L10n97

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Contents

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

Image:L10n97.gif

(Knotscape image)

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

Visit L10n97's page at Knotilus.

Visit L10n97's page at the original Knot Atlas.

[edit L10n97 Quick Notes]

[edit L10n97 Further Notes and Views]

[edit] Link Presentations

[edit Notes on L10n97's Link Presentations]

Planar diagram presentation	X₆₁₇₂ X₂₅₃₆ X_20,13,15,14 X_3,11,4,10 X_9,1,10,4 X_7,17,8,16 X_15,5,16,8 X_18,11,19,12 X_12,19,13,20 X_14,17,9,18
Gauss code	{1, -2, -4, 5}, {2, -1, -6, 7}, {-5, 4, 8, -9, 3, -10}, {-7, 6, 10, -8, 9, -3}

A Braid Representative

A Morse Link Presentation

Image:L10n97_ML.gif

[edit] Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$\frac{-t(4)^2 t(3)^2+t(1) t(4) t(3)^2+t(2) t(4)^2 t(3)+t(1) t(3)-t(1) t(4) t(3)-t(2) t(4) t(3)-t(1) t(2)+t(2) t(4)}{\sqrt{t(1)} \sqrt{t(2)} t(3) t(4)}$ (db)
Jones polynomial	$-\frac{2}{q^{9/2}}+\frac{1}{q^{7/2}}-q^{5/2}-\frac{3}{q^{5/2}}-\frac{1}{q^{13/2}}+\frac{1}{q^{11/2}}-\sqrt{q}-\frac{2}{\sqrt{q}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a 5 z 3 + a 5 z -3 + 3 a 5 z + 3 a 5 z -1 - a 3 z 5 -6 a 3 z 3 -3 a 3 z -3 -13 a 3 z -10 a 3 z -1 + a z 5 + 7 a z 3 + 3 a z -3 - z 3 a -1 - a -1 z -3 + 14 a z + 11 a z -1 -4 z a -1 -4 a -1 z -1$ (db)
Kauffman polynomial	$- z 5 a 7 + 4 z 3 a 7 -3 z a 7 + a 7 z -1 - z 6 a 6 + 3 z 4 a 6 - a 6 - z 7 a 5 + 4 z 5 a 5 -4 z 3 a 5 + 3 z a 5 -3 a 5 z -1 + a 5 z -3 -2 z 6 a 4 + 10 z 4 a 4 -16 z 2 a 4 -3 a 4 z -2 + 11 a 4 - z 7 a 3 + 7 z 5 a 3 -18 z 3 a 3 + 21 z a 3 -12 a 3 z -1 + 3 a 3 z -3 -2 z 6 a 2 + 15 z 4 a 2 -33 z 2 a 2 -6 a 2 z -2 + 24 a 2 - z 7 a + 9 z 5 a -25 z 3 a + 28 z a -14 a z -1 + 3 a z -3 - z 6 + 8 z 4 -17 z 2 -3 z -2 + 13- z 7 a -1 + 7 z 5 a -1 -15 z 3 a -1 + 13 z a -1 -6 a -1 z -1 + a -1 z -3$ (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 L10n97. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

<table border=1> <tr align=center> <td width=13.3333%><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=6.66667%>-6</td><td width=6.66667%>-5</td><td width=6.66667%>-4</td><td width=6.66667%>-3</td><td width=6.66667%>-2</td><td width=6.66667%>-1</td><td width=6.66667%>0</td><td width=6.66667%>1</td><td width=6.66667%>2</td><td width=6.66667%>3</td><td width=6.66667%>4</td><td width=13.3333%>χ</td></tr> <tr align=center><td>6</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td bgcolor=yellow> </td><td bgcolor=yellow>1</td><td>1</td></tr> <tr align=center><td>4</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td bgcolor=yellow> </td><td bgcolor=yellow> </td><td bgcolor=red>1</td><td>1</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> </td><td bgcolor=yellow>1</td><td> </td><td> </td><td>1</td></tr> <tr align=center><td>0</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td bgcolor=yellow>3</td><td bgcolor=yellow> </td><td> </td><td> </td><td> </td><td>3</td></tr> <tr align=center><td>-2</td><td> </td><td> </td><td> </td><td> </td><td> </td><td bgcolor=yellow>1</td><td bgcolor=yellow>4</td><td bgcolor=red>1</td><td> </td><td> </td><td> </td><td>2</td></tr> <tr align=center><td>-4</td><td> </td><td> </td><td> </td><td> </td><td bgcolor=yellow>3</td><td bgcolor=yellow> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td>3</td></tr> <tr align=center><td>-6</td><td> </td><td> </td><td> </td><td bgcolor=yellow>1</td><td bgcolor=yellow>4</td><td bgcolor=red>1</td><td> </td><td> </td><td> </td><td> </td><td> </td><td>2</td></tr> <tr align=center><td>-8</td><td> </td><td> </td><td bgcolor=yellow>1</td><td bgcolor=yellow>1</td><td bgcolor=red>1</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td>1</td></tr> <tr align=center><td>-10</td><td> </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>-12</td><td bgcolor=yellow>1</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>0</td></tr> <tr align=center><td>-14</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> </td><td>1</td></tr> </table>

Integral Khovanov Homology

(db, data source)

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

[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/L10n97"

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