L10a8

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L10a9

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a8 at Knotilus!

Link Presentations

[edit Notes on L10a8's Link Presentations]

Planar diagram presentation	X6172 X10,4,11,3 X12,8,13,7 X16,10,17,9 X20,17,5,18 X18,13,19,14 X14,19,15,20 X8,16,9,15 X2536 X4,12,1,11
Gauss code	{1, -9, 2, -10}, {9, -1, 3, -8, 4, -2, 10, -3, 6, -7, 8, -4, 5, -6, 7, -5}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(t(1)-1)(t(2)-1)\left(t(2)^{4}-2t(2)^{3}+4t(2)^{2}-2t(2)+1\right)}{{\sqrt {t(1)}}t(2)^{5/2}}}$ (db)
Jones polynomial	$q^{11/2}-3q^{9/2}+7q^{7/2}-11q^{5/2}+12q^{3/2}-14{\sqrt {q}}+{\frac {12}{\sqrt {q}}}-{\frac {10}{q^{3/2}}}+{\frac {6}{q^{5/2}}}-{\frac {3}{q^{7/2}}}+{\frac {1}{q^{9/2}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$-z^{7}a^{-1}+2az^{5}-5z^{5}a^{-1}+z^{5}a^{-3}-a^{3}z^{3}+7az^{3}-11z^{3}a^{-1}+3z^{3}a^{-3}-2a^{3}z+9az-11za^{-1}+4za^{-3}-a^{3}z^{-1}+4az^{-1}-4a^{-1}z^{-1}+a^{-3}z^{-1}$ (db)
Kauffman polynomial	$-az^{9}-z^{9}a^{-1}-3a^{2}z^{8}-5z^{8}a^{-2}-8z^{8}-3a^{3}z^{7}-10az^{7}-15z^{7}a^{-1}-8z^{7}a^{-3}-a^{4}z^{6}+3a^{2}z^{6}-6z^{6}a^{-4}+10z^{6}+9a^{3}z^{5}+35az^{5}+43z^{5}a^{-1}+14z^{5}a^{-3}-3z^{5}a^{-5}+3a^{4}z^{4}+9a^{2}z^{4}+14z^{4}a^{-2}+7z^{4}a^{-4}-z^{4}a^{-6}+12z^{4}-9a^{3}z^{3}-35az^{3}-43z^{3}a^{-1}-15z^{3}a^{-3}+2z^{3}a^{-5}-3a^{4}z^{2}-12a^{2}z^{2}-15z^{2}a^{-2}-5z^{2}a^{-4}+z^{2}a^{-6}-18z^{2}+4a^{3}z+17az+21za^{-1}+8za^{-3}+a^{4}+4a^{2}+4a^{-2}+a^{-4}+7-a^{3}z^{-1}-4az^{-1}-4a^{-1}z^{-1}-a^{-3}z^{-1}$ (db)

Khovanov Homology

The coefficients of the monomials $t^{r}q^{j}$ are shown, along with their alternating sums $\chi$ (fixed $j$, alternation over $r$).

\   r    \    j   \ -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 12                     1 -1 10                   2   2 8                 5 1   -4 6               6 2     4 4             6 5       -1 2           8 6         2 0         6 8           2 -2       4 6             -2 -4     2 6               4 -6   1 4                 -3 -8   2                   2 -10 1                     -1

Integral Khovanov Homology (db, data source)

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

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

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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