L11a176

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Contents

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

Image:L11a176.gif

(Knotscape image)

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

Visit L11a176's page at Knotilus.

Visit L11a176's page at the original Knot Atlas.

[edit L11a176 Quick Notes]

[edit L11a176 Further Notes and Views]

[edit] Link Presentations

[edit Notes on L11a176's Link Presentations]

Planar diagram presentation	X₈₁₉₂ X_18,9,19,10 X_20,13,21,14 X_10,4,11,3 X_14,6,15,5 X_16,8,17,7 X_22,16,7,15 X_4,12,5,11 X_12,19,13,20 X_6,21,1,22 X_2,18,3,17
Gauss code	{1, -11, 4, -8, 5, -10}, {6, -1, 2, -4, 8, -9, 3, -5, 7, -6, 11, -2, 9, -3, 10, -7}

A Braid Representative

A Morse Link Presentation

Image:L11a176_ML.gif

[edit] Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$\frac{t(1)^2 t(2)^6-t(1) t(2)^6-3 t(1)^2 t(2)^5+4 t(1) t(2)^5-t(2)^5+5 t(1)^2 t(2)^4-8 t(1) t(2)^4+3 t(2)^4-5 t(1)^2 t(2)^3+11 t(1) t(2)^3-5 t(2)^3+3 t(1)^2 t(2)^2-8 t(1) t(2)^2+5 t(2)^2-t(1)^2 t(2)+4 t(1) t(2)-3 t(2)-t(1)+1}{t(1) t(2)^3}$ (db)
Jones polynomial	$-q^{13/2}+4 q^{11/2}-9 q^{9/2}+15 q^{7/2}-21 q^{5/2}+23 q^{3/2}-24 \sqrt{q}+\frac{20}{\sqrt{q}}-\frac{15}{q^{3/2}}+\frac{9}{q^{5/2}}-\frac{4}{q^{7/2}}+\frac{1}{q^{9/2}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$- z 7 a -3 -4 z 5 a -3 -5 z 3 a -3 -2 z a -3 - a -3 z -1 + z 9 a -1 - a z 7 + 6 z 7 a -1 -4 a z 5 + 13 z 5 a -1 -5 a z 3 + 12 z 3 a -1 -2 a z + 5 z a -1 + a -1 z -1$ (db)
Kauffman polynomial	$z 5 a -7 - z 3 a -7 + 4 z 6 a -6 -5 z 4 a -6 + z 2 a -6 + 8 z 7 a -5 -12 z 5 a -5 + 6 z 3 a -5 - z a -5 + 10 z 8 a -4 + a 4 z 6 -15 z 6 a -4 -2 a 4 z 4 + 8 z 4 a -4 + a 4 z 2 - z 2 a -4 + 8 z 9 a -3 + 4 a 3 z 7 -8 z 7 a -3 -9 a 3 z 5 + z 5 a -3 + 5 a 3 z 3 + 2 z a -3 - a -3 z -1 + 3 z 10 a -2 + 7 a 2 z 8 + 10 z 8 a -2 -15 a 2 z 6 -27 z 6 a -2 + 8 a 2 z 4 + 19 z 4 a -2 - a 2 z 2 -4 z 2 a -2 + a -2 + 7 a z 9 + 15 z 9 a -1 -12 a z 7 -32 z 7 a -1 + 6 a z 5 + 29 z 5 a -1 -6 a z 3 -18 z 3 a -1 + 3 a z + 6 z a -1 - a -1 z -1 + 3 z 10 + 7 z 8 -24 z 6 + 16 z 4 -4 z 2$ (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 L11a176. 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>10</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> </td><td> </td><td bgcolor=yellow>12</td><td bgcolor=yellow>11</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>9</td><td bgcolor=yellow>13</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>3</td><td bgcolor=yellow>9</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>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>-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)

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

[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/L11a176"

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