L11a306

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Contents

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

Image:L11a306.gif

(Knotscape image)

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

Visit L11a306's page at Knotilus.

Visit L11a306's page at the original Knot Atlas.

[edit L11a306 Quick Notes]

[edit L11a306 Further Notes and Views]

[edit] Link Presentations

[edit Notes on L11a306's Link Presentations]

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

A Braid Representative

A Morse Link Presentation

Image:L11a306_ML.gif

[edit] Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$\frac{-t(1) t(2)^5-3 t(1)^2 t(2)^4+4 t(1) t(2)^4-t(2)^4-3 t(1)^3 t(2)^3+7 t(1)^2 t(2)^3-8 t(1) t(2)^3+3 t(2)^3+3 t(1)^3 t(2)^2-8 t(1)^2 t(2)^2+7 t(1) t(2)^2-3 t(2)^2-t(1)^3 t(2)+4 t(1)^2 t(2)-3 t(1) t(2)-t(1)^2}{t(1)^{3/2} t(2)^{5/2}}$ (db)
Jones polynomial	$-\frac{7}{q^{9/2}}+\frac{3}{q^{7/2}}-\frac{1}{q^{5/2}}+\frac{1}{q^{27/2}}-\frac{4}{q^{25/2}}+\frac{8}{q^{23/2}}-\frac{13}{q^{21/2}}+\frac{17}{q^{19/2}}-\frac{19}{q^{17/2}}+\frac{19}{q^{15/2}}-\frac{17}{q^{13/2}}+\frac{11}{q^{11/2}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$a 13 (- z)- a 13 z -1 + 4 a 11 z 3 + 8 a 11 z + 3 a 11 z -1 -3 a 9 z 5 -8 a 9 z 3 -6 a 9 z -2 a 9 z -1 -3 a 7 z 5 -8 a 7 z 3 -5 a 7 z - a 5 z 5 -2 a 5 z 3$ (db)
Kauffman polynomial	$a 16 z 6 -2 a 16 z 4 + a 16 z 2 + 4 a 15 z 7 -10 a 15 z 5 + 6 a 15 z 3 + 6 a 14 z 8 -13 a 14 z 6 + 5 a 14 z 4 - a 14 z 2 + a 14 + 5 a 13 z 9 -6 a 13 z 7 -5 a 13 z 5 + 2 a 13 z 3 + a 13 z - a 13 z -1 + 2 a 12 z 10 + 7 a 12 z 8 -27 a 12 z 6 + 30 a 12 z 4 -19 a 12 z 2 + 3 a 12 + 11 a 11 z 9 -28 a 11 z 7 + 34 a 11 z 5 -27 a 11 z 3 + 14 a 11 z -3 a 11 z -1 + 2 a 10 z 10 + 8 a 10 z 8 -31 a 10 z 6 + 42 a 10 z 4 -20 a 10 z 2 + 3 a 10 + 6 a 9 z 9 -12 a 9 z 7 + 14 a 9 z 5 -8 a 9 z 3 + 8 a 9 z -2 a 9 z -1 + 7 a 8 z 8 -15 a 8 z 6 + 14 a 8 z 4 -3 a 8 z 2 + 6 a 7 z 7 -14 a 7 z 5 + 13 a 7 z 3 -5 a 7 z + 3 a 6 z 6 -5 a 6 z 4 + a 5 z 5 -2 a 5 z 3$ (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 L11a306. 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%>-11</td><td width=6.25%>-10</td><td width=6.25%>-9</td><td width=6.25%>-8</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=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>-6</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>1</td><td>-2</td></tr> <tr align=center><td>-8</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td bgcolor=yellow>4</td><td bgcolor=yellow> </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> </td><td bgcolor=yellow>7</td><td bgcolor=yellow>3</td><td> </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 bgcolor=yellow>10</td><td bgcolor=yellow>4</td><td> </td><td> </td><td> </td><td>6</td></tr> <tr align=center><td>-14</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>-16</td><td> </td><td> </td><td> </td><td> </td><td> </td><td bgcolor=yellow>10</td><td bgcolor=yellow>10</td><td> </td><td> </td><td> </td><td> </td><td> </td><td>0</td></tr> <tr align=center><td>-18</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>-20</td><td> </td><td> </td><td> </td><td bgcolor=yellow>6</td><td bgcolor=yellow>10</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td>-4</td></tr> <tr align=center><td>-22</td><td> </td><td> </td><td bgcolor=yellow>3</td><td bgcolor=yellow>8</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>-24</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>-26</td><td bgcolor=yellow> </td><td bgcolor=yellow>3</td><td> </td><td> </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>-28</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)

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

[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/L11a306"

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