L11a308

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Contents

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

Image:L11a308.gif

(Knotscape image)

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

Visit L11a308's page at Knotilus.

Visit L11a308's page at the original Knot Atlas.

[edit L11a308 Quick Notes]

[edit L11a308 Further Notes and Views]

[edit] Link Presentations

[edit Notes on L11a308's Link Presentations]

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

A Braid Representative

A Morse Link Presentation

Image:L11a308_ML.gif

[edit] Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$\frac{(t(1)+t(2)-1) (t(1) t(2)+1) (t(2) t(1)-t(1)-t(2)) \left(t(2)^2-t(2)+1\right)}{t(1)^{3/2} t(2)^{5/2}}$ (db)
Jones polynomial	$-\frac{1}{\sqrt{q}}+\frac{3}{q^{3/2}}-\frac{7}{q^{5/2}}+\frac{10}{q^{7/2}}-\frac{15}{q^{9/2}}+\frac{17}{q^{11/2}}-\frac{17}{q^{13/2}}+\frac{15}{q^{15/2}}-\frac{12}{q^{17/2}}+\frac{7}{q^{19/2}}-\frac{3}{q^{21/2}}+\frac{1}{q^{23/2}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$a^9 \left(-z^5\right)-3 a^9 z^3-3 a^9 z-a^9 z^{-1} +a^7 z^7+4 a^7 z^5+7 a^7 z^3+8 a^7 z+3 a^7 z^{-1} +a^5 z^7+3 a^5 z^5-5 a^5 z-2 a^5 z^{-1} -a^3 z^5-3 a^3 z^3-2 a^3 z$ (db)
Kauffman polynomial	$a 14 z 4 - a 14 z 2 + 3 a 13 z 5 -2 a 13 z 3 + 6 a 12 z 6 -6 a 12 z 4 + 3 a 12 z 2 + 9 a 11 z 7 -15 a 11 z 5 + 13 a 11 z 3 -3 a 11 z + 9 a 10 z 8 -16 a 10 z 6 + 12 a 10 z 4 -4 a 10 z 2 + a 10 + 6 a 9 z 9 -7 a 9 z 7 -3 a 9 z 5 + 4 a 9 z 3 - a 9 z -1 + 2 a 8 z 10 + 7 a 8 z 8 -29 a 8 z 6 + 29 a 8 z 4 -17 a 8 z 2 + 3 a 8 + 10 a 7 z 9 -30 a 7 z 7 + 31 a 7 z 5 -23 a 7 z 3 + 12 a 7 z -3 a 7 z -1 + 2 a 6 z 10 + a 6 z 8 -18 a 6 z 6 + 21 a 6 z 4 -11 a 6 z 2 + 3 a 6 + 4 a 5 z 9 -13 a 5 z 7 + 12 a 5 z 5 -7 a 5 z 3 + 7 a 5 z -2 a 5 z -1 + 3 a 4 z 8 -11 a 4 z 6 + 11 a 4 z 4 -2 a 4 z 2 + a 3 z 7 -4 a 3 z 5 + 5 a 3 z 3 -2 a 3 z$ (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 L11a308. 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%>-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=6.25%>1</td><td width=6.25%>2</td><td width=12.5%>χ</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> </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>2</td><td bgcolor=yellow> </td><td>-2</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 bgcolor=yellow>5</td><td bgcolor=yellow>1</td><td> </td><td>4</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 bgcolor=yellow>6</td><td bgcolor=yellow>3</td><td> </td><td> </td><td>-3</td></tr> <tr align=center><td>-8</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td bgcolor=yellow>9</td><td bgcolor=yellow>4</td><td> </td><td> </td><td> </td><td>5</td></tr> <tr align=center><td>-10</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td bgcolor=yellow>8</td><td bgcolor=yellow>6</td><td> </td><td> </td><td> </td><td> </td><td>-2</td></tr> <tr align=center><td>-12</td><td> </td><td> </td><td> </td><td> </td><td> </td><td bgcolor=yellow>9</td><td bgcolor=yellow>9</td><td> </td><td> </td><td> </td><td> </td><td> </td><td>0</td></tr> <tr align=center><td>-14</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>-16</td><td> </td><td> </td><td> </td><td bgcolor=yellow>5</td><td bgcolor=yellow>8</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td>-3</td></tr> <tr align=center><td>-18</td><td> </td><td> </td><td bgcolor=yellow>2</td><td bgcolor=yellow>7</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>-20</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>-22</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>-24</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)

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

[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.

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