L11a19

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Contents

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

Image:L11a19.gif

(Knotscape image)

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

Visit L11a19's page at Knotilus.

Visit L11a19's page at the original Knot Atlas.

[edit L11a19 Quick Notes]

[edit L11a19 Further Notes and Views]

[edit] Link Presentations

[edit Notes on L11a19's Link Presentations]

Planar diagram presentation	X₆₁₇₂ X_10,4,11,3 X_12,8,13,7 X_16,10,17,9 X_20,14,21,13 X_22,17,5,18 X_18,21,19,22 X_14,20,15,19 X_8,16,9,15 X₂₅₃₆ X_4,12,1,11
Gauss code	{1, -10, 2, -11}, {10, -1, 3, -9, 4, -2, 11, -3, 5, -8, 9, -4, 6, -7, 8, -5, 7, -6}

A Braid Representative

A Morse Link Presentation

Image:L11a19_ML.gif

[edit] Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$\frac{(t(1)-1) (t(2)-1) \left(t(2)^4-4 t(2)^3+4 t(2)^2-4 t(2)+1\right)}{\sqrt{t(1)} t(2)^{5/2}}$ (db)
Jones polynomial	$16 q^{9/2}-18 q^{7/2}+17 q^{5/2}-\frac{1}{q^{5/2}}-16 q^{3/2}+\frac{3}{q^{3/2}}+q^{17/2}-3 q^{15/2}+7 q^{13/2}-12 q^{11/2}+11 \sqrt{q}-\frac{7}{\sqrt{q}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$z 3 a -7 + 2 z a -7 + a -7 z -1 -2 z 5 a -5 -6 z 3 a -5 -7 z a -5 -3 a -5 z -1 + z 7 a -3 + 4 z 5 a -3 + 8 z 3 a -3 + 9 z a -3 + 3 a -3 z -1 -2 z 5 a -1 + a z 3 -6 z 3 a -1 + 2 a z -6 z a -1 + a z -1 -2 a -1 z -1$ (db)
Kauffman polynomial	$- z 10 a -2 - z 10 a -4 -3 z 9 a -1 -8 z 9 a -3 -5 z 9 a -5 -10 z 8 a -2 -16 z 8 a -4 -9 z 8 a -6 -3 z 8 - a z 7 + 4 z 7 a -1 + 10 z 7 a -3 -4 z 7 a -5 -9 z 7 a -7 + 41 z 6 a -2 + 49 z 6 a -4 + 13 z 6 a -6 -6 z 6 a -8 + 11 z 6 + 4 a z 5 + 13 z 5 a -1 + 25 z 5 a -3 + 33 z 5 a -5 + 14 z 5 a -7 -3 z 5 a -9 -42 z 4 a -2 -40 z 4 a -4 -4 z 4 a -6 + 6 z 4 a -8 - z 4 a -10 -13 z 4 -6 a z 3 -25 z 3 a -1 -40 z 3 a -3 -36 z 3 a -5 -13 z 3 a -7 + 2 z 3 a -9 + 15 z 2 a -2 + 11 z 2 a -4 -4 z 2 a -6 -4 z 2 a -8 + z 2 a -10 + 5 z 2 + 4 a z + 13 z a -1 + 20 z a -3 + 16 z a -5 + 5 z a -7 -2 a -2 + 2 a -6 + a -8 - a z -1 -2 a -1 z -1 -3 a -3 z -1 -3 a -5 z -1 - a -7 z -1$ (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 L11a19. 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%>-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=6.25%>6</td><td width=6.25%>7</td><td width=12.5%>χ</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> </td><td bgcolor=yellow>1</td><td>-1</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> </td><td bgcolor=yellow>2</td><td bgcolor=yellow> </td><td>2</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> </td><td bgcolor=yellow>5</td><td bgcolor=yellow>1</td><td> </td><td>-4</td></tr> <tr align=center><td>12</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>10</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>8</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td bgcolor=yellow>9</td><td bgcolor=yellow>7</td><td> </td><td> </td><td> </td><td> </td><td>2</td></tr> <tr align=center><td>6</td><td> </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>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>2</td><td> </td><td> </td><td> </td><td bgcolor=yellow>5</td><td bgcolor=yellow>10</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td>5</td></tr> <tr align=center><td>0</td><td> </td><td> </td><td bgcolor=yellow>2</td><td bgcolor=yellow>6</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>-2</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>-4</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>-6</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)

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

[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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