L11a327

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Contents

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

Image:L11a327.gif

(Knotscape image)

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

Visit L11a327's page at Knotilus.

Visit L11a327's page at the original Knot Atlas.

[edit L11a327 Quick Notes]

[edit L11a327 Further Notes and Views]

[edit] Link Presentations

[edit Notes on L11a327's Link Presentations]

Planar diagram presentation	X_10,1,11,2 X_2,11,3,12 X_12,3,13,4 X_18,10,19,9 X_6,13,7,14 X_14,7,15,8 X_8,15,1,16 X_22,18,9,17 X_16,22,17,21 X_4,20,5,19 X_20,6,21,5
Gauss code	{1, -2, 3, -10, 11, -5, 6, -7}, {4, -1, 2, -3, 5, -6, 7, -9, 8, -4, 10, -11, 9, -8}

A Braid Representative

A Morse Link Presentation

Image:L11a327_ML.gif

[edit] Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$\frac{-u^3 v^5+u^3 v^4-u^3 v^3+u^3 v^2+u^2 v^5-4 u^2 v^4+5 u^2 v^3-5 u^2 v^2+2 u^2 v+2 u v^4-5 u v^3+5 u v^2-4 u v+u+v^3-v^2+v-1}{u^{3/2} v^{5/2}}$ (db)
Jones polynomial	$q^{9/2}-2 q^{7/2}+5 q^{5/2}-9 q^{3/2}+11 \sqrt{q}-\frac{14}{\sqrt{q}}+\frac{13}{q^{3/2}}-\frac{12}{q^{5/2}}+\frac{9}{q^{7/2}}-\frac{5}{q^{9/2}}+\frac{2}{q^{11/2}}-\frac{1}{q^{13/2}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a 3 z 7 + 6 a 3 z 5 + 13 a 3 z 3 + 10 a 3 z + 2 a 3 z -1 - a z 9 -8 a z 7 + z 7 a -1 -25 a z 5 + 6 z 5 a -1 -36 a z 3 + 13 z 3 a -1 -22 a z + 10 z a -1 -3 a z -1 + a -1 z -1$ (db)
Kauffman polynomial	$a 7 z 5 -3 a 7 z 3 + 2 a 7 z + 2 a 6 z 6 -4 a 6 z 4 + a 6 z 2 + 3 a 5 z 7 -5 a 5 z 5 + 2 a 5 z 3 - a 5 z + 4 a 4 z 8 -10 a 4 z 6 + z 6 a -4 + 14 a 4 z 4 -4 z 4 a -4 -7 a 4 z 2 + 4 z 2 a -4 + 3 a 3 z 9 -6 a 3 z 7 + 2 z 7 a -3 + 10 a 3 z 5 -6 z 5 a -3 -10 a 3 z 3 + 4 z 3 a -3 + 9 a 3 z -2 a 3 z -1 + a 2 z 10 + 4 a 2 z 8 + 3 z 8 a -2 -17 a 2 z 6 -9 z 6 a -2 + 30 a 2 z 4 + 10 z 4 a -2 -18 a 2 z 2 -7 z 2 a -2 + 3 a 2 + a -2 + 6 a z 9 + 3 z 9 a -1 -21 a z 7 -10 z 7 a -1 + 43 a z 5 + 21 z 5 a -1 -47 a z 3 -28 z 3 a -1 + 24 a z + 12 z a -1 -3 a z -1 - a -1 z -1 + z 10 + 3 z 8 -15 z 6 + 26 z 4 -21 z 2 + 3$ (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 L11a327. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

<table border=1> <tr align=center> <td width=12.5%><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=6.25%>-6</td><td width=6.25%>-5</td><td width=6.25%>-4</td><td width=6.25%>-3</td><td width=6.25%>-2</td><td width=6.25%>-1</td><td width=6.25%>0</td><td width=6.25%>1</td><td width=6.25%>2</td><td width=6.25%>3</td><td width=6.25%>4</td><td width=6.25%>5</td><td width=12.5%>χ</td></tr> <tr align=center><td>10</td><td> </td><td> </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>8</td><td> </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 bgcolor=yellow> </td><td>1</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>4</td><td bgcolor=yellow>1</td><td> </td><td>-3</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>5</td><td bgcolor=yellow>1</td><td> </td><td> </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> </td><td> </td><td>-2</td></tr> <tr align=center><td>0</td><td> </td><td> </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>-2</td><td> </td><td> </td><td> </td><td> </td><td> </td><td bgcolor=yellow>6</td><td bgcolor=yellow>7</td><td> </td><td> </td><td> </td><td> </td><td> </td><td>1</td></tr> <tr align=center><td>-4</td><td> </td><td> </td><td> </td><td> </td><td bgcolor=yellow>6</td><td bgcolor=yellow>7</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td>-1</td></tr> <tr align=center><td>-6</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> </td><td> </td><td>3</td></tr> <tr align=center><td>-8</td><td> </td><td> </td><td bgcolor=yellow>2</td><td bgcolor=yellow>6</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td>-4</td></tr> <tr align=center><td>-10</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> </td><td> </td><td>3</td></tr> <tr align=center><td>-12</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> </td><td> </td><td>-1</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> </td><td>1</td></tr> </table>

Integral Khovanov Homology

(db, data source)

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

[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.

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