L10a79

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Contents

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

Image:L10a79.gif

(Knotscape image)

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

Visit L10a79's page at Knotilus.

Visit L10a79's page at the original Knot Atlas.

[edit L10a79 Quick Notes]

[edit L10a79 Further Notes and Views]

[edit] Link Presentations

[edit Notes on L10a79's Link Presentations]

Planar diagram presentation	X₈₁₉₂ X_12,3,13,4 X_20,13,7,14 X_14,9,15,10 X_10,19,11,20 X_16,6,17,5 X_18,16,19,15 X₂₇₃₈ X_4,11,5,12 X_6,18,1,17
Gauss code	{1, -8, 2, -9, 6, -10}, {8, -1, 4, -5, 9, -2, 3, -4, 7, -6, 10, -7, 5, -3}

A Braid Representative

A Morse Link Presentation

Image:L10a79_ML.gif

[edit] Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-\frac{2 u^2 v^3-5 u^2 v^2+4 u^2 v-u^2+u v^4-5 u v^3+9 u v^2-5 u v+u-v^4+4 v^3-5 v^2+2 v}{u v^2}$ (db)
Jones polynomial	$q^{5/2}-4 q^{3/2}+8 \sqrt{q}-\frac{12}{\sqrt{q}}+\frac{14}{q^{3/2}}-\frac{16}{q^{5/2}}+\frac{13}{q^{7/2}}-\frac{11}{q^{9/2}}+\frac{7}{q^{11/2}}-\frac{3}{q^{13/2}}+\frac{1}{q^{15/2}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$- z a 7 - a 7 z -1 + 3 z 3 a 5 + 5 z a 5 + 3 a 5 z -1 -2 z 5 a 3 -5 z 3 a 3 -6 z a 3 -2 a 3 z -1 - z 5 a + z a + z 3 a -1$ (db)
Kauffman polynomial	$a 8 z 6 -3 a 8 z 4 + 3 a 8 z 2 - a 8 + 3 a 7 z 7 -8 a 7 z 5 + 7 a 7 z 3 -3 a 7 z + a 7 z -1 + 4 a 6 z 8 -7 a 6 z 6 -2 a 6 z 4 + 7 a 6 z 2 -3 a 6 + 2 a 5 z 9 + 6 a 5 z 7 -26 a 5 z 5 + 24 a 5 z 3 -10 a 5 z + 3 a 5 z -1 + 11 a 4 z 8 -20 a 4 z 6 + 3 a 4 z 4 + 6 a 4 z 2 -3 a 4 + 2 a 3 z 9 + 13 a 3 z 7 -36 a 3 z 5 + 27 a 3 z 3 -9 a 3 z + 2 a 3 z -1 + 7 a 2 z 8 -4 a 2 z 6 -7 a 2 z 4 + z 4 a -2 + 4 a 2 z 2 + 10 a z 7 -14 a z 5 + 4 z 5 a -1 + 8 a z 3 -2 z 3 a -1 -2 a z + 8 z 6 -8 z 4 + 2 z 2$ (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 L10a79. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

<table border=1> <tr align=center> <td width=13.3333%><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=6.66667%>-7</td><td width=6.66667%>-6</td><td width=6.66667%>-5</td><td width=6.66667%>-4</td><td width=6.66667%>-3</td><td width=6.66667%>-2</td><td width=6.66667%>-1</td><td width=6.66667%>0</td><td width=6.66667%>1</td><td width=6.66667%>2</td><td width=6.66667%>3</td><td width=13.3333%>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 bgcolor=yellow>1</td><td>-1</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 bgcolor=yellow>3</td><td bgcolor=yellow> </td><td>3</td></tr> <tr align=center><td>2</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>1</td><td> </td><td>-4</td></tr> <tr align=center><td>0</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td bgcolor=yellow>7</td><td bgcolor=yellow>3</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>-2</td></tr> <tr align=center><td>-4</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>-6</td><td> </td><td> </td><td> </td><td> </td><td bgcolor=yellow>6</td><td bgcolor=yellow>9</td><td> </td><td> </td><td> </td><td> </td><td> </td><td>3</td></tr> <tr align=center><td>-8</td><td> </td><td> </td><td> </td><td bgcolor=yellow>5</td><td bgcolor=yellow>7</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td>-2</td></tr> <tr align=center><td>-10</td><td> </td><td> </td><td bgcolor=yellow>2</td><td bgcolor=yellow>6</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> </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>-14</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>2</td></tr> <tr align=center><td>-16</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> </table>

Integral Khovanov Homology

(db, data source)

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

[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/L10a79"

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