L11a303

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Contents

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

Image:L11a303.gif

(Knotscape image)

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

Visit L11a303's page at Knotilus.

Visit L11a303's page at the original Knot Atlas.

[edit L11a303 Quick Notes]

[edit L11a303 Further Notes and Views]

[edit] Link Presentations

[edit Notes on L11a303's Link Presentations]

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

A Braid Representative

A Morse Link Presentation

Image:L11a303_ML.gif

[edit] Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-\frac{u^3 v^5-2 u^3 v^4+2 u^3 v^3-u^3 v^2-u^2 v^5+5 u^2 v^4-8 u^2 v^3+7 u^2 v^2-2 u^2 v-2 u v^4+7 u v^3-8 u v^2+5 u v-u-v^3+2 v^2-2 v+1}{u^{3/2} v^{5/2}}$ (db)
Jones polynomial	$q^{9/2}-3 q^{7/2}+7 q^{5/2}-12 q^{3/2}+16 \sqrt{q}-\frac{19}{\sqrt{q}}+\frac{18}{q^{3/2}}-\frac{17}{q^{5/2}}+\frac{12}{q^{7/2}}-\frac{7}{q^{9/2}}+\frac{3}{q^{11/2}}-\frac{1}{q^{13/2}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a 3 z 7 + 5 a 3 z 5 + 9 a 3 z 3 + 6 a 3 z + 2 a 3 z -1 - a z 9 -7 a z 7 + z 7 a -1 -19 a z 5 + 5 z 5 a -1 -24 a z 3 + 9 z 3 a -1 -14 a z + 6 z a -1 -3 a z -1 + a -1 z -1$ (db)
Kauffman polynomial	$-2 a 2 z 10 -2 z 10 -5 a 3 z 9 -10 a z 9 -5 z 9 a -1 -6 a 4 z 8 -5 a 2 z 8 -5 z 8 a -2 -4 z 8 -5 a 5 z 7 + 6 a 3 z 7 + 26 a z 7 + 12 z 7 a -1 -3 z 7 a -3 -3 a 6 z 6 + 8 a 4 z 6 + 14 a 2 z 6 + 13 z 6 a -2 - z 6 a -4 + 17 z 6 - a 7 z 5 + 7 a 5 z 5 -7 a 3 z 5 -38 a z 5 -15 z 5 a -1 + 8 z 5 a -3 + 5 a 6 z 4 -4 a 4 z 4 -15 a 2 z 4 -11 z 4 a -2 + 3 z 4 a -4 -20 z 4 + 2 a 7 z 3 -3 a 5 z 3 + 9 a 3 z 3 + 35 a z 3 + 16 z 3 a -1 -5 z 3 a -3 -2 a 6 z 2 + 9 a 2 z 2 + 5 z 2 a -2 -2 z 2 a -4 + 14 z 2 - a 7 z + a 5 z -7 a 3 z -16 a z -7 z a -1 -3 a 2 - a -2 -3 + 2 a 3 z -1 + 3 a z -1 + a -1 z -1$ (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 =

-1 is the signature of L11a303. 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%>-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=6.25%>5</td><td width=12.5%>χ</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> </td><td> </td><td> </td><td bgcolor=yellow>1</td><td>-1</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> </td><td bgcolor=yellow>2</td><td bgcolor=yellow> </td><td>2</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 bgcolor=yellow>5</td><td bgcolor=yellow>1</td><td> </td><td>-4</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 bgcolor=yellow>7</td><td bgcolor=yellow>2</td><td> </td><td> </td><td>5</td></tr> <tr align=center><td>2</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td bgcolor=yellow>9</td><td bgcolor=yellow>5</td><td> </td><td> </td><td> </td><td>-4</td></tr> <tr align=center><td>0</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td bgcolor=yellow>10</td><td bgcolor=yellow>7</td><td> </td><td> </td><td> </td><td> </td><td>3</td></tr> <tr align=center><td>-2</td><td> </td><td> </td><td> </td><td> </td><td> </td><td bgcolor=yellow>9</td><td bgcolor=yellow>10</td><td> </td><td> </td><td> </td><td> </td><td> </td><td>1</td></tr> <tr align=center><td>-4</td><td> </td><td> </td><td> </td><td> </td><td bgcolor=yellow>8</td><td bgcolor=yellow>9</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>9</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td>5</td></tr> <tr align=center><td>-8</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>-10</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>-12</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>-14</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)

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

[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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