L10n62

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Contents

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

Image:L10n62.gif

(Knotscape image)

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

Visit L10n62's page at Knotilus.

Visit L10n62's page at the original Knot Atlas.

[edit L10n62 Quick Notes]

[edit L10n62 Further Notes and Views]

[edit] Link Presentations

[edit Notes on L10n62's Link Presentations]

Planar diagram presentation	X_12,1,13,2 X_16,8,17,7 X_5,1,6,10 X₃₇₄₆ X_9,5,10,4 X_18,14,19,13 X_20,16,11,15 X_14,20,15,19 X_2,11,3,12 X_8,18,9,17
Gauss code	{1, -9, -4, 5, -3, 4, 2, -10, -5, 3}, {9, -1, 6, -8, 7, -2, 10, -6, 8, -7}

A Braid Representative

A Morse Link Presentation

Image:L10n62_ML.gif

[edit] Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-\frac{(t(1)-1) (t(2)-1) \left(t(2)^2 t(1)^2+t(1)^2-t(2) t(1)+t(2)^2+1\right)}{t(1)^{3/2} t(2)^{3/2}}$ (db)
Jones polynomial	$-4 q^{9/2}+3 q^{7/2}-4 q^{5/2}+q^{3/2}-q^{17/2}+3 q^{15/2}-3 q^{13/2}+4 q^{11/2}-\sqrt{q}$ (db)
Signature	5 (db)
HOMFLY-PT polynomial	$- z a -9 - a -9 z -1 + z 5 a -7 + 5 z 3 a -7 + 8 z a -7 + 5 a -7 z -1 - z 7 a -5 -6 z 5 a -5 -13 z 3 a -5 -15 z a -5 -8 a -5 z -1 + z 5 a -3 + 5 z 3 a -3 + 8 z a -3 + 4 a -3 z -1$ (db)
Kauffman polynomial	$- z 8 a -4 - z 8 a -6 - z 7 a -3 -5 z 7 a -5 -4 z 7 a -7 + 3 z 6 a -4 -2 z 6 a -6 -5 z 6 a -8 + 6 z 5 a -3 + 24 z 5 a -5 + 16 z 5 a -7 -2 z 5 a -9 + 3 z 4 a -4 + 23 z 4 a -6 + 20 z 4 a -8 -13 z 3 a -3 -35 z 3 a -5 -18 z 3 a -7 + 4 z 3 a -9 -13 z 2 a -4 -33 z 2 a -6 -23 z 2 a -8 -3 z 2 a -10 + 12 z a -3 + 23 z a -5 + 11 z a -7 - z a -9 - z a -11 + 8 a -4 + 14 a -6 + 9 a -8 + 2 a -10 -4 a -3 z -1 -8 a -5 z -1 -5 a -7 z -1 - a -9 z -1$ (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 L10n62. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

<table border=1> <tr align=center> <td width=15.3846%><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=7.69231%>-2</td><td width=7.69231%>-1</td><td width=7.69231%>0</td><td width=7.69231%>1</td><td width=7.69231%>2</td><td width=7.69231%>3</td><td width=7.69231%>4</td><td width=7.69231%>5</td><td width=7.69231%>6</td><td width=15.3846%>χ</td></tr> <tr align=center><td>18</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>16</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>14</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td bgcolor=yellow>2</td><td bgcolor=yellow>2</td><td> </td><td>0</td></tr> <tr align=center><td>12</td><td> </td><td> </td><td> </td><td> </td><td> </td><td bgcolor=yellow>3</td><td bgcolor=yellow>2</td><td> </td><td> </td><td>-1</td></tr> <tr align=center><td>10</td><td> </td><td> </td><td> </td><td> </td><td bgcolor=yellow>1</td><td bgcolor=yellow>2</td><td bgcolor=red>1</td><td> </td><td> </td><td>0</td></tr> <tr align=center><td>8</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>1</td></tr> <tr align=center><td>6</td><td> </td><td> </td><td bgcolor=yellow>2</td><td bgcolor=yellow>1</td><td> </td><td> </td><td> </td><td> </td><td> </td><td>1</td></tr> <tr align=center><td>4</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>3</td></tr> <tr align=center><td>2</td><td bgcolor=yellow> </td><td bgcolor=yellow> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td>0</td></tr> <tr align=center><td>0</td><td bgcolor=yellow>1</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)

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

[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/L10n62"

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