L11n255

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Contents

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

Image:L11n255.gif

(Knotscape image)

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

Visit L11n255's page at Knotilus.

Visit L11n255's page at the original Knot Atlas.

[edit L11n255 Quick Notes]

[edit L11n255 Further Notes and Views]

[edit] Link Presentations

[edit Notes on L11n255's Link Presentations]

Planar diagram presentation	X₆₁₇₂ X_3,11,4,10 X_11,16,12,17 X_21,18,22,19 X_13,20,14,21 X_19,12,20,13 X_17,22,18,9 X_15,8,16,5 X_7,14,8,15 X₂₅₃₆ X_9,1,10,4
Gauss code	{1, -10, -2, 11}, {10, -1, -9, 8}, {-11, 2, -3, 6, -5, 9, -8, 3, -7, 4, -6, 5, -4, 7}

A Braid Representative

A Morse Link Presentation

Image:L11n255_ML.gif

[edit] Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-\frac{2 t(1) t(2) t(3)^3-2 t(2) t(3)^3+2 t(3)^3+t(1) t(3)^2-4 t(1) t(2) t(3)^2+2 t(2) t(3)^2-4 t(3)^2-2 t(1) t(3)+4 t(1) t(2) t(3)-t(2) t(3)+4 t(3)+2 t(1)-2 t(1) t(2)-2}{\sqrt{t(1)} \sqrt{t(2)} t(3)^{3/2}}$ (db)
Jones polynomial	$-3 q -9 + 6 q -8 -9 q -7 + 11 q -6 -11 q -5 + 12 q -4 -7 q -3 + 6 q -2 -2 q -1 + 1$ (db)
Signature	-4 (db)
HOMFLY-PT polynomial	$- a 10 z -2 - a 10 + z 4 a 8 + 3 z 2 a 8 + 3 a 8 z -2 + 5 a 8 - z 6 a 6 -3 z 4 a 6 -4 z 2 a 6 -2 a 6 z -2 -5 a 6 - z 6 a 4 -3 z 4 a 4 -3 z 2 a 4 - a 4 z -2 -2 a 4 + z 4 a 2 + 3 z 2 a 2 + a 2 z -2 + 3 a 2$ (db)
Kauffman polynomial	$6 z 3 a 11 -7 z a 11 + 2 a 11 z -1 + 3 z 6 a 10 -2 z 2 a 10 - a 10 z -2 + a 10 + 7 z 7 a 9 -20 z 5 a 9 + 35 z 3 a 9 -27 z a 9 + 8 a 9 z -1 + 5 z 8 a 8 -10 z 6 a 8 + 13 z 4 a 8 -8 z 2 a 8 -3 a 8 z -2 + 5 a 8 + z 9 a 7 + 10 z 7 a 7 -34 z 5 a 7 + 46 z 3 a 7 -34 z a 7 + 10 a 7 z -1 + 7 z 8 a 6 -15 z 6 a 6 + 8 z 4 a 6 -3 z 2 a 6 -2 a 6 z -2 + 4 a 6 + z 9 a 5 + 5 z 7 a 5 -19 z 5 a 5 + 18 z 3 a 5 -10 z a 5 + 2 a 5 z -1 + 2 z 8 a 4 - z 6 a 4 -9 z 4 a 4 + 9 z 2 a 4 + a 4 z -2 -3 a 4 + 2 z 7 a 3 -5 z 5 a 3 + z 3 a 3 + 4 z a 3 -2 a 3 z -1 + z 6 a 2 -4 z 4 a 2 + 6 z 2 a 2 + a 2 z -2 -4 a 2$ (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 =

-4 is the signature of L11n255. 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%>-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=7.14286%>2</td><td width=14.2857%>χ</td></tr> <tr align=center><td>1</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>-1</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td bgcolor=yellow>1</td><td bgcolor=yellow> </td><td>-1</td></tr> <tr align=center><td>-3</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>-5</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td bgcolor=yellow>5</td><td bgcolor=yellow>4</td><td> </td><td> </td><td>-1</td></tr> <tr align=center><td>-7</td><td> </td><td> </td><td> </td><td> </td><td> </td><td bgcolor=yellow>7</td><td bgcolor=yellow>2</td><td> </td><td> </td><td> </td><td>5</td></tr> <tr align=center><td>-9</td><td> </td><td> </td><td> </td><td> </td><td bgcolor=yellow>4</td><td bgcolor=yellow>5</td><td> </td><td> </td><td> </td><td> </td><td>1</td></tr> <tr align=center><td>-11</td><td> </td><td> </td><td> </td><td bgcolor=yellow>7</td><td bgcolor=yellow>7</td><td> </td><td> </td><td> </td><td> </td><td> </td><td>0</td></tr> <tr align=center><td>-13</td><td> </td><td> </td><td bgcolor=yellow>3</td><td bgcolor=yellow>5</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td>2</td></tr> <tr align=center><td>-15</td><td> </td><td bgcolor=yellow>3</td><td bgcolor=yellow>6</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td>-3</td></tr> <tr align=center><td>-17</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>3</td></tr> <tr align=center><td>-19</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>-3</td></tr> </table>

Integral Khovanov Homology

(db, data source)

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

[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/L11n255"

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