L8a17

L8a16

L8a18

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L8a17 at Knotilus!
	L8a17 is $8_{2}^{3}$ in the Rolfsen table of links.

Link Presentations

[edit Notes on L8a17's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {t(3)^{2}t(2)^{2}+t(1)t(3)t(2)^{2}-t(3)t(2)^{2}+t(1)t(3)^{2}t(2)-t(3)^{2}t(2)+t(1)t(2)-2t(1)t(3)t(2)+2t(3)t(2)-t(2)-t(1)+t(1)t(3)-t(3)}{{\sqrt {t(1)}}t(2)t(3)}}$ (db)
Jones polynomial	$q^{-10}-2q^{-9}+4q^{-8}-4q^{-7}+6q^{-6}-4q^{-5}+4q^{-4}-2q^{-3}+q^{-2}$ (db)
Signature	-4 (db)
HOMFLY-PT polynomial	$a^{10}z^{-2}+a^{10}-3a^{8}z^{2}-2a^{8}z^{-2}-6a^{8}+2a^{6}z^{4}+6a^{6}z^{2}+a^{6}z^{-2}+5a^{6}+a^{4}z^{4}+2a^{4}z^{2}$ (db)
Kauffman polynomial	$a^{12}z^{4}-2a^{12}z^{2}+a^{12}+2a^{11}z^{5}-3a^{11}z^{3}+2a^{10}z^{6}-2a^{10}z^{4}+a^{10}z^{2}+a^{10}z^{-2}-3a^{10}+a^{9}z^{7}+2a^{9}z^{5}-6a^{9}z^{3}+6a^{9}z-2a^{9}z^{-1}+5a^{8}z^{6}-12a^{8}z^{4}+15a^{8}z^{2}+2a^{8}z^{-2}-8a^{8}+a^{7}z^{7}+2a^{7}z^{5}-6a^{7}z^{3}+6a^{7}z-2a^{7}z^{-1}+3a^{6}z^{6}-8a^{6}z^{4}+10a^{6}z^{2}+a^{6}z^{-2}-5a^{6}+2a^{5}z^{5}-3a^{5}z^{3}+a^{4}z^{4}-2a^{4}z^{2}$ (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   \ -8 -7 -6 -5 -4 -3 -2 -1 0 χ -3                 1 1 -5               2 1 -1 -7             2     2 -9           2 2     0 -11         4 2       2 -13       1 3         2 -15     3 3           0 -17   1 3             2 -19   1               -1 -21 1                 1

Integral Khovanov Homology (db, data source)

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

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.

L8a16 L8a18