L11n447

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Contents

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

Image:L11n447.gif

(Knotscape image)

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

Visit L11n447's page at Knotilus.

Visit L11n447's page at the original Knot Atlas.

[edit L11n447 Quick Notes]

[edit L11n447 Further Notes and Views]

[edit] Link Presentations

[edit Notes on L11n447's Link Presentations]

Planar diagram presentation	X₆₁₇₂ X_10,3,11,4 X_14,7,15,8 X_8,13,5,14 X_11,18,12,19 X_19,17,20,22 X_21,9,22,16 X_15,21,16,20 X_17,12,18,13 X₂₅₃₆ X_4,9,1,10
Gauss code	{1, -10, 2, -11}, {10, -1, 3, -4}, {-9, 5, -6, 8, -7, 6}, {11, -2, -5, 9, 4, -3, -8, 7}

A Braid Representative

A Morse Link Presentation

Image:L11n447_ML.gif

[edit] Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-\frac{(w-1) (x-1) \left(-u v+u x^2-u x+u+v x^2-v x+v-x^2\right)}{\sqrt{u} \sqrt{v} \sqrt{w} x^{3/2}}$ (db)
Jones polynomial	$-\frac{12}{q^{9/2}}+\frac{9}{q^{7/2}}-\frac{11}{q^{5/2}}+\frac{8}{q^{3/2}}-\frac{1}{q^{17/2}}+\frac{2}{q^{15/2}}-\frac{6}{q^{13/2}}+\frac{7}{q^{11/2}}+2 \sqrt{q}-\frac{6}{\sqrt{q}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a 9 z -1 + a 9 z -3 -4 z a 7 -7 a 7 z -1 -3 a 7 z -3 + 5 z 3 a 5 + 13 z a 5 + 12 a 5 z -1 + 3 a 5 z -3 -2 z 5 a 3 -7 z 3 a 3 -11 z a 3 -7 a 3 z -1 - a 3 z -3 + 2 z 3 a + 2 z a + a z -1$ (db)
Kauffman polynomial	$a 9 z 7 -5 a 9 z 5 + 10 a 9 z 3 - a 9 z -3 -10 a 9 z + 5 a 9 z -1 + 2 a 8 z 8 -7 a 8 z 6 + 5 a 8 z 4 + 6 a 8 z 2 + 3 a 8 z -2 -9 a 8 + a 7 z 9 + 5 a 7 z 7 -34 a 7 z 5 + 55 a 7 z 3 -3 a 7 z -3 -38 a 7 z + 14 a 7 z -1 + 8 a 6 z 8 -22 a 6 z 6 + a 6 z 4 + 27 a 6 z 2 + 6 a 6 z -2 -21 a 6 + a 5 z 9 + 15 a 5 z 7 -65 a 5 z 5 + 83 a 5 z 3 -3 a 5 z -3 -54 a 5 z + 18 a 5 z -1 + 6 a 4 z 8 -8 a 4 z 6 -18 a 4 z 4 + 33 a 4 z 2 + 3 a 4 z -2 -18 a 4 + 11 a 3 z 7 -35 a 3 z 5 + 45 a 3 z 3 - a 3 z -3 -31 a 3 z + 11 a 3 z -1 + 7 a 2 z 6 -14 a 2 z 4 + 15 a 2 z 2 -6 a 2 + a z 5 + 7 a z 3 -5 a z + 2 a z -1 + 3 z 2 -1$ (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 =

-1 is the signature of L11n447. 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%>-8</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=14.2857%>χ</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> </td><td bgcolor=yellow>2</td><td>-2</td></tr> <tr align=center><td>0</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td bgcolor=yellow>4</td><td bgcolor=yellow> </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 bgcolor=yellow>6</td><td bgcolor=yellow>4</td><td> </td><td>-2</td></tr> <tr align=center><td>-4</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td bgcolor=yellow>5</td><td bgcolor=yellow>3</td><td bgcolor=red>1</td><td> </td><td>3</td></tr> <tr align=center><td>-6</td><td> </td><td> </td><td> </td><td> </td><td> </td><td bgcolor=yellow>4</td><td bgcolor=yellow>6</td><td> </td><td> </td><td> </td><td>2</td></tr> <tr align=center><td>-8</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> </td><td>3</td></tr> <tr align=center><td>-10</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>5</td></tr> <tr align=center><td>-12</td><td> </td><td> </td><td bgcolor=yellow>1</td><td bgcolor=yellow>2</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td>-1</td></tr> <tr align=center><td>-14</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> </td><td>4</td></tr> <tr align=center><td>-16</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>-1</td></tr> <tr align=center><td>-18</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)

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

[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/L11n447"

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