L11a5

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Contents

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

Image:L11a5.gif

(Knotscape image)

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

Visit L11a5's page at Knotilus.

Visit L11a5's page at the original Knot Atlas.

[edit L11a5 Quick Notes]

[edit L11a5 Further Notes and Views]

[edit] Link Presentations

[edit Notes on L11a5's Link Presentations]

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

A Braid Representative

A Morse Link Presentation

Image:L11a5_ML.gif

[edit] Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$\frac{(u-1) (v-1) \left(3 v^2-5 v+3\right)}{\sqrt{u} v^{3/2}}$ (db)
Jones polynomial	$-11 q^{9/2}+13 q^{7/2}-14 q^{5/2}+\frac{1}{q^{5/2}}+12 q^{3/2}-\frac{4}{q^{3/2}}-q^{17/2}+3 q^{15/2}-5 q^{13/2}+8 q^{11/2}-10 \sqrt{q}+\frac{6}{\sqrt{q}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$z 5 a -1 + z 5 a -3 + z 5 a -5 - a z 3 + z 3 a -1 + 2 z 3 a -5 - z 3 a -7 - z a -3 + 2 z a -5 - z a -7 + a z -1 - a -1 z -1$ (db)
Kauffman polynomial	$-2 z 10 a -4 -2 z 10 a -6 -5 z 9 a -3 -9 z 9 a -5 -4 z 9 a -7 -6 z 8 a -2 - z 8 a -4 + 2 z 8 a -6 -3 z 8 a -8 -6 z 7 a -1 + 12 z 7 a -3 + 36 z 7 a -5 + 17 z 7 a -7 - z 7 a -9 + 9 z 6 a -2 + 18 z 6 a -4 + 16 z 6 a -6 + 13 z 6 a -8 -6 z 6 -4 a z 5 + 4 z 5 a -1 -13 z 5 a -3 -46 z 5 a -5 -21 z 5 a -7 + 4 z 5 a -9 - a 2 z 4 -4 z 4 a -2 -23 z 4 a -4 -27 z 4 a -6 -15 z 4 a -8 + 6 z 4 + 4 a z 3 + 4 z 3 a -1 + 8 z 3 a -3 + 22 z 3 a -5 + 11 z 3 a -7 -3 z 3 a -9 + z 2 a -2 + 6 z 2 a -4 + 11 z 2 a -6 + 6 z 2 a -8 -2 z a -3 -4 z a -5 -2 z a -7 + 1- a z -1 - a -1 z -1$ (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 L11a5. 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%>-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=6.25%>7</td><td width=6.25%>8</td><td width=12.5%>χ</td></tr> <tr align=center><td>18</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>16</td><td> </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 bgcolor=yellow> </td><td>-2</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 bgcolor=yellow>3</td><td bgcolor=yellow>1</td><td> </td><td>2</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 bgcolor=yellow>5</td><td bgcolor=yellow>2</td><td> </td><td> </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 bgcolor=yellow>6</td><td bgcolor=yellow>3</td><td> </td><td> </td><td> </td><td>3</td></tr> <tr align=center><td>8</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td bgcolor=yellow>7</td><td bgcolor=yellow>5</td><td> </td><td> </td><td> </td><td> </td><td>-2</td></tr> <tr align=center><td>6</td><td> </td><td> </td><td> </td><td> </td><td> </td><td bgcolor=yellow>7</td><td bgcolor=yellow>6</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>5</td><td bgcolor=yellow>7</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td>2</td></tr> <tr align=center><td>2</td><td> </td><td> </td><td> </td><td bgcolor=yellow>5</td><td bgcolor=yellow>7</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td>-2</td></tr> <tr align=center><td>0</td><td> </td><td> </td><td bgcolor=yellow>3</td><td bgcolor=yellow>7</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>-2</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> </td><td> </td><td>-2</td></tr> <tr align=center><td>-4</td><td bgcolor=yellow> </td><td bgcolor=yellow>3</td><td> </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>-6</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)

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

[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).

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