L11n334

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Contents

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

Image:L11n334.gif

(Knotscape image)

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

Visit L11n334's page at Knotilus.

Visit L11n334's page at the original Knot Atlas.

[edit L11n334 Quick Notes]

[edit L11n334 Further Notes and Views]

[edit] Link Presentations

[edit Notes on L11n334's Link Presentations]

Planar diagram presentation	X₆₁₇₂ X_11,16,12,17 X₈₄₉₃ X_2,18,3,17 X_5,14,6,15 X_18,7,19,8 X_15,12,16,5 X_13,20,14,21 X_9,13,10,22 X_21,11,22,10 X_4,19,1,20
Gauss code	{1, -4, 3, -11}, {-5, -1, 6, -3, -9, 10, -2, 7}, {-8, 5, -7, 2, 4, -6, 11, 8, -10, 9}

A Braid Representative

A Morse Link Presentation

Image:L11n334_ML.gif

[edit] Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$0$ (db)
Jones polynomial	$q 3 -2 q 2 + 2 q -1 + 2 q -1 + q -2 + 2 q -4 -2 q -5 + 2 q -6 - q -7$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$- z 2 a 6 - a 6 + z 4 a 4 + 3 z 2 a 4 + a 4 z -2 + 2 a 4 -2 a 2 z -2 - a 2 - z 4 -3 z 2 + z -2 -1 + z 2 a -2 + a -2$ (db)
Kauffman polynomial	$a 7 z 7 -5 a 7 z 5 + 6 a 7 z 3 -2 a 7 z + 2 a 6 z 8 -11 a 6 z 6 + 16 a 6 z 4 -9 a 6 z 2 + 3 a 6 + a 5 z 9 -4 a 5 z 7 -2 a 5 z 5 + 12 a 5 z 3 -7 a 5 z + 3 a 4 z 8 -20 a 4 z 6 + 37 a 4 z 4 -27 a 4 z 2 - a 4 z -2 + 9 a 4 + a 3 z 9 -5 a 3 z 7 + 13 a 3 z 3 -10 a 3 z + 2 a 3 z -1 + 2 a 2 z 8 -14 a 2 z 6 + z 6 a -2 + 26 a 2 z 4 -4 z 4 a -2 -20 a 2 z 2 + 3 z 2 a -2 -2 a 2 z -2 + 7 a 2 - a -2 + 2 a z 7 + 2 z 7 a -1 -12 a z 5 -9 z 5 a -1 + 15 a z 3 + 8 z 3 a -1 -6 a z - z a -1 + 2 a z -1 + z 8 -4 z 6 + z 4 + z 2 - z -2 + 1$ (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 =

-1 is the signature of L11n334. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

<table border=1> <tr align=center> <td width=12.5%><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=6.25%>-7</td><td width=6.25%>-6</td><td width=6.25%>-5</td><td width=6.25%>-4</td><td width=6.25%>-3</td><td width=6.25%>-2</td><td width=6.25%>-1</td><td width=6.25%>0</td><td width=6.25%>1</td><td width=6.25%>2</td><td width=6.25%>3</td><td width=6.25%>4</td><td width=12.5%>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> </td><td> </td><td bgcolor=red>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> </td><td> </td><td bgcolor=red>1</td><td> </td><td>-1</td></tr> <tr align=center><td>3</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td bgcolor=red>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> </td><td bgcolor=red>2</td><td bgcolor=red>2</td><td bgcolor=red>1</td><td> </td><td> </td><td>1</td></tr> <tr align=center><td>-1</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td bgcolor=red>1</td><td bgcolor=red>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> </td><td bgcolor=red>2</td><td bgcolor=red>2</td><td bgcolor=red>3</td><td> </td><td> </td><td> </td><td> </td><td>3</td></tr> <tr align=center><td>-5</td><td> </td><td> </td><td> </td><td> </td><td bgcolor=red>2</td><td bgcolor=red>4</td><td bgcolor=red>1</td><td> </td><td> </td><td> </td><td> </td><td> </td><td>1</td></tr> <tr align=center><td>-7</td><td> </td><td> </td><td> </td><td bgcolor=red>1</td><td bgcolor=red>1</td><td bgcolor=red>2</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td>2</td></tr> <tr align=center><td>-9</td><td> </td><td> </td><td bgcolor=red>1</td><td bgcolor=red>2</td><td bgcolor=red>1</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td>0</td></tr> <tr align=center><td>-11</td><td> </td><td bgcolor=red>1</td><td bgcolor=red>1</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> </td><td bgcolor=red>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> <tr align=center><td>-15</td><td bgcolor=red>1</td><td> </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)

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

[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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