L11a3

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Contents

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

Image:L11a3.gif

(Knotscape image)

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

Visit L11a3's page at Knotilus.

Visit L11a3's page at the original Knot Atlas.

[edit L11a3 Quick Notes]

[edit L11a3 Further Notes and Views]

[edit] Link Presentations

[edit Notes on L11a3's Link Presentations]

Planar diagram presentation	X₆₁₇₂ X_16,7,17,8 X_4,17,1,18 X_14,10,15,9 X₈₄₉₃ X_10,5,11,6 X_20,12,21,11 X_22,20,5,19 X_18,13,19,14 X_12,22,13,21 X_2,16,3,15
Gauss code	{1, -11, 5, -3}, {6, -1, 2, -5, 4, -6, 7, -10, 9, -4, 11, -2, 3, -9, 8, -7, 10, -8}

A Braid Representative

A Morse Link Presentation

Image:L11a3_ML.gif

[edit] Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-\frac{(t(1)-1) (t(2)-1) \left(t(2)^4-5 t(2)^3+7 t(2)^2-5 t(2)+1\right)}{\sqrt{t(1)} t(2)^{5/2}}$ (db)
Jones polynomial	$-q^{13/2}+5 q^{11/2}-10 q^{9/2}+16 q^{7/2}-22 q^{5/2}+24 q^{3/2}-25 \sqrt{q}+\frac{20}{\sqrt{q}}-\frac{15}{q^{3/2}}+\frac{9}{q^{5/2}}-\frac{4}{q^{7/2}}+\frac{1}{q^{9/2}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$- z 3 a -5 + a -5 z -1 + 2 z 5 a -3 - a 3 z 3 + 3 z 3 a -3 - a 3 z - z a -3 -3 a -3 z -1 - z 7 a -1 + 2 a z 5 -3 z 5 a -1 + 4 a z 3 -4 z 3 a -1 + 2 a z + 2 a -1 z -1$ (db)
Kauffman polynomial	$-2 z 10 a -2 -2 z 10 -6 a z 9 -13 z 9 a -1 -7 z 9 a -3 -7 a 2 z 8 -19 z 8 a -2 -11 z 8 a -4 -15 z 8 -4 a 3 z 7 + 4 a z 7 + 13 z 7 a -1 -5 z 7 a -3 -10 z 7 a -5 - a 4 z 6 + 15 a 2 z 6 + 43 z 6 a -2 + 12 z 6 a -4 -5 z 6 a -6 + 42 z 6 + 9 a 3 z 5 + 14 a z 5 + 16 z 5 a -1 + 26 z 5 a -3 + 14 z 5 a -5 - z 5 a -7 + 2 a 4 z 4 -9 a 2 z 4 -27 z 4 a -2 - z 4 a -4 + 5 z 4 a -6 -32 z 4 -6 a 3 z 3 -16 a z 3 -20 z 3 a -1 -15 z 3 a -3 -5 z 3 a -5 - a 4 z 2 + 2 a 2 z 2 + 7 z 2 a -2 + 3 z 2 a -4 + 7 z 2 + 2 a 3 z + 4 a z + z a -1 - z a -3 -3 a -2 -3 a -4 - a -6 + 2 a -1 z -1 + 3 a -3 z -1 + a -5 z -1$ (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 =

1 is the signature of L11a3. 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%>-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=6.25%>6</td><td width=12.5%>χ</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> </td><td> </td><td bgcolor=yellow>1</td><td>1</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> </td><td> </td><td bgcolor=yellow>4</td><td bgcolor=yellow> </td><td>-4</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 bgcolor=yellow>6</td><td bgcolor=yellow>1</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> </td><td> </td><td bgcolor=yellow>10</td><td bgcolor=yellow>4</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> </td><td bgcolor=yellow>12</td><td bgcolor=yellow>6</td><td> </td><td> </td><td> </td><td>6</td></tr> <tr align=center><td>4</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td bgcolor=yellow>12</td><td bgcolor=yellow>10</td><td> </td><td> </td><td> </td><td> </td><td>-2</td></tr> <tr align=center><td>2</td><td> </td><td> </td><td> </td><td> </td><td> </td><td bgcolor=yellow>13</td><td bgcolor=yellow>12</td><td> </td><td> </td><td> </td><td> </td><td> </td><td>1</td></tr> <tr align=center><td>0</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>-2</td><td> </td><td> </td><td> </td><td bgcolor=yellow>6</td><td bgcolor=yellow>11</td><td> </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 bgcolor=yellow>3</td><td bgcolor=yellow>9</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>-6</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>-8</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>-10</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)

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = 0$	$i = 2$
$r = -5$	${\mathbb Z}$
$r = -4$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -3$	${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = -2$	${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{6}$	${\mathbb Z}^{6}$
$r = -1$	${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{9}$	${\mathbb Z}^{9}$
$r = 0$	${\mathbb Z}^{14}\oplus{\mathbb Z}_2^{11}$	${\mathbb Z}^{13}$
$r = 1$	${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{12}$	${\mathbb Z}^{12}$
$r = 2$	${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{12}$	${\mathbb Z}^{12}$
$r = 3$	${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{10}$	${\mathbb Z}^{10}$
$r = 4$	${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6}$	${\mathbb Z}^{6}$
$r = 5$	${\mathbb Z}\oplus{\mathbb Z}_2^{4}$	${\mathbb Z}^{4}$
$r = 6$	${\mathbb Z}_2$	${\mathbb Z}$

[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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