L11a316

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Contents

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

Image:L11a316.gif

(Knotscape image)

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

Visit L11a316's page at Knotilus.

Visit L11a316's page at the original Knot Atlas.

[edit L11a316 Quick Notes]

[edit L11a316 Further Notes and Views]

[edit] Link Presentations

[edit Notes on L11a316's Link Presentations]

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

A Braid Representative

A Morse Link Presentation

Image:L11a316_ML.gif

[edit] Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$\frac{u^3 v^4-2 u^3 v^3+2 u^3 v^2-u^3 v+u^2 v^5-4 u^2 v^4+5 u^2 v^3-5 u^2 v^2+3 u^2 v-u^2-u v^5+3 u v^4-5 u v^3+5 u v^2-4 u v+u-v^4+2 v^3-2 v^2+v}{u^{3/2} v^{5/2}}$ (db)
Jones polynomial	$-\frac{14}{q^{9/2}}+\frac{10}{q^{7/2}}-\frac{7}{q^{5/2}}+\frac{3}{q^{3/2}}+\frac{1}{q^{23/2}}-\frac{3}{q^{21/2}}+\frac{6}{q^{19/2}}-\frac{10}{q^{17/2}}+\frac{14}{q^{15/2}}-\frac{16}{q^{13/2}}+\frac{15}{q^{11/2}}-\frac{1}{\sqrt{q}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$a^9 \left(-z^5\right)-3 a^9 z^3-2 a^9 z-a^9 z^{-1} +a^7 z^7+4 a^7 z^5+6 a^7 z^3+6 a^7 z+3 a^7 z^{-1} +a^5 z^7+3 a^5 z^5-4 a^5 z-2 a^5 z^{-1} -a^3 z^5-3 a^3 z^3-2 a^3 z$ (db)
Kauffman polynomial	$a 14 z 4 - a 14 z 2 + 3 a 13 z 5 -3 a 13 z 3 + 5 a 12 z 6 -5 a 12 z 4 + a 12 z 2 + 7 a 11 z 7 -11 a 11 z 5 + 8 a 11 z 3 -2 a 11 z + 8 a 10 z 8 -18 a 10 z 6 + 18 a 10 z 4 -6 a 10 z 2 + a 10 + 6 a 9 z 9 -12 a 9 z 7 + 6 a 9 z 5 + a 9 z 3 + a 9 z - a 9 z -1 + 2 a 8 z 10 + 6 a 8 z 8 -33 a 8 z 6 + 39 a 8 z 4 -18 a 8 z 2 + 3 a 8 + 10 a 7 z 9 -33 a 7 z 7 + 35 a 7 z 5 -22 a 7 z 3 + 11 a 7 z -3 a 7 z -1 + 2 a 6 z 10 + a 6 z 8 -21 a 6 z 6 + 26 a 6 z 4 -13 a 6 z 2 + 3 a 6 + 4 a 5 z 9 -13 a 5 z 7 + 11 a 5 z 5 -7 a 5 z 3 + 6 a 5 z -2 a 5 z -1 + 3 a 4 z 8 -11 a 4 z 6 + 11 a 4 z 4 -3 a 4 z 2 + a 3 z 7 -4 a 3 z 5 + 5 a 3 z 3 -2 a 3 z$ (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 =

-5 is the signature of L11a316. 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>0</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>2</td><td bgcolor=yellow> </td><td>-2</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>5</td><td bgcolor=yellow>1</td><td> </td><td>4</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 bgcolor=yellow>6</td><td bgcolor=yellow>3</td><td> </td><td> </td><td>-3</td></tr> <tr align=center><td>-8</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td bgcolor=yellow>8</td><td bgcolor=yellow>4</td><td> </td><td> </td><td> </td><td>4</td></tr> <tr align=center><td>-10</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td bgcolor=yellow>7</td><td bgcolor=yellow>6</td><td> </td><td> </td><td> </td><td> </td><td>-1</td></tr> <tr align=center><td>-12</td><td> </td><td> </td><td> </td><td> </td><td> </td><td bgcolor=yellow>9</td><td bgcolor=yellow>8</td><td> </td><td> </td><td> </td><td> </td><td> </td><td>1</td></tr> <tr align=center><td>-14</td><td> </td><td> </td><td> </td><td> </td><td bgcolor=yellow>6</td><td bgcolor=yellow>8</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td>2</td></tr> <tr align=center><td>-16</td><td> </td><td> </td><td> </td><td bgcolor=yellow>4</td><td bgcolor=yellow>8</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td>-4</td></tr> <tr align=center><td>-18</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>-20</td><td> </td><td bgcolor=yellow>1</td><td bgcolor=yellow>4</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>-22</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>-24</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)

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

[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.

Retrieved from "http://katlas.math.toronto.edu/wiki/L11a316"

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