L10n76

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Contents

1 Link Presentations
2 Polynomial invariants
3 Khovanov Homology
4 Computer Talk
5 Modifying This Page

Image:L10n76.gif

(Knotscape image)

See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L10n76's page at Knotilus.

Visit L10n76's page at the original Knot Atlas.

[edit L10n76 Quick Notes]

[edit L10n76 Further Notes and Views]

[edit] Link Presentations

[edit Notes on L10n76's Link Presentations]

Planar diagram presentation	X₆₁₇₂ X_5,14,6,15 X₈₄₉₃ X_2,16,3,15 X_16,7,17,8 X_9,18,10,19 X_4,17,1,18 X_19,12,20,5 X_11,20,12,13 X_13,10,14,11
Gauss code	{1, -4, 3, -7}, {-2, -1, 5, -3, -6, 10, -9, 8}, {-10, 2, 4, -5, 7, 6, -8, 9}

A Braid Representative

A Morse Link Presentation

Image:L10n76_ML.gif

[edit] Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-\frac{(u-1) (v-1) (w-1) (v+w)}{\sqrt{u} v w}$ (db)
Jones polynomial	$q -9 -2 q -8 + 4 q -7 -4 q -6 + 6 q -5 -5 q -4 + 5 q -3 -3 q -2 + 2 q -1$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$z 2 a 8 + a 8 z -2 + a 8 - z 4 a 6 -2 z 2 a 6 -2 a 6 z -2 -3 a 6 - z 4 a 4 - z 2 a 4 + a 4 z -2 + 2 z 2 a 2 + 2 a 2$ (db)
Kauffman polynomial	$a 10 z 6 -4 a 10 z 4 + 5 a 10 z 2 -2 a 10 + 2 a 9 z 7 -7 a 9 z 5 + 5 a 9 z 3 + a 9 z + a 8 z 8 -8 a 8 z 4 + 6 a 8 z 2 - a 8 z -2 + a 8 + 4 a 7 z 7 -11 a 7 z 5 + 8 a 7 z 3 -6 a 7 z + 2 a 7 z -1 + a 6 z 8 + a 6 z 6 -6 a 6 z 4 -2 a 6 z -2 + 5 a 6 + 2 a 5 z 7 -3 a 5 z 5 + 4 a 5 z 3 -6 a 5 z + 2 a 5 z -1 + 2 a 4 z 6 -2 a 4 z 4 + 2 a 4 z 2 - a 4 z -2 + a 4 + a 3 z 5 + a 3 z 3 + a 3 z + 3 a 2 z 2 -2 a 2$ (db)

[edit] Khovanov Homology

The coefficients of the monomials

t r q j

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 -2 r = s + 1

or

j -2 r = s -1

, where

s =

-2 is the signature of L10n76. 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>-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>3</td><td bgcolor=yellow>3</td><td> </td><td> </td><td>0</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>1</td><td bgcolor=yellow>3</td><td> </td><td> </td><td> </td><td> </td><td>2</td></tr> <tr align=center><td>-13</td><td> </td><td> </td><td bgcolor=yellow>3</td><td bgcolor=yellow>3</td><td> </td><td> </td><td> </td><td> </td><td> </td><td>0</td></tr> <tr align=center><td>-15</td><td> </td><td bgcolor=yellow>1</td><td bgcolor=yellow>3</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> </td><td bgcolor=yellow>1</td><td> </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> </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)

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

[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

Read me first: Modifying Knot Pages

See/edit the Link Page master template (intermediate).

See/edit the Link_Splice_Base (expert).

Back to the top.

Retrieved from "http://katlas.math.toronto.edu/wiki/L10n76"

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