L11a28

From Knot Atlas

Jump to: navigation, search

Contents

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

Image:L11a28.gif

(Knotscape image)

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

Visit L11a28's page at Knotilus.

Visit L11a28's page at the original Knot Atlas.

[edit L11a28 Quick Notes]

[edit L11a28 Further Notes and Views]

[edit] Link Presentations

[edit Notes on L11a28's Link Presentations]

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

A Braid Representative

A Morse Link Presentation

Image:L11a28_ML.gif

[edit] Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-\frac{(t(1)-1) (t(2)-1)^3 \left(t(2)^2-3 t(2)+1\right)}{\sqrt{t(1)} t(2)^{5/2}}$ (db)
Jones polynomial	$-10 q^{9/2}+\frac{1}{q^{9/2}}+17 q^{7/2}-\frac{4}{q^{7/2}}-23 q^{5/2}+\frac{9}{q^{5/2}}+26 q^{3/2}-\frac{17}{q^{3/2}}-q^{13/2}+4 q^{11/2}-26 \sqrt{q}+\frac{22}{\sqrt{q}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$- z 3 a -5 - z a -5 - a -5 z -1 + 2 z 5 a -3 - a 3 z 3 + 4 z 3 a -3 - a 3 z + 5 z a -3 + 3 a -3 z -1 - z 7 a -1 + 2 a z 5 -3 z 5 a -1 + 4 a z 3 -6 z 3 a -1 + 4 a z -7 z a -1 + 2 a z -1 -4 a -1 z -1$ (db)
Kauffman polynomial	$z 5 a -7 - z 3 a -7 + 4 z 6 a -6 -4 z 4 a -6 + z 2 a -6 + 9 z 7 a -5 -13 z 5 a -5 + 10 z 3 a -5 -4 z a -5 + a -5 z -1 + 12 z 8 a -4 + a 4 z 6 -18 z 6 a -4 -2 a 4 z 4 + 13 z 4 a -4 + a 4 z 2 -5 z 2 a -4 + a -4 + 8 z 9 a -3 + 4 a 3 z 7 + 4 z 7 a -3 -9 a 3 z 5 -32 z 5 a -3 + 8 a 3 z 3 + 34 z 3 a -3 -3 a 3 z -16 z a -3 + 3 a -3 z -1 + 2 z 10 a -2 + 7 a 2 z 8 + 24 z 8 a -2 -13 a 2 z 6 -58 z 6 a -2 + 7 a 2 z 4 + 45 z 4 a -2 - a 2 z 2 -17 z 2 a -2 + a 2 + 3 a -2 + 6 a z 9 + 14 z 9 a -1 + a z 7 -8 z 7 a -1 -27 a z 5 -36 z 5 a -1 + 30 a z 3 + 45 z 3 a -1 -13 a z -22 z a -1 + 2 a z -1 + 4 a -1 z -1 + 2 z 10 + 19 z 8 -50 z 6 + 37 z 4 -13 z 2 + 2$ (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 L11a28. 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%>-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=6.25%>6</td><td width=12.5%>χ</td></tr> <tr align=center><td>14</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>12</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> </td><td>-3</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>7</td><td bgcolor=yellow>1</td><td> </td><td>6</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 bgcolor=yellow>10</td><td bgcolor=yellow>3</td><td> </td><td> </td><td>-7</td></tr> <tr align=center><td>6</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td bgcolor=yellow>13</td><td bgcolor=yellow>7</td><td> </td><td> </td><td> </td><td>6</td></tr> <tr align=center><td>4</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td bgcolor=yellow>13</td><td bgcolor=yellow>10</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>13</td><td bgcolor=yellow>13</td><td> </td><td> </td><td> </td><td> </td><td> </td><td>0</td></tr> <tr align=center><td>0</td><td> </td><td> </td><td> </td><td> </td><td bgcolor=yellow>11</td><td bgcolor=yellow>15</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td>4</td></tr> <tr align=center><td>-2</td><td> </td><td> </td><td> </td><td bgcolor=yellow>6</td><td bgcolor=yellow>11</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td>-5</td></tr> <tr align=center><td>-4</td><td> </td><td> </td><td bgcolor=yellow>3</td><td bgcolor=yellow>11</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td>8</td></tr> <tr align=center><td>-6</td><td> </td><td bgcolor=yellow>1</td><td bgcolor=yellow>6</td><td> </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>-8</td><td bgcolor=yellow> </td><td bgcolor=yellow>3</td><td> </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>-10</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)

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = 0$	$i = 2$
$r = -5$	${\mathbb Z}$
$r = -4$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -3$	${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = -2$	${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{6}$	${\mathbb Z}^{6}$
$r = -1$	${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{11}$	${\mathbb Z}^{11}$
$r = 0$	${\mathbb Z}^{15}\oplus{\mathbb Z}_2^{11}$	${\mathbb Z}^{13}$
$r = 1$	${\mathbb Z}^{13}\oplus{\mathbb Z}_2^{13}$	${\mathbb Z}^{13}$
$r = 2$	${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{13}$	${\mathbb Z}^{13}$
$r = 3$	${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{10}$	${\mathbb Z}^{10}$
$r = 4$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{7}$	${\mathbb Z}^{7}$
$r = 5$	${\mathbb Z}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = 6$	${\mathbb Z}_2$	${\mathbb Z}$

[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.

Retrieved from "http://katlas.math.toronto.edu/wiki/L11a28"

Category: Knot Page

Views

Personal tools

Log in / create account

Search

Toolbox

tour