L11a342

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Contents

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

Image:L11a342.gif

(Knotscape image)

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

Visit L11a342's page at Knotilus.

Visit L11a342's page at the original Knot Atlas.

[edit L11a342 Quick Notes]

[edit L11a342 Further Notes and Views]

[edit] Link Presentations

[edit Notes on L11a342's Link Presentations]

Planar diagram presentation	X_12,1,13,2 X_16,7,17,8 X_10,5,1,6 X₆₃₇₄ X_4,9,5,10 X_20,14,21,13 X_22,17,11,18 X_18,21,19,22 X_14,20,15,19 X_2,11,3,12 X_8,15,9,16
Gauss code	{1, -10, 4, -5, 3, -4, 2, -11, 5, -3}, {10, -1, 6, -9, 11, -2, 7, -8, 9, -6, 8, -7}

A Braid Representative

A Morse Link Presentation

Image:L11a342_ML.gif

[edit] Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$\frac{2 u^3 v^2-4 u^3 v+u^3+2 u^2 v^3-10 u^2 v^2+10 u^2 v-2 u^2-2 u v^3+10 u v^2-10 u v+2 u+v^3-4 v^2+2 v}{u^{3/2} v^{3/2}}$ (db)
Jones polynomial	$\frac{20}{q^{9/2}}-\frac{18}{q^{7/2}}+\frac{13}{q^{5/2}}-\frac{8}{q^{3/2}}+\frac{1}{q^{21/2}}-\frac{3}{q^{19/2}}+\frac{7}{q^{17/2}}-\frac{13}{q^{15/2}}+\frac{17}{q^{13/2}}-\frac{20}{q^{11/2}}-\sqrt{q}+\frac{3}{\sqrt{q}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$- a 11 z -1 + 4 a 9 z + 4 a 9 z -1 -6 a 7 z 3 -11 a 7 z -6 a 7 z -1 + 3 a 5 z 5 + 8 a 5 z 3 + 10 a 5 z + 5 a 5 z -1 + a 3 z 5 - a 3 z 3 -4 a 3 z -2 a 3 z -1 - a z 3 - a z$ (db)
Kauffman polynomial	$a 12 z 6 -3 a 12 z 4 + 3 a 12 z 2 - a 12 + 3 a 11 z 7 -8 a 11 z 5 + 8 a 11 z 3 -4 a 11 z + a 11 z -1 + 4 a 10 z 8 -5 a 10 z 6 -5 a 10 z 4 + 9 a 10 z 2 -3 a 10 + 3 a 9 z 9 + 5 a 9 z 7 -27 a 9 z 5 + 31 a 9 z 3 -18 a 9 z + 4 a 9 z -1 + a 8 z 10 + 11 a 8 z 8 -22 a 8 z 6 + 4 a 8 z 4 + 7 a 8 z 2 -3 a 8 + 7 a 7 z 9 + 4 a 7 z 7 -41 a 7 z 5 + 52 a 7 z 3 -29 a 7 z + 6 a 7 z -1 + a 6 z 10 + 14 a 6 z 8 -28 a 6 z 6 + 15 a 6 z 4 - a 6 + 4 a 5 z 9 + 8 a 5 z 7 -33 a 5 z 5 + 41 a 5 z 3 -23 a 5 z + 5 a 5 z -1 + 7 a 4 z 8 -9 a 4 z 6 + 5 a 4 z 4 - a 4 + 6 a 3 z 7 -10 a 3 z 5 + 10 a 3 z 3 -7 a 3 z + 2 a 3 z -1 + 3 a 2 z 6 -4 a 2 z 4 + a 2 z 2 + a z 5 -2 a z 3 + a z$ (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 =

-3 is the signature of L11a342. 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>2</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>0</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>-2</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>-4</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td bgcolor=yellow>8</td><td bgcolor=yellow>3</td><td> </td><td> </td><td>-5</td></tr> <tr align=center><td>-6</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td bgcolor=yellow>10</td><td bgcolor=yellow>5</td><td> </td><td> </td><td> </td><td>5</td></tr> <tr align=center><td>-8</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td bgcolor=yellow>10</td><td bgcolor=yellow>8</td><td> </td><td> </td><td> </td><td> </td><td>-2</td></tr> <tr align=center><td>-10</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>-12</td><td> </td><td> </td><td> </td><td> </td><td bgcolor=yellow>8</td><td bgcolor=yellow>11</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td>3</td></tr> <tr align=center><td>-14</td><td> </td><td> </td><td> </td><td bgcolor=yellow>5</td><td bgcolor=yellow>9</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td>-4</td></tr> <tr align=center><td>-16</td><td> </td><td> </td><td bgcolor=yellow>2</td><td bgcolor=yellow>8</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>-18</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>-20</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>-22</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)

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

[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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