L11a29

From Knot Atlas

Jump to: navigation, search

Contents

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

Image:L11a29.gif

(Knotscape image)

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

Visit L11a29's page at Knotilus.

Visit L11a29's page at the original Knot Atlas.

[edit L11a29 Quick Notes]

[edit L11a29 Further Notes and Views]

[edit] Link Presentations

[edit Notes on L11a29's Link Presentations]

Planar diagram presentation	X₆₁₇₂ X_10,4,11,3 X_20,10,21,9 X_18,14,19,13 X_14,8,15,7 X_8,18,9,17 X_12,20,13,19 X_22,16,5,15 X_16,22,17,21 X₂₅₃₆ X_4,12,1,11
Gauss code	{1, -10, 2, -11}, {10, -1, 5, -6, 3, -2, 11, -7, 4, -5, 8, -9, 6, -4, 7, -3, 9, -8}

A Braid Representative

A Morse Link Presentation

Image:L11a29_ML.gif

[edit] Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-\frac{(t(1)-1) (t(2)-1) \left(5 t(2)^2-8 t(2)+5\right)}{\sqrt{t(1)} t(2)^{3/2}}$ (db)
Jones polynomial	$22 q^{9/2}-21 q^{7/2}+14 q^{5/2}-8 q^{3/2}+q^{21/2}-4 q^{19/2}+10 q^{17/2}-15 q^{15/2}+21 q^{13/2}-24 q^{11/2}+3 \sqrt{q}-\frac{1}{\sqrt{q}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$z 3 a -9 + a -9 z -1 - z 5 a -7 + z 3 a -7 + z a -7 - a -7 z -1 -3 z 5 a -5 -6 z 3 a -5 -5 z a -5 -2 a -5 z -1 - z 5 a -3 + z 3 a -3 + 3 z a -3 + 2 a -3 z -1 + z 3 a -1 + z a -1$ (db)
Kauffman polynomial	$-2 z 10 a -6 -2 z 10 a -8 -6 z 9 a -5 -13 z 9 a -7 -7 z 9 a -9 -8 z 8 a -4 -16 z 8 a -6 -16 z 8 a -8 -8 z 8 a -10 -6 z 7 a -3 -2 z 7 a -5 + 17 z 7 a -7 + 9 z 7 a -9 -4 z 7 a -11 -3 z 6 a -2 + 11 z 6 a -4 + 42 z 6 a -6 + 48 z 6 a -8 + 19 z 6 a -10 - z 6 a -12 - z 5 a -1 + 9 z 5 a -3 + 23 z 5 a -5 + 8 z 5 a -7 + 3 z 5 a -9 + 8 z 5 a -11 + 4 z 4 a -2 -7 z 4 a -4 -37 z 4 a -6 -44 z 4 a -8 -16 z 4 a -10 + 2 z 4 a -12 + 2 z 3 a -1 -9 z 3 a -3 -31 z 3 a -5 -19 z 3 a -7 -3 z 3 a -9 -4 z 3 a -11 - z 2 a -2 -2 z 2 a -4 + 14 z 2 a -6 + 25 z 2 a -8 + 9 z 2 a -10 - z 2 a -12 - z a -1 + 7 z a -3 + 15 z a -5 + 9 z a -7 + 2 z a -9 + 2 a -4 -4 a -6 -9 a -8 -4 a -10 -2 a -3 z -1 -2 a -5 z -1 + a -7 z -1 + a -9 z -1$ (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 =

3 is the signature of L11a29. 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%>-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=6.25%>8</td><td width=6.25%>9</td><td width=12.5%>χ</td></tr> <tr align=center><td>22</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>20</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>18</td><td> </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>1</td><td> </td><td>-6</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 bgcolor=yellow>8</td><td bgcolor=yellow>3</td><td> </td><td> </td><td>5</td></tr> <tr align=center><td>14</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td bgcolor=yellow>13</td><td bgcolor=yellow>7</td><td> </td><td> </td><td> </td><td>-6</td></tr> <tr align=center><td>12</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td bgcolor=yellow>11</td><td bgcolor=yellow>8</td><td> </td><td> </td><td> </td><td> </td><td>3</td></tr> <tr align=center><td>10</td><td> </td><td> </td><td> </td><td> </td><td> </td><td bgcolor=yellow>11</td><td bgcolor=yellow>13</td><td> </td><td> </td><td> </td><td> </td><td> </td><td>2</td></tr> <tr align=center><td>8</td><td> </td><td> </td><td> </td><td> </td><td bgcolor=yellow>10</td><td bgcolor=yellow>11</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td>-1</td></tr> <tr align=center><td>6</td><td> </td><td> </td><td> </td><td bgcolor=yellow>4</td><td bgcolor=yellow>11</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td>7</td></tr> <tr align=center><td>4</td><td> </td><td> </td><td bgcolor=yellow>4</td><td bgcolor=yellow>10</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td>-6</td></tr> <tr align=center><td>2</td><td> </td><td bgcolor=yellow>1</td><td bgcolor=yellow>6</td><td> </td><td> </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 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>-2</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)

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = 2$	$i = 4$
$r = -2$	${\mathbb Z}$
$r = -1$	${\mathbb Z}^{2}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = 0$	${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{2}$	${\mathbb Z}^{4}$
$r = 1$	${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{4}$	${\mathbb Z}^{4}$
$r = 2$	${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{10}$	${\mathbb Z}^{10}$
$r = 3$	${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{11}$	${\mathbb Z}^{11}$
$r = 4$	${\mathbb Z}^{13}\oplus{\mathbb Z}_2^{11}$	${\mathbb Z}^{11}$
$r = 5$	${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{13}$	${\mathbb Z}^{13}$
$r = 6$	${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{8}$	${\mathbb Z}^{8}$
$r = 7$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{7}$	${\mathbb Z}^{7}$
$r = 8$	${\mathbb Z}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = 9$	${\mathbb Z}_2$	${\mathbb Z}$

[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/L11a29"

Category: Knot Page

Views

Personal tools

Log in / create account

Search

Toolbox

tour