L11a315

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Contents

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

Image:L11a315.gif

(Knotscape image)

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

Visit L11a315's page at Knotilus.

Visit L11a315's page at the original Knot Atlas.

[edit L11a315 Quick Notes]

[edit L11a315 Further Notes and Views]

[edit] Link Presentations

[edit Notes on L11a315's Link Presentations]

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

A Braid Representative

A Morse Link Presentation

Image:L11a315_ML.gif

[edit] Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-\frac{2 u^3 v^4-2 u^3 v^3+u^3 v^2+u^2 v^5-4 u^2 v^4+7 u^2 v^3-5 u^2 v^2+2 u^2 v+2 u v^4-5 u v^3+7 u v^2-4 u v+u+v^3-2 v^2+2 v}{u^{3/2} v^{5/2}}$ (db)
Jones polynomial	$\frac{2}{q^{9/2}}-\frac{1}{q^{7/2}}+\frac{1}{q^{29/2}}-\frac{3}{q^{27/2}}+\frac{6}{q^{25/2}}-\frac{11}{q^{23/2}}+\frac{14}{q^{21/2}}-\frac{15}{q^{19/2}}+\frac{15}{q^{17/2}}-\frac{13}{q^{15/2}}+\frac{9}{q^{13/2}}-\frac{6}{q^{11/2}}$ (db)
Signature	-7 (db)
HOMFLY-PT polynomial	$- z 3 a 13 -3 z a 13 - a 13 z -1 + 3 z 5 a 11 + 12 z 3 a 11 + 13 z a 11 + 3 a 11 z -1 -2 z 7 a 9 -10 z 5 a 9 -16 z 3 a 9 -10 z a 9 -2 a 9 z -1 - z 7 a 7 -5 z 5 a 7 -8 z 3 a 7 -4 z a 7$ (db)
Kauffman polynomial	$a 18 z 4 - a 18 z 2 + 3 a 17 z 5 -3 a 17 z 3 + a 17 z + 5 a 16 z 6 -4 a 16 z 4 + a 16 z 2 + 7 a 15 z 7 -9 a 15 z 5 + 5 a 15 z 3 + a 15 z + 7 a 14 z 8 -11 a 14 z 6 + 8 a 14 z 4 -3 a 14 z 2 + a 14 + 4 a 13 z 9 -13 a 13 z 5 + 9 a 13 z 3 + a 13 z - a 13 z -1 + a 12 z 10 + 10 a 12 z 8 -35 a 12 z 6 + 39 a 12 z 4 -22 a 12 z 2 + 3 a 12 + 7 a 11 z 9 -19 a 11 z 7 + 20 a 11 z 5 -24 a 11 z 3 + 16 a 11 z -3 a 11 z -1 + a 10 z 10 + 5 a 10 z 8 -26 a 10 z 6 + 32 a 10 z 4 -17 a 10 z 2 + 3 a 10 + 3 a 9 z 9 -11 a 9 z 7 + 16 a 9 z 5 -17 a 9 z 3 + 11 a 9 z -2 a 9 z -1 + 2 a 8 z 8 -7 a 8 z 6 + 6 a 8 z 4 + a 7 z 7 -5 a 7 z 5 + 8 a 7 z 3 -4 a 7 z$ (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 L11a315. 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>4</td><td bgcolor=yellow> </td><td> </td><td>4</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>-14</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td bgcolor=yellow>8</td><td bgcolor=yellow>4</td><td> </td><td> </td><td> </td><td>4</td></tr> <tr align=center><td>-16</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>-18</td><td> </td><td> </td><td> </td><td> </td><td> </td><td bgcolor=yellow>8</td><td bgcolor=yellow>8</td><td> </td><td> </td><td> </td><td> </td><td> </td><td>0</td></tr> <tr align=center><td>-20</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>-22</td><td> </td><td> </td><td> </td><td bgcolor=yellow>5</td><td bgcolor=yellow>8</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td>-3</td></tr> <tr align=center><td>-24</td><td> </td><td> </td><td bgcolor=yellow>2</td><td bgcolor=yellow>7</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>-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>2</td><td> </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>-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)

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

[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/L11a315"

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