L10a2

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Contents

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

Image:L10a2.gif

(Knotscape image)

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

Visit L10a2's page at Knotilus.

Visit L10a2's page at the original Knot Atlas.

[edit L10a2 Quick Notes]

[edit L10a2 Further Notes and Views]

[edit] Link Presentations

[edit Notes on L10a2's Link Presentations]

Planar diagram presentation	X₆₁₇₂ X_16,7,17,8 X_4,17,1,18 X_14,10,15,9 X₈₄₉₃ X_10,5,11,6 X_20,13,5,14 X_18,11,19,12 X_12,19,13,20 X_2,16,3,15
Gauss code	{1, -10, 5, -3}, {6, -1, 2, -5, 4, -6, 8, -9, 7, -4, 10, -2, 3, -8, 9, -7}

A Braid Representative

A Morse Link Presentation

Image:L10a2_ML.gif

[edit] Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-\frac{(t(1)-1) (t(2)-1) \left(t(2)^4-3 t(2)^3+3 t(2)^2-3 t(2)+1\right)}{\sqrt{t(1)} t(2)^{5/2}}$ (db)
Jones polynomial	$\frac{14}{q^{9/2}}-\frac{15}{q^{7/2}}+\frac{13}{q^{5/2}}+q^{3/2}-\frac{11}{q^{3/2}}+\frac{1}{q^{17/2}}-\frac{4}{q^{15/2}}+\frac{8}{q^{13/2}}-\frac{11}{q^{11/2}}-4 \sqrt{q}+\frac{6}{\sqrt{q}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a^7 \left(-z^3\right)-a^7 z+2 a^5 z^5+5 a^5 z^3+2 a^5 z-a^5 z^{-1} -a^3 z^7-4 a^3 z^5-5 a^3 z^3+3 a^3 z^{-1} +a z^5+2 a z^3-a z-2 a z^{-1}$ (db)
Kauffman polynomial	$a 10 z 4 + 4 a 9 z 5 -2 a 9 z 3 + 8 a 8 z 6 -9 a 8 z 4 + 3 a 8 z 2 + 9 a 7 z 7 -11 a 7 z 5 + 5 a 7 z 3 -2 a 7 z + 6 a 6 z 8 - a 6 z 6 -9 a 6 z 4 + 2 a 6 z 2 + a 6 + 2 a 5 z 9 + 11 a 5 z 7 -30 a 5 z 5 + 20 a 5 z 3 -4 a 5 z - a 5 z -1 + 11 a 4 z 8 -23 a 4 z 6 + 12 a 4 z 4 -4 a 4 z 2 + 3 a 4 + 2 a 3 z 9 + 6 a 3 z 7 -27 a 3 z 5 + 22 a 3 z 3 - a 3 z -3 a 3 z -1 + 5 a 2 z 8 -13 a 2 z 6 + 9 a 2 z 4 -3 a 2 z 2 + 3 a 2 + 4 a z 7 -12 a z 5 + 9 a z 3 + a z -2 a z -1 + z 6 -2 z 4$ (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 =

-3 is the signature of L10a2. 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>4</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>2</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>0</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> </td><td>-2</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>8</td><td bgcolor=yellow>3</td><td> </td><td> </td><td>5</td></tr> <tr align=center><td>-4</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td bgcolor=yellow>7</td><td bgcolor=yellow>5</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> </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>-8</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>1</td></tr> <tr align=center><td>-10</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>-12</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>3</td></tr> <tr align=center><td>-14</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>-16</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>3</td></tr> <tr align=center><td>-18</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)

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

[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.

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