L11a307

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Contents

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

Image:L11a307.gif

(Knotscape image)

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

Visit L11a307's page at Knotilus.

Visit L11a307's page at the original Knot Atlas.

[edit L11a307 Quick Notes]

[edit L11a307 Further Notes and Views]

[edit] Link Presentations

[edit Notes on L11a307's Link Presentations]

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

A Braid Representative

A Morse Link Presentation

Image:L11a307_ML.gif

[edit] Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-\frac{u^3 v^3-3 u^3 v^2+3 u^3 v-3 u^2 v^3+9 u^2 v^2-8 u^2 v+3 u^2+3 u v^3-8 u v^2+9 u v-3 u+3 v^2-3 v+1}{u^{3/2} v^{3/2}}$ (db)
Jones polynomial	$q^{3/2}-4 \sqrt{q}+\frac{7}{\sqrt{q}}-\frac{12}{q^{3/2}}+\frac{17}{q^{5/2}}-\frac{19}{q^{7/2}}+\frac{19}{q^{9/2}}-\frac{17}{q^{11/2}}+\frac{12}{q^{13/2}}-\frac{8}{q^{15/2}}+\frac{3}{q^{17/2}}-\frac{1}{q^{19/2}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$z a 9 + a 9 z -1 -3 z 3 a 7 -5 z a 7 - a 7 z -1 + 3 z 5 a 5 + 8 z 3 a 5 + 5 z a 5 - z 7 a 3 -4 z 5 a 3 -7 z 3 a 3 -5 z a 3 + z 5 a + 2 z 3 a$ (db)
Kauffman polynomial	$a 11 z 5 -2 a 11 z 3 + a 11 z + 3 a 10 z 6 -4 a 10 z 4 + a 10 z 2 + 6 a 9 z 7 -11 a 9 z 5 + 11 a 9 z 3 -7 a 9 z + a 9 z -1 + 6 a 8 z 8 -5 a 8 z 6 -3 a 8 z 4 + 4 a 8 z 2 - a 8 + 4 a 7 z 9 + 3 a 7 z 7 -14 a 7 z 5 + 12 a 7 z 3 -5 a 7 z + a 7 z -1 + a 6 z 10 + 13 a 6 z 8 -31 a 6 z 6 + 27 a 6 z 4 -9 a 6 z 2 + 8 a 5 z 9 -8 a 5 z 7 -8 a 5 z 5 + 10 a 5 z 3 - a 5 z + a 4 z 10 + 13 a 4 z 8 -40 a 4 z 6 + 41 a 4 z 4 -16 a 4 z 2 + 4 a 3 z 9 - a 3 z 7 -17 a 3 z 5 + 18 a 3 z 3 -4 a 3 z + 6 a 2 z 8 -16 a 2 z 6 + 13 a 2 z 4 -4 a 2 z 2 + 4 a z 7 -11 a z 5 + 7 a z 3 + z 6 -2 z 4$ (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 L11a307. 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>4</td><td bgcolor=yellow>1</td><td> </td><td>-3</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>8</td><td bgcolor=yellow>3</td><td> </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 bgcolor=yellow>10</td><td bgcolor=yellow>5</td><td> </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 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>-8</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>-10</td><td> </td><td> </td><td> </td><td> </td><td bgcolor=yellow>7</td><td bgcolor=yellow>9</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td>-2</td></tr> <tr align=center><td>-12</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>-14</td><td> </td><td> </td><td bgcolor=yellow>3</td><td bgcolor=yellow>7</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>-16</td><td> </td><td bgcolor=yellow> </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>5</td></tr> <tr align=center><td>-18</td><td bgcolor=yellow>1</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>-2</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)

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

[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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