L11a33

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Contents

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

Image:L11a33.gif

(Knotscape image)

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

Visit L11a33's page at Knotilus.

Visit L11a33's page at the original Knot Atlas.

[edit L11a33 Quick Notes]

[edit L11a33 Further Notes and Views]

[edit] Link Presentations

[edit Notes on L11a33's Link Presentations]

Planar diagram presentation	X₆₁₇₂ X_10,3,11,4 X_18,11,19,12 X_16,7,17,8 X_8,17,9,18 X_20,13,21,14 X_22,15,5,16 X_12,19,13,20 X_14,21,15,22 X₂₅₃₆ X_4,9,1,10
Gauss code	{1, -10, 2, -11}, {10, -1, 4, -5, 11, -2, 3, -8, 6, -9, 7, -4, 5, -3, 8, -6, 9, -7}

A Braid Representative

A Morse Link Presentation

Image:L11a33_ML.gif

[edit] Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$\frac{-2 u v^6+3 u v^5-3 u v^4+3 u v^3-3 u v^2+2 u v-u-v^7+2 v^6-3 v^5+3 v^4-3 v^3+3 v^2-2 v}{\sqrt{u} v^{7/2}}$ (db)
Jones polynomial	$\frac{2}{q^{9/2}}-\frac{1}{q^{7/2}}+\frac{1}{q^{29/2}}-\frac{2}{q^{27/2}}+\frac{5}{q^{25/2}}-\frac{7}{q^{23/2}}+\frac{9}{q^{21/2}}-\frac{11}{q^{19/2}}+\frac{10}{q^{17/2}}-\frac{9}{q^{15/2}}+\frac{6}{q^{13/2}}-\frac{5}{q^{11/2}}$ (db)
Signature	-7 (db)
HOMFLY-PT polynomial	$- z 3 a 13 -4 z a 13 -3 a 13 z -1 + 3 z 5 a 11 + 14 z 3 a 11 + 18 z a 11 + 7 a 11 z -1 -2 z 7 a 9 -11 z 5 a 9 -19 z 3 a 9 -13 z a 9 -4 a 9 z -1 - z 7 a 7 -5 z 5 a 7 -7 z 3 a 7 -3 z a 7$ (db)
Kauffman polynomial	$- z 4 a 18 + 2 z 2 a 18 - a 18 -2 z 5 a 17 + 2 z 3 a 17 -3 z 6 a 16 + 3 z 4 a 16 - z 2 a 16 -3 z 7 a 15 + 2 z 5 a 15 -3 z 8 a 14 + 4 z 6 a 14 -3 z 4 a 14 + z 2 a 14 -3 z 9 a 13 + 9 z 7 a 13 -17 z 5 a 13 + 20 z 3 a 13 -12 z a 13 + 3 a 13 z -1 - z 10 a 12 -3 z 8 a 12 + 22 z 6 a 12 -39 z 4 a 12 + 29 z 2 a 12 -7 a 12 -6 z 9 a 11 + 27 z 7 a 11 -49 z 5 a 11 + 53 z 3 a 11 -31 z a 11 + 7 a 11 z -1 - z 10 a 10 -2 z 8 a 10 + 23 z 6 a 10 -39 z 4 a 10 + 26 z 2 a 10 -7 a 10 -3 z 9 a 9 + 14 z 7 a 9 -23 z 5 a 9 + 24 z 3 a 9 -16 z a 9 + 4 a 9 z -1 -2 z 8 a 8 + 8 z 6 a 8 -7 z 4 a 8 + z 2 a 8 - z 7 a 7 + 5 z 5 a 7 -7 z 3 a 7 + 3 z a 7$ (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 =

-7 is the signature of L11a33. 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%>-11</td><td width=6.25%>-10</td><td width=6.25%>-9</td><td width=6.25%>-8</td><td width=6.25%>-7</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=12.5%>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> </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>2</td><td bgcolor=yellow>1</td><td>-1</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>3</td><td bgcolor=yellow> </td><td> </td><td>3</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>3</td><td bgcolor=yellow>2</td><td> </td><td> </td><td>-1</td></tr> <tr align=center><td>-14</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>-16</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td bgcolor=yellow>5</td><td bgcolor=yellow>4</td><td> </td><td> </td><td> </td><td> </td><td>-1</td></tr> <tr align=center><td>-18</td><td> </td><td> </td><td> </td><td> </td><td> </td><td bgcolor=yellow>6</td><td bgcolor=yellow>5</td><td> </td><td> </td><td> </td><td> </td><td> </td><td>1</td></tr> <tr align=center><td>-20</td><td> </td><td> </td><td> </td><td> </td><td bgcolor=yellow>3</td><td bgcolor=yellow>5</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td>2</td></tr> <tr align=center><td>-22</td><td> </td><td> </td><td> </td><td bgcolor=yellow>4</td><td bgcolor=yellow>6</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td>-2</td></tr> <tr align=center><td>-24</td><td> </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>2</td></tr> <tr align=center><td>-26</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>-28</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>-30</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)

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = -8$	$i = -6$
$r = -11$	${\mathbb Z}$
$r = -10$	${\mathbb Z}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -9$	${\mathbb Z}^{4}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -8$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4}$	${\mathbb Z}^{4}$
$r = -7$	${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = -6$	${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6}$	${\mathbb Z}^{6}$
$r = -5$	${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5}$	${\mathbb Z}^{5}$
$r = -4$	${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5}$	${\mathbb Z}^{6}$
$r = -3$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = -2$	${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = -1$	${\mathbb Z}_2^{2}$	${\mathbb Z}^{2}$
$r = 0$	${\mathbb Z}$	${\mathbb Z}$

[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/L11a33"

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