L11a42

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Contents

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

Image:L11a42.gif

(Knotscape image)

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

Visit L11a42's page at Knotilus.

Visit L11a42's page at the original Knot Atlas.

[edit L11a42 Quick Notes]

[edit L11a42 Further Notes and Views]

[edit] Link Presentations

[edit Notes on L11a42's Link Presentations]

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

A Braid Representative

A Morse Link Presentation

Image:L11a42_ML.gif

[edit] Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$\frac{(u-1) (v-1) \left(v^4-4 v^3+4 v^2-4 v+1\right)}{\sqrt{u} v^{5/2}}$ (db)
Jones polynomial	$q^{9/2}-\frac{7}{q^{9/2}}-3 q^{7/2}+\frac{12}{q^{7/2}}+6 q^{5/2}-\frac{16}{q^{5/2}}-12 q^{3/2}+\frac{18}{q^{3/2}}-\frac{1}{q^{13/2}}+\frac{3}{q^{11/2}}+15 \sqrt{q}-\frac{18}{\sqrt{q}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a z 7 -2 a 3 z 5 + 4 a z 5 -2 z 5 a -1 + a 5 z 3 -6 a 3 z 3 + 8 a z 3 -6 z 3 a -1 + z 3 a -3 + 2 a 5 z -7 a 3 z + 9 a z -6 z a -1 + 2 z a -3 + a 5 z -1 -3 a 3 z -1 + 4 a z -1 -2 a -1 z -1$ (db)
Kauffman polynomial	$- a 2 z 10 - z 10 -3 a 3 z 9 -6 a z 9 -3 z 9 a -1 -5 a 4 z 8 -9 a 2 z 8 -4 z 8 a -2 -8 z 8 -5 a 5 z 7 -7 a 3 z 7 - z 7 a -1 -3 z 7 a -3 -3 a 6 z 6 + 2 a 4 z 6 + 14 a 2 z 6 + 8 z 6 a -2 - z 6 a -4 + 18 z 6 - a 7 z 5 + 7 a 5 z 5 + 23 a 3 z 5 + 24 a z 5 + 18 z 5 a -1 + 9 z 5 a -3 + 5 a 6 z 4 + 7 a 4 z 4 -2 z 4 a -2 + 3 z 4 a -4 -7 z 4 + 2 a 7 z 3 -4 a 5 z 3 -24 a 3 z 3 -33 a z 3 -24 z 3 a -1 -9 z 3 a -3 -3 a 6 z 2 -8 a 4 z 2 -8 a 2 z 2 -2 z 2 a -2 -2 z 2 a -4 -3 z 2 - a 7 z + 2 a 5 z + 13 a 3 z + 19 a z + 13 z a -1 + 4 z a -3 + a 6 + 3 a 4 + 3 a 2 + 2- a 5 z -1 -3 a 3 z -1 -4 a z -1 -2 a -1 z -1$ (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 L11a42. 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>10</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> </td><td>2</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 bgcolor=yellow>4</td><td bgcolor=yellow>1</td><td> </td><td>-3</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 bgcolor=yellow>8</td><td bgcolor=yellow>2</td><td> </td><td> </td><td>6</td></tr> <tr align=center><td>2</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td bgcolor=yellow>7</td><td bgcolor=yellow>4</td><td> </td><td> </td><td> </td><td>-3</td></tr> <tr align=center><td>0</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td bgcolor=yellow>11</td><td bgcolor=yellow>8</td><td> </td><td> </td><td> </td><td> </td><td>3</td></tr> <tr align=center><td>-2</td><td> </td><td> </td><td> </td><td> </td><td> </td><td bgcolor=yellow>9</td><td bgcolor=yellow>9</td><td> </td><td> </td><td> </td><td> </td><td> </td><td>0</td></tr> <tr align=center><td>-4</td><td> </td><td> </td><td> </td><td> </td><td bgcolor=yellow>7</td><td bgcolor=yellow>9</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td>-2</td></tr> <tr align=center><td>-6</td><td> </td><td> </td><td> </td><td bgcolor=yellow>5</td><td bgcolor=yellow>9</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td>4</td></tr> <tr align=center><td>-8</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>-10</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>-12</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>-14</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)

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

[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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