L8a2

L8a1

L8a3

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L8a2 at Knotilus!
	L8a2 is $8_{10}^{2}$ in the Rolfsen table of links.

Link Presentations

[edit Notes on L8a2's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(t(1)-1)(t(2)-1)^{3}}{{\sqrt {t(1)}}t(2)^{3/2}}}$ (db)
Jones polynomial	$-q^{9/2}+3q^{7/2}-5q^{5/2}+5q^{3/2}-6{\sqrt {q}}+{\frac {5}{\sqrt {q}}}-{\frac {4}{q^{3/2}}}+{\frac {2}{q^{5/2}}}-{\frac {1}{q^{7/2}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$z^{5}a^{-1}-2az^{3}+3z^{3}a^{-1}-z^{3}a^{-3}+a^{3}z-4az+4za^{-1}-za^{-3}+a^{3}z^{-1}-2az^{-1}+2a^{-1}z^{-1}-a^{-3}z^{-1}$ (db)
Kauffman polynomial	$z^{3}a^{-5}+3z^{4}a^{-4}-z^{2}a^{-4}+a^{3}z^{5}+5z^{5}a^{-3}-3a^{3}z^{3}-6z^{3}a^{-3}+3a^{3}z+3za^{-3}-a^{3}z^{-1}-a^{-3}z^{-1}+2a^{2}z^{6}+4z^{6}a^{-2}-5a^{2}z^{4}-4z^{4}a^{-2}+3a^{2}z^{2}+z^{2}a^{-2}+az^{7}+z^{7}a^{-1}+3az^{5}+7z^{5}a^{-1}-13az^{3}-17z^{3}a^{-1}+10az+10za^{-1}-2az^{-1}-2a^{-1}z^{-1}+6z^{6}-12z^{4}+5z^{2}-1$ (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   \ -4 -3 -2 -1 0 1 2 3 4 χ 10                 1 1 8               2   -2 6             3 1   2 4           2 2     0 2         4 3       1 0       3 4         1 -2     1 2           -1 -4   1 3             2 -6   1               -1 -8 1                 1

Integral Khovanov Homology (db, data source)

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

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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