L11a289

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Contents

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

Image:L11a289.gif

(Knotscape image)

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

Visit L11a289's page at Knotilus.

Visit L11a289's page at the original Knot Atlas.

[edit L11a289 Quick Notes]

[edit L11a289 Further Notes and Views]

Celtic or pseudo-Celtic linear decorative knot

Decorative variant with big loops at ends

(Also see Detecting a Link Using the Multivariable Alexander Polynomial.)

[edit] Link Presentations

[edit Notes on L11a289's Link Presentations]

Planar diagram presentation	X_10,1,11,2 X_12,4,13,3 X_22,12,9,11 X_2,9,3,10 X_20,14,21,13 X_14,7,15,8 X_18,16,19,15 X_16,6,17,5 X_6,18,7,17 X_4,19,5,20 X_8,22,1,21
Gauss code	{1, -4, 2, -10, 8, -9, 6, -11}, {4, -1, 3, -2, 5, -6, 7, -8, 9, -7, 10, -5, 11, -3}

A Braid Representative

A Morse Link Presentation

Image:L11a289_ML.gif

[edit] Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$\frac{(u-1) (v-1) \left(u^2 v^2-2 u^2 v+u^2-2 u v^2+2 u v-2 u+v^2-2 v+1\right)}{u^{3/2} v^{3/2}}$ (db)
Jones polynomial	$16 q^{9/2}-18 q^{7/2}+17 q^{5/2}-\frac{1}{q^{5/2}}-15 q^{3/2}+\frac{3}{q^{3/2}}+q^{17/2}-4 q^{15/2}+8 q^{13/2}-12 q^{11/2}+10 \sqrt{q}-\frac{7}{\sqrt{q}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$z 3 a -7 + z a -7 -2 z 5 a -5 -5 z 3 a -5 -3 z a -5 + z 7 a -3 + 4 z 5 a -3 + 7 z 3 a -3 + 5 z a -3 -2 z 5 a -1 + a z 3 -6 z 3 a -1 + 2 a z -5 z a -1 + a z -1 - a -1 z -1$ (db)
Kauffman polynomial	$z 4 a -10 + 4 z 5 a -9 -2 z 3 a -9 + 8 z 6 a -8 -8 z 4 a -8 + 3 z 2 a -8 + 10 z 7 a -7 -12 z 5 a -7 + 6 z 3 a -7 -2 z a -7 + 8 z 8 a -6 -4 z 6 a -6 -9 z 4 a -6 + 4 z 2 a -6 + 4 z 9 a -5 + 9 z 7 a -5 -34 z 5 a -5 + 24 z 3 a -5 -6 z a -5 + z 10 a -4 + 13 z 8 a -4 -33 z 6 a -4 + 16 z 4 a -4 + 7 z 9 a -3 -7 z 7 a -3 -24 z 5 a -3 + 32 z 3 a -3 -10 z a -3 + z 10 a -2 + 8 z 8 a -2 -32 z 6 a -2 + 28 z 4 a -2 -4 z 2 a -2 + 3 z 9 a -1 + a z 7 -5 z 7 a -1 -4 a z 5 -10 z 5 a -1 + 6 a z 3 + 22 z 3 a -1 -4 a z -10 z a -1 + a z -1 + a -1 z -1 + 3 z 8 -11 z 6 + 12 z 4 -3 z 2 -1$ (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 =

3 is the signature of L11a289. 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%>-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=6.25%>5</td><td width=6.25%>6</td><td width=6.25%>7</td><td width=12.5%>χ</td></tr> <tr align=center><td>18</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=yellow>1</td><td>-1</td></tr> <tr align=center><td>16</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td bgcolor=yellow>3</td><td bgcolor=yellow> </td><td>3</td></tr> <tr align=center><td>14</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td bgcolor=yellow>5</td><td bgcolor=yellow>1</td><td> </td><td>-4</td></tr> <tr align=center><td>12</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td bgcolor=yellow>7</td><td bgcolor=yellow>3</td><td> </td><td> </td><td>4</td></tr> <tr align=center><td>10</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td bgcolor=yellow>9</td><td bgcolor=yellow>5</td><td> </td><td> </td><td> </td><td>-4</td></tr> <tr align=center><td>8</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td bgcolor=yellow>9</td><td bgcolor=yellow>7</td><td> </td><td> </td><td> </td><td> </td><td>2</td></tr> <tr align=center><td>6</td><td> </td><td> </td><td> </td><td> </td><td> </td><td bgcolor=yellow>8</td><td bgcolor=yellow>9</td><td> </td><td> </td><td> </td><td> </td><td> </td><td>1</td></tr> <tr align=center><td>4</td><td> </td><td> </td><td> </td><td> </td><td bgcolor=yellow>7</td><td bgcolor=yellow>9</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td>-2</td></tr> <tr align=center><td>2</td><td> </td><td> </td><td> </td><td bgcolor=yellow>5</td><td bgcolor=yellow>10</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td>5</td></tr> <tr align=center><td>0</td><td> </td><td> </td><td bgcolor=yellow>2</td><td bgcolor=yellow>5</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td>-3</td></tr> <tr align=center><td>-2</td><td> </td><td bgcolor=yellow>1</td><td bgcolor=yellow>5</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td>4</td></tr> <tr align=center><td>-4</td><td bgcolor=yellow> </td><td bgcolor=yellow>2</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td>-2</td></tr> <tr align=center><td>-6</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> </td><td>1</td></tr> </table>

Integral Khovanov Homology

(db, data source)

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = 2$	$i = 4$
$r = -4$	${\mathbb Z}$
$r = -3$	${\mathbb Z}^{2}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -2$	${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2}$	${\mathbb Z}^{2}$
$r = -1$	${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5}$	${\mathbb Z}^{5}$
$r = 0$	${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{5}$	${\mathbb Z}^{7}$
$r = 1$	${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{8}$	${\mathbb Z}^{8}$
$r = 2$	${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{9}$	${\mathbb Z}^{9}$
$r = 3$	${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{9}$	${\mathbb Z}^{9}$
$r = 4$	${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{7}$	${\mathbb Z}^{7}$
$r = 5$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5}$	${\mathbb Z}^{5}$
$r = 6$	${\mathbb Z}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = 7$	${\mathbb Z}_2$	${\mathbb Z}$

[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/L11a289"

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