L11a314

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Contents

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

Image:L11a314.gif

(Knotscape image)

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

Visit L11a314's page at Knotilus.

Visit L11a314's page at the original Knot Atlas.

[edit L11a314 Quick Notes]

[edit L11a314 Further Notes and Views]

[edit] Link Presentations

[edit Notes on L11a314's Link Presentations]

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

A Braid Representative

A Morse Link Presentation

Image:L11a314_ML.gif

[edit] Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$\frac{u^3 v^4-2 u^3 v^3+2 u^3 v^2-u^3 v+u^2 v^5-5 u^2 v^4+11 u^2 v^3-10 u^2 v^2+5 u^2 v-u^2-u v^5+5 u v^4-10 u v^3+11 u v^2-5 u v+u-v^4+2 v^3-2 v^2+v}{u^{3/2} v^{5/2}}$ (db)
Jones polynomial	$25 q^{9/2}-26 q^{7/2}+21 q^{5/2}-16 q^{3/2}+\frac{1}{q^{3/2}}-q^{19/2}+4 q^{17/2}-10 q^{15/2}+17 q^{13/2}-22 q^{11/2}+9 \sqrt{q}-\frac{4}{\sqrt{q}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$z 7 a -3 + z 7 a -5 - z 5 a -1 + 3 z 5 a -3 + 3 z 5 a -5 - z 5 a -7 -2 z 3 a -1 + 4 z 3 a -3 + 4 z 3 a -5 -2 z 3 a -7 - z a -1 + 3 z a -3 + 2 z a -5 -2 z a -7 + a -3 z -1 - a -5 z -1$ (db)
Kauffman polynomial	$-3 z 10 a -4 -3 z 10 a -6 -7 z 9 a -3 -16 z 9 a -5 -9 z 9 a -7 -7 z 8 a -2 -10 z 8 a -4 -15 z 8 a -6 -12 z 8 a -8 -4 z 7 a -1 + 9 z 7 a -3 + 28 z 7 a -5 + 6 z 7 a -7 -9 z 7 a -9 + 14 z 6 a -2 + 30 z 6 a -4 + 38 z 6 a -6 + 19 z 6 a -8 -4 z 6 a -10 - z 6 + 9 z 5 a -1 + 2 z 5 a -3 -14 z 5 a -5 + 7 z 5 a -7 + 13 z 5 a -9 - z 5 a -11 -7 z 4 a -2 -19 z 4 a -4 -27 z 4 a -6 -13 z 4 a -8 + 4 z 4 a -10 + 2 z 4 -6 z 3 a -1 -4 z 3 a -3 + 5 z 3 a -5 -6 z 3 a -7 -8 z 3 a -9 + z 3 a -11 + 3 z 2 a -4 + 8 z 2 a -6 + 5 z 2 a -8 - z 2 a -10 - z 2 + z a -1 + 2 z a -3 - z a -5 + 2 z a -9 + a -4 - a -3 z -1 - a -5 z -1$ (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 L11a314. 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%>-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=6.25%>6</td><td width=6.25%>7</td><td width=6.25%>8</td><td width=12.5%>χ</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> </td><td bgcolor=yellow>1</td><td>1</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> </td><td bgcolor=yellow>3</td><td bgcolor=yellow> </td><td>-3</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> </td><td bgcolor=yellow>7</td><td bgcolor=yellow>1</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> </td><td> </td><td bgcolor=yellow>10</td><td bgcolor=yellow>3</td><td> </td><td> </td><td>-7</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>12</td><td bgcolor=yellow>7</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>14</td><td bgcolor=yellow>11</td><td> </td><td> </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 bgcolor=yellow>12</td><td bgcolor=yellow>11</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> </td><td bgcolor=yellow>9</td><td bgcolor=yellow>14</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td>5</td></tr> <tr align=center><td>4</td><td> </td><td> </td><td> </td><td bgcolor=yellow>7</td><td bgcolor=yellow>12</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td>-5</td></tr> <tr align=center><td>2</td><td> </td><td> </td><td bgcolor=yellow>3</td><td bgcolor=yellow>10</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td>7</td></tr> <tr align=center><td>0</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>-2</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>-4</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)

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = 2$	$i = 4$
$r = -3$	${\mathbb Z}$
$r = -2$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -1$	${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = 0$	${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{6}$	${\mathbb Z}^{7}$
$r = 1$	${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{9}$	${\mathbb Z}^{9}$
$r = 2$	${\mathbb Z}^{14}\oplus{\mathbb Z}_2^{12}$	${\mathbb Z}^{12}$
$r = 3$	${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{14}$	${\mathbb Z}^{14}$
$r = 4$	${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{11}$	${\mathbb Z}^{12}$
$r = 5$	${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{10}$	${\mathbb Z}^{10}$
$r = 6$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{7}$	${\mathbb Z}^{7}$
$r = 7$	${\mathbb Z}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = 8$	${\mathbb Z}_2$	${\mathbb Z}$

[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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