L11a335

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Contents

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

Image:L11a335.gif

(Knotscape image)

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

Visit L11a335's page at Knotilus.

Visit L11a335's page at the original Knot Atlas.

[edit L11a335 Quick Notes]

[edit L11a335 Further Notes and Views]

[edit] Link Presentations

[edit Notes on L11a335's Link Presentations]

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

A Braid Representative

A Morse Link Presentation

Image:L11a335_ML.gif

[edit] Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-\frac{u^3 v^4-3 u^3 v^3+3 u^3 v^2+u^2 v^5-5 u^2 v^4+12 u^2 v^3-11 u^2 v^2+4 u^2 v+4 u v^4-11 u v^3+12 u v^2-5 u v+u+3 v^3-3 v^2+v}{u^{3/2} v^{5/2}}$ (db)
Jones polynomial	$\frac{26}{q^{9/2}}-\frac{26}{q^{7/2}}+\frac{22}{q^{5/2}}+q^{3/2}-\frac{16}{q^{3/2}}-\frac{1}{q^{19/2}}+\frac{4}{q^{17/2}}-\frac{11}{q^{15/2}}+\frac{17}{q^{13/2}}-\frac{23}{q^{11/2}}-4 \sqrt{q}+\frac{9}{\sqrt{q}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$z 5 a 7 + 2 z 3 a 7 + 3 z a 7 + 2 a 7 z -1 - z 7 a 5 -3 z 5 a 5 -5 z 3 a 5 -6 z a 5 -3 a 5 z -1 - z 7 a 3 -3 z 5 a 3 -4 z 3 a 3 -2 z a 3 + a 3 z -1 + z 5 a + 2 z 3 a + z a$ (db)
Kauffman polynomial	$- z 5 a 11 + z 3 a 11 -4 z 6 a 10 + 3 z 4 a 10 -10 z 7 a 9 + 16 z 5 a 9 -13 z 3 a 9 + 6 z a 9 -13 z 8 a 8 + 20 z 6 a 8 -10 z 4 a 8 + z 2 a 8 -10 z 9 a 7 + 9 z 7 a 7 + z 5 a 7 + 3 z 3 a 7 -6 z a 7 + 2 a 7 z -1 -3 z 10 a 6 -18 z 8 a 6 + 48 z 6 a 6 -39 z 4 a 6 + 15 z 2 a 6 -3 a 6 -17 z 9 a 5 + 31 z 7 a 5 -20 z 5 a 5 + 16 z 3 a 5 -12 z a 5 + 3 a 5 z -1 -3 z 10 a 4 -12 z 8 a 4 + 39 z 6 a 4 -36 z 4 a 4 + 17 z 2 a 4 -3 a 4 -7 z 9 a 3 + 8 z 7 a 3 + 5 z 5 a 3 -7 z 3 a 3 + z a 3 + a 3 z -1 -7 z 8 a 2 + 14 z 6 a 2 -8 z 4 a 2 + 2 z 2 a 2 - a 2 -4 z 7 a + 9 z 5 a -6 z 3 a + z a - z 6 + 2 z 4 - z 2$ (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 L11a335. 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%>-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=6.25%>3</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>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>3</td><td bgcolor=yellow> </td><td>3</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 bgcolor=yellow>6</td><td bgcolor=yellow>1</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> </td><td bgcolor=yellow>10</td><td bgcolor=yellow>3</td><td> </td><td> </td><td>7</td></tr> <tr align=center><td>-4</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>-6</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td bgcolor=yellow>13</td><td bgcolor=yellow>9</td><td> </td><td> </td><td> </td><td> </td><td>4</td></tr> <tr align=center><td>-8</td><td> </td><td> </td><td> </td><td> </td><td> </td><td bgcolor=yellow>13</td><td bgcolor=yellow>13</td><td> </td><td> </td><td> </td><td> </td><td> </td><td>0</td></tr> <tr align=center><td>-10</td><td> </td><td> </td><td> </td><td> </td><td bgcolor=yellow>10</td><td bgcolor=yellow>13</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td>-3</td></tr> <tr align=center><td>-12</td><td> </td><td> </td><td> </td><td bgcolor=yellow>7</td><td bgcolor=yellow>13</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td>6</td></tr> <tr align=center><td>-14</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>-16</td><td> </td><td bgcolor=yellow> </td><td bgcolor=yellow>7</td><td> </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>-18</td><td bgcolor=yellow>1</td><td bgcolor=yellow>4</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>-20</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)

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

[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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