L10n66

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Contents

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

Image:L10n66.gif

(Knotscape image)

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

Visit L10n66's page at Knotilus.

Visit L10n66's page at the original Knot Atlas.

[edit L10n66 Quick Notes]

[edit L10n66 Further Notes and Views]

[edit] Link Presentations

[edit Notes on L10n66's Link Presentations]

Planar diagram presentation	X₆₁₇₂ X_3,11,4,10 X_16,12,17,11 X_18,13,19,14 X_20,18,9,17 X_12,19,13,20 X_15,8,16,5 X_7,14,8,15 X₂₅₃₆ X_9,1,10,4
Gauss code	{1, -9, -2, 10}, {9, -1, -8, 7}, {-10, 2, 3, -6, 4, 8, -7, -3, 5, -4, 6, -5}

A Braid Representative

A Morse Link Presentation

Image:L10n66_ML.gif

[edit] Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$\frac{-u v w^2+u v w+u w^3-2 u w^2+2 v w-v-w^2+w}{\sqrt{u} \sqrt{v} w^{3/2}}$ (db)
Jones polynomial	$q -6 - q -5 + 2 q -4 + q 3 - q -3 - q 2 + q -2 + q + 1$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$a 6 z -2 + a 6 -2 a 4 z 2 -3 a 4 z -2 -5 a 4 + a 2 z 4 + 5 a 2 z 2 + 4 a 2 z -2 + z 2 a -2 + a -2 z -2 + 8 a 2 + 2 a -2 - z 4 -5 z 2 -3 z -2 -6$ (db)
Kauffman polynomial	$a 5 z 7 + a 3 z 7 + a z 7 + z 7 a -1 + a 6 z 6 + 3 a 4 z 6 + 3 a 2 z 6 + z 6 a -2 + 2 z 6 -4 a 5 z 5 -5 a 3 z 5 -6 a z 5 -5 z 5 a -1 -5 a 6 z 4 -16 a 4 z 4 -20 a 2 z 4 -5 z 4 a -2 -14 z 4 + 2 a 5 z 3 + 4 a 3 z 3 + 6 a z 3 + 4 z 3 a -1 + 7 a 6 z 2 + 23 a 4 z 2 + 35 a 2 z 2 + 6 z 2 a -2 + 25 z 2 + a 5 z + a 3 z + a z + z a -1 -4 a 6 -14 a 4 -21 a 2 -4 a -2 -14- a 5 z -1 - a 3 z -1 - a z -1 - a -1 z -1 + a 6 z -2 + 3 a 4 z -2 + 4 a 2 z -2 + a -2 z -2 + 3 z -2$ (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 =

0 is the signature of L10n66. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

<table border=1> <tr align=center> <td width=13.3333%><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=6.66667%>-6</td><td width=6.66667%>-5</td><td width=6.66667%>-4</td><td width=6.66667%>-3</td><td width=6.66667%>-2</td><td width=6.66667%>-1</td><td width=6.66667%>0</td><td width=6.66667%>1</td><td width=6.66667%>2</td><td width=6.66667%>3</td><td width=6.66667%>4</td><td width=13.3333%>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 bgcolor=yellow> </td><td bgcolor=yellow>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 bgcolor=yellow> </td><td bgcolor=yellow> </td><td> </td><td>0</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> </td><td bgcolor=yellow>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 bgcolor=yellow>3</td><td bgcolor=yellow>1</td><td> </td><td> </td><td> </td><td>2</td></tr> <tr align=center><td>-1</td><td> </td><td> </td><td> </td><td> </td><td> </td><td bgcolor=yellow>1</td><td bgcolor=yellow>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 bgcolor=yellow>1</td><td bgcolor=yellow>2</td><td bgcolor=red>2</td><td> </td><td> </td><td> </td><td> </td><td>1</td></tr> <tr align=center><td>-5</td><td> </td><td> </td><td> </td><td bgcolor=yellow>1</td><td bgcolor=yellow>1</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td>0</td></tr> <tr align=center><td>-7</td><td> </td><td> </td><td bgcolor=yellow>1</td><td bgcolor=yellow>1</td><td bgcolor=red>1</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td>1</td></tr> <tr align=center><td>-9</td><td> </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> </td><td>1</td></tr> <tr align=center><td>-11</td><td bgcolor=yellow> </td><td bgcolor=yellow> </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 bgcolor=yellow>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> </table>

Integral Khovanov Homology

(db, data source)

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

[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.

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