L11n421

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Contents

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

Image:L11n421.gif

(Knotscape image)

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

Visit L11n421's page at Knotilus.

Visit L11n421's page at the original Knot Atlas.

[edit L11n421 Quick Notes]

[edit L11n421 Further Notes and Views]

[edit] Link Presentations

[edit Notes on L11n421's Link Presentations]

Planar diagram presentation	X₈₁₉₂ X_16,8,17,7 X_3,10,4,11 X_17,2,18,3 X_9,19,10,18 X_20,12,21,11 X_14,6,15,5 X_22,15,13,16 X_6,14,1,13 X_4,19,5,20 X_12,22,7,21
Gauss code	{1, 4, -3, -10, 7, -9}, {2, -1, -5, 3, 6, -11}, {9, -7, 8, -2, -4, 5, 10, -6, 11, -8}

A Braid Representative

A Morse Link Presentation

Image:L11n421_ML.gif

[edit] Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$\frac{t(1) t(2)^2 t(3)^3-t(2)^2 t(3)^3-2 t(1) t(2)^2 t(3)^2+t(2)^2 t(3)^2+t(1) t(2) t(3)^2-t(1)^2 t(3)+2 t(1) t(3)-t(1) t(2) t(3)+t(1)^2-t(1)}{t(1) t(2) t(3)^{3/2}}$ (db)
Jones polynomial	$q 5 - q 4 + 3 q 3 -3 q 2 + 4 q -3 + 4 q -1 -2 q -2 + 2 q -3 - q -4$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$z 2 a -4 + a -4 z -2 + 3 a -4 - a 2 z 4 -2 z 4 a -2 -3 a 2 z 2 -8 z 2 a -2 -2 a -2 z -2 - a 2 -8 a -2 + z 6 + 5 z 4 + 8 z 2 + z -2 + 6$ (db)
Kauffman polynomial	$z 2 a -6 - a -6 + z 3 a -5 + 2 z 4 a -4 -4 z 2 a -4 - a -4 z -2 + 4 a -4 + a 3 z 7 + z 7 a -3 -5 a 3 z 5 -4 z 5 a -3 + 6 a 3 z 3 + 7 z 3 a -3 - a 3 z -6 z a -3 + 2 a -3 z -1 + 2 a 2 z 8 + 2 z 8 a -2 -11 a 2 z 6 -11 z 6 a -2 + 18 a 2 z 4 + 23 z 4 a -2 -11 a 2 z 2 -25 z 2 a -2 -2 a -2 z -2 + 2 a 2 + 12 a -2 + a z 9 + z 9 a -1 -3 a z 7 -3 z 7 a -1 -3 a z 5 -2 z 5 a -1 + 9 a z 3 + 9 z 3 a -1 -3 a z -8 z a -1 + 2 a -1 z -1 + 4 z 8 -22 z 6 + 39 z 4 -31 z 2 - z -2 + 10$ (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 L11n421. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

<table border=1> <tr align=center> <td width=14.2857%><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=7.14286%>-5</td><td width=7.14286%>-4</td><td width=7.14286%>-3</td><td width=7.14286%>-2</td><td width=7.14286%>-1</td><td width=7.14286%>0</td><td width=7.14286%>1</td><td width=7.14286%>2</td><td width=7.14286%>3</td><td width=7.14286%>4</td><td width=14.2857%>χ</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> </td><td bgcolor=yellow>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> </td><td bgcolor=yellow>1</td><td bgcolor=yellow>1</td><td>0</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> </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>2</td><td bgcolor=yellow>2</td><td> </td><td> </td><td>0</td></tr> <tr align=center><td>3</td><td> </td><td> </td><td> </td><td> </td><td> </td><td bgcolor=yellow>2</td><td bgcolor=yellow>1</td><td> </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>2</td><td bgcolor=yellow>3</td><td> </td><td> </td><td> </td><td> </td><td>1</td></tr> <tr align=center><td>-1</td><td> </td><td> </td><td> </td><td bgcolor=yellow>2</td><td bgcolor=yellow>1</td><td> </td><td> </td><td> </td><td> </td><td> </td><td>1</td></tr> <tr align=center><td>-3</td><td> </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>-5</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>0</td></tr> <tr align=center><td>-7</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>1</td></tr> <tr align=center><td>-9</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> </table>

Integral Khovanov Homology

(db, data source)

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

[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.

Retrieved from "http://katlas.math.toronto.edu/wiki/L11n421"

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