L10n72

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Contents

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

Image:L10n72.gif

(Knotscape image)

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

Visit L10n72's page at Knotilus.

Visit L10n72's page at the original Knot Atlas.

[edit L10n72 Quick Notes]

[edit L10n72 Further Notes and Views]

[edit] Link Presentations

[edit Notes on L10n72's Link Presentations]

Planar diagram presentation	X₆₁₇₂ X_12,4,13,3 X_7,17,8,16 X_9,11,10,20 X_11,18,12,19 X_15,9,16,8 X_19,5,20,10 X_17,14,18,15 X₂₅₃₆ X_4,14,1,13
Gauss code	{1, -9, 2, -10}, {9, -1, -3, 6, -4, 7}, {-5, -2, 10, 8, -6, 3, -8, 5, -7, 4}

A Braid Representative

A Morse Link Presentation

Image:L10n72_ML.gif

[edit] Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-\frac{u v^2+u v w^2-u v w-u w^2+u w-v^2 w+v^2+v w-v-w^2}{\sqrt{u} v w}$ (db)
Jones polynomial	$q 6 + 2 q 3 - q 2 + 3 q -2 + 2 q -1 -2 q -2 + q -3$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$a -6 z -2 + a -6 - z 2 a -4 -2 a -4 z -2 -4 a -4 + a 2 z 2 + 2 z 2 a -2 + a -2 z -2 + a 2 + 4 a -2 - z 4 -3 z 2 -2$ (db)
Kauffman polynomial	$z 8 a -2 + z 8 + 2 a z 7 + 3 z 7 a -1 + z 7 a -3 + a 2 z 6 -5 z 6 a -2 + z 6 a -6 -3 z 6 -9 a z 5 -15 z 5 a -1 -6 z 5 a -3 -4 a 2 z 4 + 6 z 4 a -2 -2 z 4 a -4 -6 z 4 a -6 -2 z 4 + 9 a z 3 + 19 z 3 a -1 + 8 z 3 a -3 -2 z 3 a -5 + 3 a 2 z 2 - z 2 a -2 + 6 z 2 a -4 + 9 z 2 a -6 + 5 z 2 -2 a z -6 z a -1 + 4 z a -5 - a 2 -3 a -2 -6 a -4 -4 a -6 -1-2 a -3 z -1 -2 a -5 z -1 + a -2 z -2 + 2 a -4 z -2 + a -6 z -2$ (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 =

2 is the signature of L10n72. 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%>-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=6.66667%>5</td><td width=6.66667%>6</td><td width=13.3333%>χ</td></tr> <tr align=center><td>13</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>11</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 bgcolor=red>1</td><td>1</td></tr> <tr align=center><td>9</td><td> </td><td> </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>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> </td><td> </td><td>2</td></tr> <tr align=center><td>5</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>3</td><td bgcolor=yellow>1</td><td> </td><td> </td><td> </td><td> </td><td> </td><td>2</td></tr> <tr align=center><td>1</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> </td><td> </td><td>1</td></tr> <tr align=center><td>-1</td><td> </td><td> </td><td bgcolor=yellow>1</td><td bgcolor=yellow>2</td><td bgcolor=red>1</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td>0</td></tr> <tr align=center><td>-3</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> </td><td> </td><td>0</td></tr> <tr align=center><td>-5</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> </td><td> </td><td>-1</td></tr> <tr align=center><td>-7</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)

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

[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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