L11a73

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Contents

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

Image:L11a73.gif

(Knotscape image)

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

Visit L11a73's page at Knotilus.

Visit L11a73's page at the original Knot Atlas.

[edit L11a73 Quick Notes]

[edit L11a73 Further Notes and Views]

[edit] Link Presentations

[edit Notes on L11a73's Link Presentations]

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

A Braid Representative

A Morse Link Presentation

Image:L11a73_ML.gif

[edit] Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-\frac{2 t(2)^5+4 t(1) t(2)^4-5 t(2)^4-7 t(1) t(2)^3+7 t(2)^3+7 t(1) t(2)^2-7 t(2)^2-5 t(1) t(2)+4 t(2)+2 t(1)}{\sqrt{t(1)} t(2)^{5/2}}$ (db)
Jones polynomial	$-\frac{1}{q^{5/2}}+\frac{3}{q^{7/2}}-\frac{7}{q^{9/2}}+\frac{10}{q^{11/2}}-\frac{15}{q^{13/2}}+\frac{16}{q^{15/2}}-\frac{16}{q^{17/2}}+\frac{13}{q^{19/2}}-\frac{10}{q^{21/2}}+\frac{6}{q^{23/2}}-\frac{2}{q^{25/2}}+\frac{1}{q^{27/2}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$a 13 (- z)-2 a 13 z -1 + 3 a 11 z 3 + 7 a 11 z + 4 a 11 z -1 -2 a 9 z 5 -4 a 9 z 3 - a 9 z - a 9 z -1 -3 a 7 z 5 -9 a 7 z 3 -7 a 7 z - a 7 z -1 - a 5 z 5 -2 a 5 z 3$ (db)
Kauffman polynomial	$a 16 z 6 -4 a 16 z 4 + 5 a 16 z 2 -2 a 16 + 2 a 15 z 7 -5 a 15 z 5 + 2 a 15 z 3 + a 15 z + 3 a 14 z 8 -6 a 14 z 6 + a 14 z 4 + 2 a 14 z 2 - a 14 + 3 a 13 z 9 -6 a 13 z 7 + 9 a 13 z 5 -16 a 13 z 3 + 9 a 13 z -2 a 13 z -1 + a 12 z 10 + 6 a 12 z 8 -22 a 12 z 6 + 32 a 12 z 4 -25 a 12 z 2 + 6 a 12 + 7 a 11 z 9 -18 a 11 z 7 + 29 a 11 z 5 -31 a 11 z 3 + 17 a 11 z -4 a 11 z -1 + a 10 z 10 + 9 a 10 z 8 -30 a 10 z 6 + 41 a 10 z 4 -22 a 10 z 2 + 5 a 10 + 4 a 9 z 9 -4 a 9 z 7 - a 9 z 5 + 4 a 9 z 3 + 2 a 9 z - a 9 z -1 + 6 a 8 z 8 -12 a 8 z 6 + 9 a 8 z 4 - a 8 + 6 a 7 z 7 -15 a 7 z 5 + 15 a 7 z 3 -7 a 7 z + a 7 z -1 + 3 a 6 z 6 -5 a 6 z 4 + a 5 z 5 -2 a 5 z 3$ (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 =

-5 is the signature of L11a73. 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>-4</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>-6</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>1</td><td>-2</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 bgcolor=yellow>4</td><td bgcolor=yellow> </td><td> </td><td>4</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 bgcolor=yellow>6</td><td bgcolor=yellow>3</td><td> </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 bgcolor=yellow>9</td><td bgcolor=yellow>4</td><td> </td><td> </td><td> </td><td>5</td></tr> <tr align=center><td>-14</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td bgcolor=yellow>8</td><td bgcolor=yellow>7</td><td> </td><td> </td><td> </td><td> </td><td>-1</td></tr> <tr align=center><td>-16</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>-18</td><td> </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>3</td></tr> <tr align=center><td>-20</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>-22</td><td> </td><td> </td><td bgcolor=yellow>1</td><td bgcolor=yellow>5</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>-24</td><td> </td><td bgcolor=yellow>1</td><td bgcolor=yellow>5</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>-26</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>-28</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)

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

[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.

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