L10n85

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Contents

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

Image:L10n85.gif

(Knotscape image)

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

Visit L10n85's page at Knotilus.

Visit L10n85's page at the original Knot Atlas.

[edit L10n85 Quick Notes]

[edit L10n85 Further Notes and Views]

Link L10n85.

A graph, L10n85.

A graph, L10n85.

A part of a link and a part of a graph.

[edit] Link Presentations

[edit Notes on L10n85's Link Presentations]

Planar diagram presentation	X₆₁₇₂ X_10,3,11,4 X_20,14,15,13 X_7,16,8,17 X_15,8,16,9 X_18,12,19,11 X_12,20,13,19 X_14,18,5,17 X₂₅₃₆ X_4,9,1,10
Gauss code	{1, -9, 2, -10}, {-5, 4, 8, -6, 7, -3}, {9, -1, -4, 5, 10, -2, 6, -7, 3, -8}

A Braid Representative

A Morse Link Presentation

Image:L10n85_ML.gif

[edit] Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-\frac{(w-1) \left(u v^2 w-u v w+2 u v-u-v^2 w+2 v w-v+1\right)}{\sqrt{u} v w}$ (db)
Jones polynomial	$q 4 + 2 q -4 -3 q 3 -3 q -3 + 5 q 2 + 6 q -2 -6 q -6 q -1 + 8$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$- z 6 + a 2 z 4 + z 4 a -2 -4 z 4 + a 2 z 2 + 2 z 2 a -2 -5 z 2 + a 4 -2 a 2 + a -2 + a 4 z -2 -2 a 2 z -2 + z -2$ (db)
Kauffman polynomial	$3 a 4 z 4 + z 4 a -4 -7 a 4 z 2 - z 2 a -4 - a 4 z -2 + 4 a 4 + a 3 z 7 - a 3 z 5 + 3 z 5 a -3 + 2 a 3 z 3 -4 z 3 a -3 -5 a 3 z + 2 a 3 z -1 + a 2 z 8 - a 2 z 6 + 4 z 6 a -2 + 4 a 2 z 4 -6 z 4 a -2 -8 a 2 z 2 + 3 z 2 a -2 -2 a 2 z -2 + 6 a 2 - a -2 + 4 a z 7 + 3 z 7 a -1 -7 a z 5 -3 z 5 a -1 + 7 a z 3 + z 3 a -1 -5 a z + 2 a z -1 + z 8 + 3 z 6 -6 z 4 + 3 z 2 - z -2 + 2$ (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 L10n85. 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%>-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=7.69231%>1</td><td width=7.69231%>2</td><td width=7.69231%>3</td><td width=7.69231%>4</td><td width=15.3846%>χ</td></tr> <tr align=center><td>9</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td bgcolor=yellow>1</td><td>1</td></tr> <tr align=center><td>7</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td bgcolor=yellow>2</td><td bgcolor=yellow> </td><td>-2</td></tr> <tr align=center><td>5</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>2</td></tr> <tr align=center><td>3</td><td> </td><td> </td><td> </td><td> </td><td> </td><td bgcolor=yellow>4</td><td bgcolor=yellow>3</td><td> </td><td> </td><td>-1</td></tr> <tr align=center><td>1</td><td> </td><td> </td><td> </td><td> </td><td bgcolor=yellow>4</td><td bgcolor=yellow>2</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>3</td><td bgcolor=yellow>5</td><td> </td><td> </td><td> </td><td> </td><td>2</td></tr> <tr align=center><td>-3</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>-5</td><td> </td><td bgcolor=yellow>1</td><td bgcolor=yellow>4</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td>3</td></tr> <tr align=center><td>-7</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>-9</td><td bgcolor=yellow>2</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td>2</td></tr> </table>

Integral Khovanov Homology

(db, data source)

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

[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.

Retrieved from "http://katlas.math.toronto.edu/wiki/L10n85"

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