L10n58

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Contents

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

Image:L10n58.gif

(Knotscape image)

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

Visit L10n58's page at Knotilus.

Visit L10n58's page at the original Knot Atlas.

[edit L10n58 Quick Notes]

[edit L10n58 Further Notes and Views]

[edit] Link Presentations

[edit Notes on L10n58's Link Presentations]

Planar diagram presentation	X_12,1,13,2 X_7,17,8,16 X_5,1,6,10 X₃₇₄₆ X_9,5,10,4 X_17,11,18,20 X_13,19,14,18 X_19,15,20,14 X_2,11,3,12 X_15,9,16,8
Gauss code	{1, -9, -4, 5, -3, 4, -2, 10, -5, 3}, {9, -1, -7, 8, -10, 2, -6, 7, -8, 6}

A Braid Representative

A Morse Link Presentation

Image:L10n58_ML.gif

[edit] Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-\frac{(u-1) (v-1) \left(u^2 v+u v^2-2 u v+u+v\right)}{u^{3/2} v^{3/2}}$ (db)
Jones polynomial	$8 q 9 / 2 -8 q 7 / 2 + 4 q 5 / 2 -3 q 3 / 2 - q 19 / 2 + 3 q 17 / 2 -5 q 15 / 2 + 8 q 13 / 2 -8 q 11 / 2$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$- z a -9 - a -9 z -1 + 3 z 3 a -7 + 7 z a -7 + 5 a -7 z -1 -2 z 5 a -5 -8 z 3 a -5 -13 z a -5 -8 a -5 z -1 + 3 z 3 a -3 + 7 z a -3 + 4 a -3 z -1$ (db)
Kauffman polynomial	$z 5 a -11 -2 z 3 a -11 + z a -11 + 3 z 6 a -10 -7 z 4 a -10 + 5 z 2 a -10 -2 a -10 + 3 z 7 a -9 -3 z 5 a -9 -4 z 3 a -9 + z a -9 + a -9 z -1 + z 8 a -8 + 8 z 6 a -8 -26 z 4 a -8 + 24 z 2 a -8 -9 a -8 + 7 z 7 a -7 -14 z 5 a -7 + 12 z 3 a -7 -9 z a -7 + 5 a -7 z -1 + z 8 a -6 + 8 z 6 a -6 -25 z 4 a -6 + 31 z 2 a -6 -14 a -6 + 4 z 7 a -5 -10 z 5 a -5 + 20 z 3 a -5 -19 z a -5 + 8 a -5 z -1 + 3 z 6 a -4 -6 z 4 a -4 + 12 z 2 a -4 -8 a -4 + 6 z 3 a -3 -10 z a -3 + 4 a -3 z -1$ (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 =

3 is the signature of L10n58. 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%>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=7.69231%>7</td><td width=7.69231%>8</td><td width=15.3846%>χ</td></tr> <tr align=center><td>20</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>18</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>16</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td bgcolor=yellow>3</td><td bgcolor=yellow>1</td><td> </td><td>2</td></tr> <tr align=center><td>14</td><td> </td><td> </td><td> </td><td> </td><td> </td><td bgcolor=yellow>5</td><td bgcolor=yellow>2</td><td> </td><td> </td><td>-3</td></tr> <tr align=center><td>12</td><td> </td><td> </td><td> </td><td> </td><td bgcolor=yellow>3</td><td bgcolor=yellow>3</td><td> </td><td> </td><td> </td><td>0</td></tr> <tr align=center><td>10</td><td> </td><td> </td><td> </td><td bgcolor=yellow>5</td><td bgcolor=yellow>5</td><td> </td><td> </td><td> </td><td> </td><td>0</td></tr> <tr align=center><td>8</td><td> </td><td> </td><td bgcolor=yellow>3</td><td bgcolor=yellow>3</td><td> </td><td> </td><td> </td><td> </td><td> </td><td>0</td></tr> <tr align=center><td>6</td><td> </td><td bgcolor=yellow>1</td><td bgcolor=yellow>5</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td>4</td></tr> <tr align=center><td>4</td><td bgcolor=yellow>2</td><td bgcolor=yellow>3</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td>-1</td></tr> <tr align=center><td>2</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> </table>

Integral Khovanov Homology

(db, data source)

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

[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/L10n58"

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