L10n78

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L10n77

L10n79

Contents

Image:L10n78.gif
(Knotscape image) 		See the full Thistlethwaite Link Table (up to 11 crossings).

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[edit L10n78 Quick Notes]
[edit L10n78 Further Notes and Views]


[edit] Link Presentations

[edit Notes on L10n78's Link Presentations]

Planar diagram presentation X6172 X5,14,6,15 X3849 X2,16,3,15 X16,7,17,8 X9,18,10,19 X11,20,12,13 X13,12,14,5 X17,1,18,4 X19,10,20,11
Gauss code {1, -4, -3, 9}, {-2, -1, 5, 3, -6, 10, -7, 8}, {-8, 2, 4, -5, -9, 6, -10, 7}
A Braid Representative
Image:BraidPart3.gifImage:BraidPart0.gifImage:BraidPart3.gifImage:BraidPart0.gifImage:BraidPart1.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gif
Image:BraidPart4.gifImage:BraidPart1.gifImage:BraidPart4.gifImage:BraidPart3.gifImage:BraidPart2.gifImage:BraidPart3.gifImage:BraidPart3.gifImage:BraidPart3.gifImage:BraidPart3.gifImage:BraidPart3.gif
Image:BraidPart0.gifImage:BraidPart2.gifImage:BraidPart0.gifImage:BraidPart4.gifImage:BraidPart0.gifImage:BraidPart4.gifImage:BraidPart4.gifImage:BraidPart4.gifImage:BraidPart4.gifImage:BraidPart4.gif
A Morse Link Presentation Image:L10n78_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+t(1) t(3)^2 t(2)^2-t(3)^2 t(2)^2+t(3) t(2)^2-t(1) t(3)^2 t(2)+t(1) t(3) t(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 2q−2−q−3 + 3q−4−3q−5 + 4q−6−3q−7 + 2q−8−q−9 + q−10 (db)
Signature -4 (db)
HOMFLY-PT polynomial a8z4 + 4a8z2 + a8z−2 + 4a8−a6z6−6a6z4−13a6z2−2a6z−2−11a6 + 2a4z4 + 8a4z2 + a4z−2 + 7a4 (db)
Kauffman polynomial z4a12−3z2a12 + a12 + z5a11−2z3a11 + z6a10−2z4a10 + z2a10 + z7a9−3z5a9 + 4z3a9 + z8a8−5z6a8 + 11z4a8−8z2a8−a8z−2 + 5a8 + 2z7a7−9z5a7 + 17z3a7−11za7 + 2a7z−1 + z8a6−6z6a6 + 17z4a6−23z2a6−2a6z−2 + 13a6 + z7a5−5z5a5 + 11z3a5−11za5 + 2a5z−1 + 3z4a4−11z2a4−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 L10n78. 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>1</td><td bgcolor=yellow>2</td><td>1</td></tr> <tr align=center><td>-7</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>-9</td><td> </td><td> </td><td> </td><td> </td><td> </td><td bgcolor=yellow>1</td><td bgcolor=yellow>1</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>3</td><td bgcolor=yellow>2</td><td> </td><td> </td><td> </td><td>1</td></tr> <tr align=center><td>-13</td><td> </td><td> </td><td> </td><td bgcolor=yellow>1</td><td bgcolor=yellow>2</td><td> </td><td> </td><td> </td><td> </td><td>1</td></tr> <tr align=center><td>-15</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>-17</td><td> </td><td bgcolor=yellow> </td><td bgcolor=yellow>1</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td>1</td></tr> <tr align=center><td>-19</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>-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}
r = −6 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r = −5 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r = −4 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{3}
r = −3 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r = −2 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r = −1 {\mathbb Z}_2 {\mathbb Z}
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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See/edit the Link Page master template (intermediate).

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L10n77

L10n79

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