L8a1

L7n2

L8a2

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L8a1 at Knotilus!
	L8a1 is $8_{13}^{2}$ in the Rolfsen table of links.

Link Presentations

[edit Notes on L8a1's Link Presentations]

Planar diagram presentation	X6172 X14,7,15,8 X4,15,1,16 X12,10,13,9 X8493 X10,5,11,6 X16,11,5,12 X2,14,3,13
Gauss code	{1, -8, 5, -3}, {6, -1, 2, -5, 4, -6, 7, -4, 8, -2, 3, -7}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(t(1)-1)(t(2)-1)\left(t(2)^{2}-3t(2)+1\right)}{{\sqrt {t(1)}}t(2)^{3/2}}}$ (db)
Jones polynomial	$q^{5/2}-4q^{3/2}+5{\sqrt {q}}-{\frac {7}{\sqrt {q}}}+{\frac {7}{q^{3/2}}}-{\frac {7}{q^{5/2}}}+{\frac {5}{q^{7/2}}}-{\frac {3}{q^{9/2}}}+{\frac {1}{q^{11/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{5}(-z)+2a^{3}z^{3}+2a^{3}z-az^{5}-2az^{3}+z^{3}a^{-1}-az+az^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$a^{6}z^{4}-a^{6}z^{2}+3a^{5}z^{5}-4a^{5}z^{3}+2a^{5}z+4a^{4}z^{6}-5a^{4}z^{4}+2a^{4}z^{2}+2a^{3}z^{7}+4a^{3}z^{5}-10a^{3}z^{3}+4a^{3}z+9a^{2}z^{6}-14a^{2}z^{4}+z^{4}a^{-2}+5a^{2}z^{2}+2az^{7}+5az^{5}+4z^{5}a^{-1}-11az^{3}-5z^{3}a^{-1}+2az+az^{-1}+a^{-1}z^{-1}+5z^{6}-7z^{4}+2z^{2}-1$ (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   \ -5 -4 -3 -2 -1 0 1 2 3 χ 6                 1 -1 4               3   3 2             2 1   -1 0           5 3     2 -2         4 4       0 -4       3 3         0 -6     2 4           2 -8   1 3             -2 -10   2               2 -12 1                 -1

Integral Khovanov Homology (db, data source)

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

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.

L7n2 L8a2