L8a18

L8a17

L8a19

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L8a18 at Knotilus!
	L8a18 is $8_{1}^{3}$ in the Rolfsen table of links.

Link Presentations

[edit Notes on L8a18's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {uv^{2}w^{2}-uvw^{2}+uvw-uw+u-v^{2}w^{2}+v^{2}w-vw+v-1}{{\sqrt {u}}vw}}$ (db)
Jones polynomial	$q^{8}-2q^{7}+3q^{6}-3q^{5}+4q^{4}-2q^{3}+3q^{2}-q+1$ (db)
Signature	4 (db)
HOMFLY-PT polynomial	$z^{4}a^{-6}+3z^{2}a^{-6}+a^{-6}z^{-2}+2a^{-6}-z^{6}a^{-4}-5z^{4}a^{-4}-8z^{2}a^{-4}-2a^{-4}z^{-2}-6a^{-4}+z^{4}a^{-2}+4z^{2}a^{-2}+a^{-2}z^{-2}+4a^{-2}$ (db)
Kauffman polynomial	$z^{2}a^{-10}+2z^{3}a^{-9}+3z^{4}a^{-8}-3z^{2}a^{-8}+a^{-8}+3z^{5}a^{-7}-4z^{3}a^{-7}+3z^{6}a^{-6}-8z^{4}a^{-6}+6z^{2}a^{-6}+a^{-6}z^{-2}-3a^{-6}+z^{7}a^{-5}-7z^{3}a^{-5}+6za^{-5}-2a^{-5}z^{-1}+4z^{6}a^{-4}-16z^{4}a^{-4}+18z^{2}a^{-4}+2a^{-4}z^{-2}-8a^{-4}+z^{7}a^{-3}-3z^{5}a^{-3}-z^{3}a^{-3}+6za^{-3}-2a^{-3}z^{-1}+z^{6}a^{-2}-5z^{4}a^{-2}+8z^{2}a^{-2}+a^{-2}z^{-2}-5a^{-2}$ (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   \ -2 -1 0 1 2 3 4 5 6 χ 17                 1 1 15               2 1 -1 13             1     1 11           2 2     0 9         2 1       1 7       1 3         2 5     2 1           1 3   1 3             2 1                   0 -1 1                 1

Integral Khovanov Homology (db, data source)

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

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.

L8a17 L8a19