L11a318

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Contents

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

Image:L11a318.gif

(Knotscape image)

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

Visit L11a318's page at Knotilus.

Visit L11a318's page at the original Knot Atlas.

[edit L11a318 Quick Notes]

[edit L11a318 Further Notes and Views]

[edit] Link Presentations

[edit Notes on L11a318's Link Presentations]

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

A Braid Representative

A Morse Link Presentation

Image:L11a318_ML.gif

[edit] Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$\frac{(t(1)-1) (t(2)-1) \left(t(1) t(2)^4+t(1)^2 t(2)^3+t(1) t(2)^3+t(2)^3+t(1) t(2)^2+t(1)^2 t(2)+t(1) t(2)+t(2)+t(1)\right)}{t(1)^{3/2} t(2)^{5/2}}$ (db)
Jones polynomial	$-\frac{6}{q^{9/2}}+\frac{5}{q^{7/2}}-\frac{5}{q^{5/2}}-q^{3/2}+\frac{4}{q^{3/2}}+\frac{1}{q^{19/2}}-\frac{2}{q^{17/2}}+\frac{3}{q^{15/2}}-\frac{4}{q^{13/2}}+\frac{5}{q^{11/2}}+\sqrt{q}-\frac{3}{\sqrt{q}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$- z 5 a 7 -4 z 3 a 7 -3 z a 7 + z 7 a 5 + 5 z 5 a 5 + 6 z 3 a 5 - a 5 z -1 + z 7 a 3 + 6 z 5 a 3 + 12 z 3 a 3 + 10 z a 3 + 3 a 3 z -1 - z 5 a -5 z 3 a -7 z a -2 a z -1$ (db)
Kauffman polynomial	$a 12 z 2 + 2 a 11 z 3 + 3 a 10 z 4 -2 a 10 z 2 + 4 a 9 z 5 -6 a 9 z 3 + 5 a 8 z 6 -12 a 8 z 4 + 3 a 8 z 2 + 6 a 7 z 7 -22 a 7 z 5 + 20 a 7 z 3 -6 a 7 z + 5 a 6 z 8 -21 a 6 z 6 + 23 a 6 z 4 -8 a 6 z 2 + a 6 + 3 a 5 z 9 -12 a 5 z 7 + 8 a 5 z 5 + 5 a 5 z 3 - a 5 z -1 + a 4 z 10 - a 4 z 8 -16 a 4 z 6 + 37 a 4 z 4 -21 a 4 z 2 + 3 a 4 + 4 a 3 z 9 -26 a 3 z 7 + 57 a 3 z 5 -51 a 3 z 3 + 19 a 3 z -3 a 3 z -1 + a 2 z 10 -6 a 2 z 8 + 10 a 2 z 6 - a 2 z 4 -7 a 2 z 2 + 3 a 2 + a z 9 -8 a z 7 + 23 a z 5 -28 a z 3 + 13 a z -2 a z -1$ (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 =

-5 is the signature of L11a318. 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>4</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>2</td><td> </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> </td><td>0</td></tr> <tr align=center><td>0</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>1</td><td> </td><td>2</td></tr> <tr align=center><td>-2</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> </td><td> </td><td> </td><td>-1</td></tr> <tr align=center><td>-4</td><td> </td><td> </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> </td><td>1</td></tr> <tr align=center><td>-6</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td bgcolor=yellow>3</td><td bgcolor=yellow>3</td><td> </td><td> </td><td> </td><td> </td><td>0</td></tr> <tr align=center><td>-8</td><td> </td><td> </td><td> </td><td> </td><td> </td><td bgcolor=yellow>3</td><td bgcolor=yellow>2</td><td> </td><td> </td><td> </td><td> </td><td> </td><td>1</td></tr> <tr align=center><td>-10</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> </td><td> </td><td>1</td></tr> <tr align=center><td>-12</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> </td><td> </td><td> </td><td>-1</td></tr> <tr align=center><td>-14</td><td> </td><td> </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> </td><td>1</td></tr> <tr align=center><td>-16</td><td> </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> </td><td> </td><td>-1</td></tr> <tr align=center><td>-18</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> </td><td>1</td></tr> <tr align=center><td>-20</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)

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

[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.

Retrieved from "http://katlas.math.toronto.edu/wiki/L11a318"

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