L11a344

From Knot Atlas

Jump to: navigation, search

Contents

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

Image:L11a344.gif

(Knotscape image)

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

Visit L11a344's page at Knotilus.

Visit L11a344's page at the original Knot Atlas.

[edit L11a344 Quick Notes]

[edit L11a344 Further Notes and Views]

[edit] Link Presentations

[edit Notes on L11a344's Link Presentations]

Planar diagram presentation	X_12,1,13,2 X_18,7,19,8 X_10,5,1,6 X₆₃₇₄ X_4,9,5,10 X_20,14,21,13 X_22,16,11,15 X_14,22,15,21 X_16,20,17,19 X_2,11,3,12 X_8,17,9,18
Gauss code	{1, -10, 4, -5, 3, -4, 2, -11, 5, -3}, {10, -1, 6, -8, 7, -9, 11, -2, 9, -6, 8, -7}

A Braid Representative

A Morse Link Presentation

Image:L11a344_ML.gif

[edit] Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$\frac{t(2)^3 t(1)^3-2 t(2)^2 t(1)^3+2 t(2) t(1)^3-2 t(1)^3-4 t(2)^3 t(1)^2+6 t(2)^2 t(1)^2-6 t(2) t(1)^2+4 t(1)^2+4 t(2)^3 t(1)-6 t(2)^2 t(1)+6 t(2) t(1)-4 t(1)-2 t(2)^3+2 t(2)^2-2 t(2)+1}{t(1)^{3/2} t(2)^{3/2}}$ (db)
Jones polynomial	$-\frac{10}{q^{9/2}}-q^{7/2}+\frac{13}{q^{7/2}}+4 q^{5/2}-\frac{18}{q^{5/2}}-8 q^{3/2}+\frac{17}{q^{3/2}}+\frac{1}{q^{15/2}}-\frac{2}{q^{13/2}}+\frac{5}{q^{11/2}}+13 \sqrt{q}-\frac{16}{\sqrt{q}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$- z a 7 -2 a 7 z -1 + 3 z 3 a 5 + 9 z a 5 + 7 a 5 z -1 -3 z 5 a 3 -11 z 3 a 3 -15 z a 3 -7 a 3 z -1 + z 7 a + 4 z 5 a + 7 z 3 a + 6 z a + 2 a z -1 - z 5 a -1 -2 z 3 a -1 - z a -1$ (db)
Kauffman polynomial	$a 8 z 6 -4 a 8 z 4 + 5 a 8 z 2 -2 a 8 + 2 a 7 z 7 -6 a 7 z 5 + 6 a 7 z 3 -4 a 7 z + 2 a 7 z -1 + 2 a 6 z 8 -13 a 6 z 4 + 18 a 6 z 2 -8 a 6 + 2 a 5 z 9 + a 5 z 7 -11 a 5 z 5 + 17 a 5 z 3 -16 a 5 z + 7 a 5 z -1 + a 4 z 10 + 4 a 4 z 8 -5 a 4 z 6 -11 a 4 z 4 + 24 a 4 z 2 -13 a 4 + 6 a 3 z 9 -6 a 3 z 7 -5 a 3 z 5 + z 5 a -3 + 15 a 3 z 3 - z 3 a -3 -15 a 3 z + 7 a 3 z -1 + a 2 z 10 + 9 a 2 z 8 -16 a 2 z 6 + 4 z 6 a -2 + 2 a 2 z 4 -6 z 4 a -2 + 13 a 2 z 2 + 2 z 2 a -2 -8 a 2 + 4 a z 9 + 2 a z 7 + 7 z 7 a -1 -12 a z 5 -11 z 5 a -1 + 9 a z 3 + 4 z 3 a -1 -4 a z - z a -1 + 2 a z -1 + 7 z 8 -8 z 6 -2 z 4 + 4 z 2 -2$ (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 L11a344. 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>3</td><td bgcolor=yellow> </td><td>-3</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>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>0</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td bgcolor=yellow>8</td><td bgcolor=yellow>5</td><td> </td><td> </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 bgcolor=yellow>10</td><td bgcolor=yellow>9</td><td> </td><td> </td><td> </td><td> </td><td>-1</td></tr> <tr align=center><td>-4</td><td> </td><td> </td><td> </td><td> </td><td> </td><td bgcolor=yellow>8</td><td bgcolor=yellow>7</td><td> </td><td> </td><td> </td><td> </td><td> </td><td>1</td></tr> <tr align=center><td>-6</td><td> </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>5</td></tr> <tr align=center><td>-8</td><td> </td><td> </td><td> </td><td bgcolor=yellow>5</td><td bgcolor=yellow>8</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> </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>5</td></tr> <tr align=center><td>-12</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>-14</td><td bgcolor=yellow> </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>1</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)

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

[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.

Retrieved from "http://katlas.math.toronto.edu/wiki/L11a344"

Category: Knot Page

Views

Personal tools

Log in / create account

Search

Toolbox

tour