L11n430

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Contents

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

Image:L11n430.gif

(Knotscape image)

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

Visit L11n430's page at Knotilus.

Visit L11n430's page at the original Knot Atlas.

[edit L11n430 Quick Notes]

[edit L11n430 Further Notes and Views]

[edit] Link Presentations

[edit Notes on L11n430's Link Presentations]

Planar diagram presentation	X₈₁₉₂ X_9,20,10,21 X_14,5,15,6 X_12,14,7,13 X_16,8,17,7 X_22,18,13,17 X_3,10,4,11 X_18,11,19,12 X_6,15,1,16 X_4,20,5,19 X_21,3,22,2
Gauss code	{1, 11, -7, -10, 3, -9}, {5, -1, -2, 7, 8, -4}, {4, -3, 9, -5, 6, -8, 10, 2, -11, -6}

A Braid Representative

A Morse Link Presentation

Image:L11n430_ML.gif

[edit] Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-\frac{(v-1) (w-1) \left(u^2 w^2-u v w+u v-u w^2+u w-v\right)}{u v w^{3/2}}$ (db)
Jones polynomial	$- q 3 + 3 q 2 -5 q + 8-7 q -1 + 9 q -2 -6 q -3 + 5 q -4 -3 q -5 + q -6$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$a 4 z 4 + 2 a 4 z 2 + a 4 z -2 + a 4 - a 2 z 6 -4 a 2 z 4 -6 a 2 z 2 - z 2 a -2 -2 a 2 z -2 -4 a 2 - a -2 + 2 z 4 + 5 z 2 + z -2 + 4$ (db)
Kauffman polynomial	$a 6 z 6 -3 a 6 z 4 + a 6 z 2 + 3 a 5 z 7 -10 a 5 z 5 + 6 a 5 z 3 + 4 a 4 z 8 -15 a 4 z 6 + 16 a 4 z 4 -9 a 4 z 2 - a 4 z -2 + 3 a 4 + 2 a 3 z 9 -4 a 3 z 7 -3 a 3 z 5 + 5 a 3 z 3 + z 3 a -3 -2 a 3 z - z a -3 + 2 a 3 z -1 + 7 a 2 z 8 -29 a 2 z 6 + 43 a 2 z 4 + 3 z 4 a -2 -27 a 2 z 2 -4 z 2 a -2 -2 a 2 z -2 + 7 a 2 + 2 a -2 + 2 a z 9 -6 a z 7 + z 7 a -1 + 6 a z 5 - z 5 a -1 + 2 a z 3 + 4 z 3 a -1 -4 a z -3 z a -1 + 2 a z -1 + 3 z 8 -13 z 6 + 27 z 4 -21 z 2 - z -2 + 7$ (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 =

0 is the signature of L11n430. 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%>-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=7.14286%>3</td><td width=14.2857%>χ</td></tr> <tr align=center><td>7</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>5</td><td> </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>3</td><td> </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>1</td><td> </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>-1</td><td> </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>1</td></tr> <tr align=center><td>-3</td><td> </td><td> </td><td> </td><td> </td><td bgcolor=yellow>5</td><td bgcolor=yellow>3</td><td> </td><td> </td><td> </td><td> </td><td>2</td></tr> <tr align=center><td>-5</td><td> </td><td> </td><td> </td><td bgcolor=yellow>3</td><td bgcolor=yellow>6</td><td> </td><td> </td><td> </td><td> </td><td> </td><td>3</td></tr> <tr align=center><td>-7</td><td> </td><td> </td><td bgcolor=yellow>2</td><td bgcolor=yellow>3</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td>-1</td></tr> <tr align=center><td>-9</td><td> </td><td bgcolor=yellow>1</td><td bgcolor=yellow>3</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td>2</td></tr> <tr align=center><td>-11</td><td bgcolor=yellow> </td><td bgcolor=yellow>2</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td>-2</td></tr> <tr align=center><td>-13</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)

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

[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/L11n430"

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