (db)
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| HOMFLY-PT polynomial
| −z2a6−a6 + z4a4 + 3z2a4 + a4z−2 + 2a4−2a2z−2−a2−z4−3z2 + z−2−1 + z2a−2 + a−2 (db)
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| Kauffman polynomial
| a7z7−5a7z5 + 6a7z3−2a7z + 2a6z8−11a6z6 + 16a6z4−9a6z2 + 3a6 + a5z9−4a5z7−2a5z5 + 12a5z3−7a5z + 3a4z8−20a4z6 + 37a4z4−27a4z2−a4z−2 + 9a4 + a3z9−5a3z7 + 13a3z3−10a3z + 2a3z−1 + 2a2z8−14a2z6 + z6a−2 + 26a2z4−4z4a−2−20a2z2 + 3z2a−2−2a2z−2 + 7a2−a−2 + 2az7 + 2z7a−1−12az5−9z5a−1 + 15az3 + 8z3a−1−6az−za−1 + 2az−1 + z8−4z6 + z4 + z2−z−2 + 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 = -1 is the signature of L11n334. 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=12.5%><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=6.25%>-7</td><td width=6.25%>-6</td><td width=6.25%>-5</td><td width=6.25%>-4</td><td width=6.25%>-3</td><td width=6.25%>-2</td><td width=6.25%>-1</td><td width=6.25%>0</td><td width=6.25%>1</td><td width=6.25%>2</td><td width=6.25%>3</td><td width=6.25%>4</td><td width=12.5%>χ</td></tr>
<tr align=center><td>7</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td bgcolor=red>1</td><td>1</td></tr>
<tr align=center><td>5</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td bgcolor=red>1</td><td> </td><td>-1</td></tr>
<tr align=center><td>3</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td bgcolor=red>1</td><td bgcolor=red>1</td><td> </td><td>0</td></tr>
<tr align=center><td>1</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td bgcolor=red>2</td><td bgcolor=red>2</td><td bgcolor=red>1</td><td> </td><td> </td><td>1</td></tr>
<tr align=center><td>-1</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td bgcolor=red>1</td><td bgcolor=red>3</td><td bgcolor=red>1</td><td> </td><td> </td><td> </td><td>1</td></tr>
<tr align=center><td>-3</td><td> </td><td> </td><td> </td><td> </td><td> </td><td bgcolor=red>2</td><td bgcolor=red>2</td><td bgcolor=red>3</td><td> </td><td> </td><td> </td><td> </td><td>3</td></tr>
<tr align=center><td>-5</td><td> </td><td> </td><td> </td><td> </td><td bgcolor=red>2</td><td bgcolor=red>4</td><td bgcolor=red>1</td><td> </td><td> </td><td> </td><td> </td><td> </td><td>1</td></tr>
<tr align=center><td>-7</td><td> </td><td> </td><td> </td><td bgcolor=red>1</td><td bgcolor=red>1</td><td bgcolor=red>2</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td>2</td></tr>
<tr align=center><td>-9</td><td> </td><td> </td><td bgcolor=red>1</td><td bgcolor=red>2</td><td bgcolor=red>1</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td>0</td></tr>
<tr align=center><td>-11</td><td> </td><td bgcolor=red>1</td><td bgcolor=red>1</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td>0</td></tr>
<tr align=center><td>-13</td><td> </td><td bgcolor=red>1</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td>1</td></tr>
<tr align=center><td>-15</td><td bgcolor=red>1</td><td> </td><td> </td><td> </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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