L11n161

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Contents

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

Image:L11n161.gif

(Knotscape image)

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

Visit L11n161's page at Knotilus.

Visit L11n161's page at the original Knot Atlas.

[edit L11n161 Quick Notes]

[edit L11n161 Further Notes and Views]

[edit] Link Presentations

[edit Notes on L11n161's Link Presentations]

Planar diagram presentation	X₈₁₉₂ X_16,7,17,8 X_10,4,11,3 X_2,15,3,16 X_14,10,15,9 X_18,11,19,12 X_5,13,6,12 X_6,21,1,22 X_13,20,14,21 X_22,17,7,18 X_19,4,20,5
Gauss code	{1, -4, 3, 11, -7, -8}, {2, -1, 5, -3, 6, 7, -9, -5, 4, -2, 10, -6, -11, 9, 8, -10}

A Braid Representative

A Morse Link Presentation

Image:L11n161_ML.gif

[edit] Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-\frac{2 t(2)^2 t(1)^2-3 t(2) t(1)^2+t(1)^2-2 t(2)^2 t(1)+5 t(2) t(1)-2 t(1)+t(2)^2-3 t(2)+2}{t(1) t(2)}$ (db)
Jones polynomial	$-\frac{7}{q^{9/2}}+\frac{6}{q^{7/2}}-\frac{7}{q^{5/2}}+\frac{4}{q^{3/2}}-\frac{1}{q^{17/2}}+\frac{3}{q^{15/2}}-\frac{4}{q^{13/2}}+\frac{6}{q^{11/2}}+\sqrt{q}-\frac{3}{\sqrt{q}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a 7 z 3 + a 7 z - a 7 z -1 - a 5 z 5 -2 a 5 z 3 + a 5 z + 3 a 5 z -1 - a 3 z 5 -3 a 3 z 3 -4 a 3 z -2 a 3 z -1 + a z 3 + a z$ (db)
Kauffman polynomial	$a 9 z 7 -4 a 9 z 5 + 4 a 9 z 3 - a 9 z + 3 a 8 z 8 -14 a 8 z 6 + 18 a 8 z 4 -6 a 8 z 2 - a 8 + 2 a 7 z 9 -6 a 7 z 7 - a 7 z 5 + 7 a 7 z 3 -2 a 7 z + a 7 z -1 + 7 a 6 z 8 -30 a 6 z 6 + 34 a 6 z 4 -9 a 6 z 2 -3 a 6 + 2 a 5 z 9 -4 a 5 z 7 -6 a 5 z 5 + 10 a 5 z 3 -5 a 5 z + 3 a 5 z -1 + 4 a 4 z 8 -15 a 4 z 6 + 16 a 4 z 4 -4 a 4 z 2 -3 a 4 + 3 a 3 z 7 -9 a 3 z 5 + 10 a 3 z 3 -6 a 3 z + 2 a 3 z -1 + a 2 z 6 + 3 a z 3 -2 a z + z 2$ (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 L11n161. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

<table border=1> <tr align=center> <td width=14.2857%><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=7.14286%>-8</td><td width=7.14286%>-7</td><td width=7.14286%>-6</td><td width=7.14286%>-5</td><td width=7.14286%>-4</td><td width=7.14286%>-3</td><td width=7.14286%>-2</td><td width=7.14286%>-1</td><td width=7.14286%>0</td><td width=7.14286%>1</td><td width=14.2857%>χ</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 bgcolor=yellow>1</td><td>-1</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 bgcolor=yellow>2</td><td bgcolor=yellow> </td><td>2</td></tr> <tr align=center><td>-2</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td bgcolor=yellow>3</td><td bgcolor=yellow>2</td><td> </td><td>-1</td></tr> <tr align=center><td>-4</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> </td><td>3</td></tr> <tr align=center><td>-6</td><td> </td><td> </td><td> </td><td> </td><td> </td><td bgcolor=yellow>3</td><td bgcolor=yellow>4</td><td> </td><td> </td><td> </td><td>1</td></tr> <tr align=center><td>-8</td><td> </td><td> </td><td> </td><td> </td><td bgcolor=yellow>4</td><td bgcolor=yellow>3</td><td> </td><td> </td><td> </td><td> </td><td>1</td></tr> <tr align=center><td>-10</td><td> </td><td> </td><td> </td><td bgcolor=yellow>2</td><td bgcolor=yellow>3</td><td> </td><td> </td><td> </td><td> </td><td> </td><td>1</td></tr> <tr align=center><td>-12</td><td> </td><td> </td><td bgcolor=yellow>2</td><td bgcolor=yellow>4</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td>-2</td></tr> <tr align=center><td>-14</td><td> </td><td bgcolor=yellow>1</td><td bgcolor=yellow>2</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td>1</td></tr> <tr align=center><td>-16</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>-2</td></tr> <tr align=center><td>-18</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>1</td></tr> </table>

Integral Khovanov Homology

(db, data source)

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

[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/L11n161"

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