L11a1

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Contents

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

Image:L11a1.gif

(Knotscape image)

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

Visit L11a1's page at Knotilus.

Visit L11a1's page at the original Knot Atlas.

[edit L11a1 Quick Notes]

[edit L11a1 Further Notes and Views]

Link L11a1.

A graph, link L11a1.

A graph, link L11a1.

A part of a link and a part of a graph.

[edit] Link Presentations

[edit Notes on L11a1's Link Presentations]

Planar diagram presentation	X₆₁₇₂ X_14,7,15,8 X_4,15,1,16 X_10,6,11,5 X₈₄₉₃ X_20,12,21,11 X_22,17,5,18 X_18,21,19,22 X_12,20,13,19 X_16,10,17,9 X_2,14,3,13
Gauss code	{1, -11, 5, -3}, {4, -1, 2, -5, 10, -4, 6, -9, 11, -2, 3, -10, 7, -8, 9, -6, 8, -7}

A Braid Representative

A Morse Link Presentation

Image:L11a1_ML.gif

[edit] Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-\frac{(u-1) (v-1) \left(v^4-5 v^3+7 v^2-5 v+1\right)}{\sqrt{u} v^{5/2}}$ (db)
Jones polynomial	$-q^{13/2}+4 q^{11/2}-9 q^{9/2}+15 q^{7/2}-21 q^{5/2}+24 q^{3/2}-25 \sqrt{q}+\frac{21}{\sqrt{q}}-\frac{16}{q^{3/2}}+\frac{10}{q^{5/2}}-\frac{5}{q^{7/2}}+\frac{1}{q^{9/2}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$- z 7 a -1 + 2 a z 5 -3 z 5 a -1 + 2 z 5 a -3 - a 3 z 3 + 3 a z 3 -4 z 3 a -1 + 4 z 3 a -3 - z 3 a -5 - a z + 2 z a -3 - z a -5 + a 3 z -1 -2 a z -1 + 2 a -1 z -1 - a -3 z -1$ (db)
Kauffman polynomial	$-2 z 10 a -2 -2 z 10 -7 a z 9 -13 z 9 a -1 -6 z 9 a -3 -9 a 2 z 8 -17 z 8 a -2 -9 z 8 a -4 -17 z 8 -5 a 3 z 7 + 5 a z 7 + 12 z 7 a -1 -6 z 7 a -3 -8 z 7 a -5 - a 4 z 6 + 20 a 2 z 6 + 36 z 6 a -2 + 9 z 6 a -4 -4 z 6 a -6 + 44 z 6 + 10 a 3 z 5 + 14 a z 5 + 16 z 5 a -1 + 25 z 5 a -3 + 12 z 5 a -5 - z 5 a -7 + a 4 z 4 -11 a 2 z 4 -19 z 4 a -2 - z 4 a -4 + 5 z 4 a -6 -25 z 4 -4 a 3 z 3 -9 a z 3 -14 z 3 a -1 -18 z 3 a -3 -8 z 3 a -5 + z 3 a -7 + 2 z 2 a -2 - z 2 a -4 -2 z 2 a -6 + z 2 - a 3 z -4 a z -2 z a -1 + 3 z a -3 + 2 z a -5 + 1 + a 3 z -1 + 2 a z -1 + 2 a -1 z -1 + a -3 z -1$ (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 L11a1. 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%>-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=6.25%>6</td><td width=12.5%>χ</td></tr> <tr align=center><td>14</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>12</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td bgcolor=yellow>3</td><td bgcolor=yellow> </td><td>-3</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 bgcolor=yellow>6</td><td bgcolor=yellow>1</td><td> </td><td>5</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 bgcolor=yellow>9</td><td bgcolor=yellow>3</td><td> </td><td> </td><td>-6</td></tr> <tr align=center><td>6</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td bgcolor=yellow>12</td><td bgcolor=yellow>6</td><td> </td><td> </td><td> </td><td>6</td></tr> <tr align=center><td>4</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td bgcolor=yellow>12</td><td bgcolor=yellow>9</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>13</td><td bgcolor=yellow>12</td><td> </td><td> </td><td> </td><td> </td><td> </td><td>1</td></tr> <tr align=center><td>0</td><td> </td><td> </td><td> </td><td> </td><td bgcolor=yellow>10</td><td bgcolor=yellow>14</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td>4</td></tr> <tr align=center><td>-2</td><td> </td><td> </td><td> </td><td bgcolor=yellow>6</td><td bgcolor=yellow>11</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td>-5</td></tr> <tr align=center><td>-4</td><td> </td><td> </td><td bgcolor=yellow>4</td><td bgcolor=yellow>10</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td>6</td></tr> <tr align=center><td>-6</td><td> </td><td bgcolor=yellow>1</td><td bgcolor=yellow>6</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td>-5</td></tr> <tr align=center><td>-8</td><td bgcolor=yellow> </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> </td><td>4</td></tr> <tr align=center><td>-10</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)

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = 0$	$i = 2$
$r = -5$	${\mathbb Z}$
$r = -4$	${\mathbb Z}^{4}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -3$	${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4}$	${\mathbb Z}^{4}$
$r = -2$	${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{6}$	${\mathbb Z}^{6}$
$r = -1$	${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{10}$	${\mathbb Z}^{10}$
$r = 0$	${\mathbb Z}^{14}\oplus{\mathbb Z}_2^{11}$	${\mathbb Z}^{13}$
$r = 1$	${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{12}$	${\mathbb Z}^{12}$
$r = 2$	${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{12}$	${\mathbb Z}^{12}$
$r = 3$	${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{9}$	${\mathbb Z}^{9}$
$r = 4$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6}$	${\mathbb Z}^{6}$
$r = 5$	${\mathbb Z}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = 6$	${\mathbb Z}_2$	${\mathbb Z}$

[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/L11a1"

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