L10n81

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L10n80

L10n82

Contents

Image:L10n81.gif
(Knotscape image) 		See the full Thistlethwaite Link Table (up to 11 crossings).

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[edit L10n81 Further Notes and Views]


[edit] Link Presentations

[edit Notes on L10n81's Link Presentations]

Planar diagram presentation X6172 X2,16,3,15 X3,10,4,11 X5,14,6,15 X11,20,12,13 X13,12,14,5 X19,1,20,4 X8,17,9,18 X16,7,17,8 X18,9,19,10
Gauss code {1, -2, -3, 7}, {-4, -1, 9, -8, 10, 3, -5, 6}, {-6, 4, 2, -9, 8, -10, -7, 5}
A Braid Representative
Image:BraidPart3.gifImage:BraidPart0.gifImage:BraidPart3.gifImage:BraidPart3.gifImage:BraidPart3.gifImage:BraidPart0.gifImage:BraidPart1.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gif
Image:BraidPart4.gifImage:BraidPart1.gifImage:BraidPart4.gifImage:BraidPart4.gifImage:BraidPart4.gifImage:BraidPart3.gifImage:BraidPart2.gifImage:BraidPart3.gifImage:BraidPart3.gifImage:BraidPart3.gif
Image:BraidPart0.gifImage:BraidPart2.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart4.gifImage:BraidPart0.gifImage:BraidPart4.gifImage:BraidPart4.gifImage:BraidPart4.gif
A Morse Link Presentation Image:L10n81_ML.gif

[edit] Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{t(3)^2 t(2)^3-t(1) t(3)^3 t(2)^2+2 t(1) t(3)^2 t(2)^2-2 t(3)^2 t(2)^2-t(1) t(3) t(2)^2+2 t(3) t(2)^2-2 t(1) t(3)^2 t(2)+t(3)^2 t(2)+2 t(1) t(3) t(2)-2 t(3) t(2)+t(2)-t(1) t(3)}{\sqrt{t(1)} t(2)^{3/2} t(3)^{3/2}} (db)
Jones polynomial q−10−2q−9 + 5q−8−6q−7 + 6q−6−6q−5 + 6q−4−2q−3 + 2q−2 (db)
Signature -4 (db)
HOMFLY-PT polynomial a8z4 + 3a8z2 + a8z−2 + 4a8−a6z6−5a6z4−11a6z2−2a6z−2−11a6 + 2a4z4 + 7a4z2 + a4z−2 + 7a4 (db)
Kauffman polynomial a12z4−2a12z2 + a12 + 2a11z5−2a11z3 + 3a10z6−4a10z4 + 2a10z2 + 2a9z7−2a9z3 + a8z8 + 4a8z4−7a8z2−a8z−2 + 5a8 + 3a7z7−5a7z5 + 9a7z3−11a7z + 2a7z−1 + a6z8−3a6z6 + 12a6z4−20a6z2−2a6z−2 + 13a6 + a5z7−3a5z5 + 9a5z3−11a5z + 2a5z−1 + 3a4z4−9a4z2−a4z−2 + 8a4 (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 = -4 is the signature of L10n81. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.    <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>-3</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>-5</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td bgcolor=yellow>2</td><td bgcolor=yellow>2</td><td>0</td></tr> <tr align=center><td>-7</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td bgcolor=yellow>4</td><td bgcolor=yellow> </td><td> </td><td>4</td></tr> <tr align=center><td>-9</td><td> </td><td> </td><td> </td><td> </td><td> </td><td bgcolor=yellow>2</td><td bgcolor=yellow>2</td><td> </td><td> </td><td>0</td></tr> <tr align=center><td>-11</td><td> </td><td> </td><td> </td><td> </td><td bgcolor=yellow>4</td><td bgcolor=yellow>4</td><td> </td><td> </td><td> </td><td>0</td></tr> <tr align=center><td>-13</td><td> </td><td> </td><td> </td><td bgcolor=yellow>3</td><td bgcolor=yellow>3</td><td> </td><td> </td><td> </td><td> </td><td>0</td></tr> <tr align=center><td>-15</td><td> </td><td> </td><td bgcolor=yellow>2</td><td bgcolor=yellow>3</td><td> </td><td> </td><td> </td><td> </td><td> </td><td>-1</td></tr> <tr align=center><td>-17</td><td> </td><td bgcolor=yellow> </td><td bgcolor=yellow>3</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td>3</td></tr> <tr align=center><td>-19</td><td bgcolor=yellow>1</td><td bgcolor=yellow>2</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td>-1</td></tr> <tr align=center><td>-21</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>
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i = −5 i = −3
r = −8 {\mathbb Z} {\mathbb Z}
r = −7 {\mathbb Z}^{2}
r = −6 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r = −5 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r = −4 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{4}
r = −3 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r = −2 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r = −1 {\mathbb Z}_2^{2} {\mathbb Z}^{2}
r = 0 {\mathbb Z}^{2} {\mathbb Z}^{2}

[edit] Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

[edit] Modifying This Page

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L10n80

L10n82

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