L11n412

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Contents

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

Image:L11n412.gif

(Knotscape image)

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

Visit L11n412's page at Knotilus.

Visit L11n412's page at the original Knot Atlas.

[edit L11n412 Quick Notes]

[edit L11n412 Further Notes and Views]

[edit] Link Presentations

[edit Notes on L11n412's Link Presentations]

Planar diagram presentation	X₈₁₉₂ X_5,15,6,14 X_10,3,11,4 X_13,5,14,4 X₂₇₃₈ X_6,9,1,10 X_11,18,12,19 X_17,12,18,7 X_15,20,16,21 X_19,22,20,13 X_21,16,22,17
Gauss code	{1, -5, 3, 4, -2, -6}, {5, -1, 6, -3, -7, 8}, {-4, 2, -9, 11, -8, 7, -10, 9, -11, 10}

A Braid Representative

A Morse Link Presentation

Image:L11n412_ML.gif

[edit] Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-\frac{t(1)^2 t(3)^3-2 t(1)^2 t(3)^2+t(1) t(2)^2 t(3)^2-2 t(2)^2 t(3)^2+t(1) t(3)^2+t(1)^2 t(2) t(3)^2-2 t(1) t(2) t(3)^2+t(2) t(3)^2+2 t(1)^2 t(3)-t(1) t(2)^2 t(3)+2 t(2)^2 t(3)-t(1) t(3)-t(1)^2 t(2) t(3)+2 t(1) t(2) t(3)-t(2) t(3)-t(2)^2}{t(1) t(2) t(3)^{3/2}}$ (db)
Jones polynomial	$- q -10 + 2 q -9 -4 q -8 + 6 q -7 -6 q -6 + 8 q -5 -6 q -4 + 6 q -3 -3 q -2 + 2 q -1$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$- a 10 z -2 - a 10 + 3 z 2 a 8 + 4 a 8 z -2 + 6 a 8 -2 z 4 a 6 -6 z 2 a 6 -5 a 6 z -2 -9 a 6 - z 4 a 4 + 2 a 4 z -2 + 2 a 4 + 2 z 2 a 2 + 2 a 2$ (db)
Kauffman polynomial	$a 11 z 7 -5 a 11 z 5 + 8 a 11 z 3 -5 a 11 z + a 11 z -1 + 2 a 10 z 8 -9 a 10 z 6 + 12 a 10 z 4 -7 a 10 z 2 - a 10 z -2 + 4 a 10 + a 9 z 9 + a 9 z 7 -20 a 9 z 5 + 35 a 9 z 3 -21 a 9 z + 5 a 9 z -1 + 6 a 8 z 8 -26 a 8 z 6 + 38 a 8 z 4 -32 a 8 z 2 -4 a 8 z -2 + 17 a 8 + a 7 z 9 + 4 a 7 z 7 -28 a 7 z 5 + 44 a 7 z 3 -33 a 7 z + 9 a 7 z -1 + 4 a 6 z 8 -15 a 6 z 6 + 26 a 6 z 4 -32 a 6 z 2 -5 a 6 z -2 + 20 a 6 + 4 a 5 z 7 -12 a 5 z 5 + 18 a 5 z 3 -16 a 5 z + 5 a 5 z -1 + 2 a 4 z 6 -4 a 4 z 2 -2 a 4 z -2 + 6 a 4 + a 3 z 5 + a 3 z 3 + a 3 z + 3 a 2 z 2 -2 a 2$ (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 =

-2 is the signature of L11n412. 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%>-9</td><td width=7.14286%>-8</td><td width=7.14286%>-7</td><td width=7.14286%>-6</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=14.2857%>χ</td></tr> <tr align=center><td>-1</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td bgcolor=yellow>2</td><td>2</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 bgcolor=yellow>3</td><td bgcolor=yellow>2</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 bgcolor=yellow>3</td><td bgcolor=yellow> </td><td> </td><td>3</td></tr> <tr align=center><td>-7</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td bgcolor=yellow>4</td><td bgcolor=yellow>4</td><td> </td><td> </td><td>0</td></tr> <tr align=center><td>-9</td><td> </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>-11</td><td> </td><td> </td><td> </td><td> </td><td bgcolor=yellow>2</td><td bgcolor=yellow>4</td><td> </td><td> </td><td> </td><td> </td><td>2</td></tr> <tr align=center><td>-13</td><td> </td><td> </td><td> </td><td bgcolor=yellow>4</td><td bgcolor=yellow>4</td><td> </td><td> </td><td> </td><td> </td><td> </td><td>0</td></tr> <tr align=center><td>-15</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>-17</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> </td><td>-2</td></tr> <tr align=center><td>-19</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>-21</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)

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

[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/L11n412"

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