L11a35

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Contents

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

Image:L11a35.gif

(Knotscape image)

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

Visit L11a35's page at Knotilus.

Visit L11a35's page at the original Knot Atlas.

[edit L11a35 Quick Notes]

[edit L11a35 Further Notes and Views]

[edit] Link Presentations

[edit Notes on L11a35's Link Presentations]

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

A Braid Representative

A Morse Link Presentation

Image:L11a35_ML.gif

[edit] Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$\frac{t(2)^5+2 t(1) t(2)^4-6 t(2)^4-7 t(1) t(2)^3+9 t(2)^3+9 t(1) t(2)^2-7 t(2)^2-6 t(1) t(2)+2 t(2)+t(1)}{\sqrt{t(1)} t(2)^{5/2}}$ (db)
Jones polynomial	$-3 q^{9/2}+\frac{3}{q^{9/2}}+6 q^{7/2}-\frac{7}{q^{7/2}}-10 q^{5/2}+\frac{10}{q^{5/2}}+14 q^{3/2}-\frac{14}{q^{3/2}}+q^{11/2}-\frac{1}{q^{11/2}}-16 \sqrt{q}+\frac{15}{\sqrt{q}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$a 5 z + a 5 z -1 + z a -5 -3 a 3 z 3 -2 z 3 a -3 -5 a 3 z -2 a 3 z -1 - z a -3 + a -3 z -1 + 2 a z 5 + z 5 a -1 + 5 a z 3 - z 3 a -1 + 6 a z + 3 a z -1 -4 z a -1 -3 a -1 z -1$ (db)
Kauffman polynomial	$z 4 a -6 - z 2 a -6 + a 5 z 7 -4 a 5 z 5 + 3 z 5 a -5 + 6 a 5 z 3 -3 z 3 a -5 -4 a 5 z + z a -5 + a 5 z -1 + 3 a 4 z 8 -11 a 4 z 6 + 5 z 6 a -4 + 12 a 4 z 4 -5 z 4 a -4 -4 a 4 z 2 + 3 z 2 a -4 - a -4 + 3 a 3 z 9 -5 a 3 z 7 + 6 z 7 a -3 -11 a 3 z 5 -5 z 5 a -3 + 22 a 3 z 3 + z 3 a -3 -11 a 3 z - z a -3 + 2 a 3 z -1 + a -3 z -1 + a 2 z 10 + 8 a 2 z 8 + 6 z 8 a -2 -36 a 2 z 6 -5 z 6 a -2 + 38 a 2 z 4 -5 z 4 a -2 -14 a 2 z 2 + 8 z 2 a -2 + 2 a 2 -2 a -2 + 7 a z 9 + 4 z 9 a -1 -11 a z 7 + z 7 a -1 -18 a z 5 -19 z 5 a -1 + 35 a z 3 + 23 z 3 a -1 -18 a z -13 z a -1 + 3 a z -1 + 3 a -1 z -1 + z 10 + 11 z 8 -35 z 6 + 27 z 4 -6 z 2$ (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 L11a35. 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%>-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=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=12.5%>χ</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> </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> </td><td bgcolor=yellow>2</td><td bgcolor=yellow> </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>1</td><td> </td><td>-3</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 bgcolor=yellow>6</td><td bgcolor=yellow>2</td><td> </td><td> </td><td>4</td></tr> <tr align=center><td>4</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>2</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td bgcolor=yellow>8</td><td bgcolor=yellow>6</td><td> </td><td> </td><td> </td><td> </td><td>2</td></tr> <tr align=center><td>0</td><td> </td><td> </td><td> </td><td> </td><td> </td><td bgcolor=yellow>8</td><td bgcolor=yellow>9</td><td> </td><td> </td><td> </td><td> </td><td> </td><td>1</td></tr> <tr align=center><td>-2</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>-4</td><td> </td><td> </td><td> </td><td bgcolor=yellow>4</td><td bgcolor=yellow>8</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td>4</td></tr> <tr align=center><td>-6</td><td> </td><td> </td><td bgcolor=yellow>3</td><td bgcolor=yellow>6</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>-8</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>-10</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>-12</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)

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

[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.

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