L11a8

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Contents

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

Image:L11a8.gif

(Knotscape image)

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

Visit L11a8's page at Knotilus.

Visit L11a8's page at the original Knot Atlas.

[edit L11a8 Quick Notes]

[edit L11a8 Further Notes and Views]

[edit] Link Presentations

[edit Notes on L11a8's Link Presentations]

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

A Braid Representative

A Morse Link Presentation

Image:L11a8_ML.gif

[edit] Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$\frac{(t(1)-1) (t(2)-1) \left(t(2)^4-5 t(2)^3+9 t(2)^2-5 t(2)+1\right)}{\sqrt{t(1)} t(2)^{5/2}}$ (db)
Jones polynomial	$-q^{7/2}+5 q^{5/2}-11 q^{3/2}+17 \sqrt{q}-\frac{25}{\sqrt{q}}+\frac{27}{q^{3/2}}-\frac{27}{q^{5/2}}+\frac{23}{q^{7/2}}-\frac{17}{q^{9/2}}+\frac{10}{q^{11/2}}-\frac{4}{q^{13/2}}+\frac{1}{q^{15/2}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$- z a 7 + 3 z 3 a 5 + 2 z a 5 - a 5 z -1 -3 z 5 a 3 -5 z 3 a 3 + 3 a 3 z -1 + z 7 a + 3 z 5 a + 4 z 3 a - z a -2 a z -1 - z 5 a -1 - z 3 a -1$ (db)
Kauffman polynomial	$a 8 z 6 -2 a 8 z 4 + a 8 z 2 + 4 a 7 z 7 -8 a 7 z 5 + 6 a 7 z 3 -2 a 7 z + 8 a 6 z 8 -17 a 6 z 6 + 14 a 6 z 4 -6 a 6 z 2 + a 6 + 7 a 5 z 9 -3 a 5 z 7 -18 a 5 z 5 + 18 a 5 z 3 -4 a 5 z - a 5 z -1 + 2 a 4 z 10 + 22 a 4 z 8 -59 a 4 z 6 + 47 a 4 z 4 -16 a 4 z 2 + 3 a 4 + 15 a 3 z 9 -8 a 3 z 7 -35 a 3 z 5 + z 5 a -3 + 30 a 3 z 3 - a 3 z -3 a 3 z -1 + 2 a 2 z 10 + 27 a 2 z 8 -61 a 2 z 6 + 5 z 6 a -2 + 39 a 2 z 4 -4 z 4 a -2 -11 a 2 z 2 + 3 a 2 + 8 a z 9 + 10 a z 7 + 11 z 7 a -1 -41 a z 5 -15 z 5 a -1 + 25 a z 3 + 7 z 3 a -1 + a z -2 a z -1 + 13 z 8 -15 z 6 + 4 z 4 -2 z 2$ (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 =

-1 is the signature of L11a8. 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%>-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=6.25%>4</td><td width=12.5%>χ</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> </td><td> </td><td> </td><td bgcolor=yellow>1</td><td>1</td></tr> <tr align=center><td>6</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>4</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>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>4</td><td> </td><td> </td><td>-6</td></tr> <tr align=center><td>0</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td bgcolor=yellow>15</td><td bgcolor=yellow>7</td><td> </td><td> </td><td> </td><td>8</td></tr> <tr align=center><td>-2</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td bgcolor=yellow>14</td><td bgcolor=yellow>12</td><td> </td><td> </td><td> </td><td> </td><td>-2</td></tr> <tr align=center><td>-4</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>-6</td><td> </td><td> </td><td> </td><td> </td><td bgcolor=yellow>10</td><td bgcolor=yellow>14</td><td> </td><td> </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 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>-10</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>-12</td><td> </td><td bgcolor=yellow>1</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>-6</td></tr> <tr align=center><td>-14</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>-16</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)

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = -2$	$i = 0$
$r = -7$	${\mathbb Z}$
$r = -6$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -5$	${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = -4$	${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{7}$	${\mathbb Z}^{7}$
$r = -3$	${\mathbb Z}^{13}\oplus{\mathbb Z}_2^{10}$	${\mathbb Z}^{10}$
$r = -2$	${\mathbb Z}^{14}\oplus{\mathbb Z}_2^{13}$	${\mathbb Z}^{13}$
$r = -1$	${\mathbb Z}^{13}\oplus{\mathbb Z}_2^{14}$	${\mathbb Z}^{14}$
$r = 0$	${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{13}$	${\mathbb Z}^{15}$
$r = 1$	${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{10}$	${\mathbb Z}^{10}$
$r = 2$	${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{7}$	${\mathbb Z}^{7}$
$r = 3$	${\mathbb Z}\oplus{\mathbb Z}_2^{4}$	${\mathbb Z}^{4}$
$r = 4$	${\mathbb Z}_2$	${\mathbb Z}$

[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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