L11n417

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Contents

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

Image:L11n417.gif

(Knotscape image)

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

Visit L11n417's page at Knotilus.

Visit L11n417's page at the original Knot Atlas.

[edit L11n417 Quick Notes]

[edit L11n417 Further Notes and Views]

[edit] Link Presentations

[edit Notes on L11n417's Link Presentations]

Planar diagram presentation	X₈₁₉₂ X_10,3,11,4 X_15,21,16,20 X_5,15,6,14 X_13,5,14,4 X_19,7,20,12 X_11,19,12,18 X_17,13,18,22 X_21,17,22,16 X₂₇₃₈ X_6,9,1,10
Gauss code	{1, -10, 2, 5, -4, -11}, {10, -1, 11, -2, -7, 6}, {-5, 4, -3, 9, -8, 7, -6, 3, -9, 8}

A Braid Representative

A Morse Link Presentation

Image:L11n417_ML.gif

[edit] Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-\frac{u^2 v w^3-u^2 w^3+u^2 w^2-u^2 w+u^2+u v^2 w^3-u v w^3+u v-u-v^2 w^3+v^2 w^2-v^2 w+v^2-v}{u v w^{3/2}}$ (db)
Jones polynomial	$- q 7 + 2 q 6 -3 q 5 + 4 q 4 -4 q 3 + 5 q 2 -3 q + 4- q -1 + q -2$ (db)
Signature	4 (db)
HOMFLY-PT polynomial	$- z 6 a -2 - z 6 a -4 -5 z 4 a -2 -4 z 4 a -4 + z 4 a -6 + z 4 -9 z 2 a -2 -2 z 2 a -4 + 4 z 2 a -6 + 4 z 2 -10 a -2 + 4 a -4 + 2 a -6 - a -8 + 5-5 a -2 z -2 + 4 a -4 z -2 - a -6 z -2 + 2 z -2$ (db)
Kauffman polynomial	$z a -9 + 2 z 2 a -8 - a -8 + z 5 a -7 - z 3 a -7 -2 z a -7 + a -7 z -1 + 3 z 6 a -6 -10 z 4 a -6 + 6 z 2 a -6 - a -6 z -2 + 5 z 7 a -5 -23 z 5 a -5 + 31 z 3 a -5 -19 z a -5 + 5 a -5 z -1 + 4 z 8 a -4 -19 z 6 a -4 + 27 z 4 a -4 -21 z 2 a -4 -4 a -4 z -2 + 13 a -4 + z 9 a -3 + z 7 a -3 -25 z 5 a -3 + 50 z 3 a -3 -35 z a -3 + 9 a -3 z -1 + 5 z 8 a -2 -29 z 6 a -2 + 55 z 4 a -2 -46 z 2 a -2 -5 a -2 z -2 + 22 a -2 + z 9 a -1 -4 z 7 a -1 - z 5 a -1 + 18 z 3 a -1 -19 z a -1 + 5 a -1 z -1 + z 8 -7 z 6 + 18 z 4 -21 z 2 -2 z -2 + 11$ (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 L11n417. 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%>-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=7.14286%>3</td><td width=7.14286%>4</td><td width=7.14286%>5</td><td width=14.2857%>χ</td></tr> <tr align=center><td>15</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>13</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>11</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>9</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td bgcolor=yellow>2</td><td bgcolor=yellow>2</td><td bgcolor=red>1</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>3</td><td bgcolor=yellow>3</td><td> </td><td> </td><td> </td><td>0</td></tr> <tr align=center><td>5</td><td> </td><td> </td><td> </td><td> </td><td bgcolor=yellow>2</td><td bgcolor=yellow>2</td><td bgcolor=red>1</td><td> </td><td> </td><td> </td><td>1</td></tr> <tr align=center><td>3</td><td> </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>2</td></tr> <tr align=center><td>1</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> </td><td>1</td></tr> <tr align=center><td>-1</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>3</td></tr> <tr align=center><td>-3</td><td bgcolor=yellow> </td><td bgcolor=yellow> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td>0</td></tr> <tr align=center><td>-5</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)

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

[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/L11n417"

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