L8a3

L8a2

L8a4

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L8a3 at Knotilus!
	L8a3 is $8_{9}^{2}$ in the Rolfsen table of links.

Link Presentations

[edit Notes on L8a3's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {2uv^{2}-4uv+u+v^{3}-4v^{2}+2v}{{\sqrt {u}}v^{3/2}}}$ (db)
Jones polynomial	$-{\frac {4}{q^{9/2}}}+{\frac {4}{q^{7/2}}}-{\frac {5}{q^{5/2}}}-q^{3/2}+{\frac {5}{q^{3/2}}}-{\frac {1}{q^{13/2}}}+{\frac {2}{q^{11/2}}}+2{\sqrt {q}}-{\frac {4}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{7}z^{-1}-3za^{5}-2a^{5}z^{-1}+2z^{3}a^{3}+3za^{3}+2a^{3}z^{-1}+z^{3}a-za-az^{-1}-za^{-1}$ (db)
Kauffman polynomial	$-z^{5}a^{7}+3z^{3}a^{7}-3za^{7}+a^{7}z^{-1}-2z^{6}a^{6}+5z^{4}a^{6}-2z^{2}a^{6}-z^{7}a^{5}-2z^{5}a^{5}+11z^{3}a^{5}-8za^{5}+2a^{5}z^{-1}-5z^{6}a^{4}+11z^{4}a^{4}-6z^{2}a^{4}+a^{4}-z^{7}a^{3}-4z^{5}a^{3}+12z^{3}a^{3}-9za^{3}+2a^{3}z^{-1}-3z^{6}a^{2}+4z^{4}a^{2}-3z^{2}a^{2}-3z^{5}a+3z^{3}a-3za+az^{-1}-2z^{4}+z^{2}-z^{3}a^{-1}+za^{-1}$ (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   \ -6 -5 -4 -3 -2 -1 0 1 2 χ 4                 1 1 2               1   -1 0             3 1   2 -2           3 2     -1 -4         2 2       0 -6       2 3         1 -8     2 2           0 -10   1 3             2 -12   1               -1 -14 1                 1

Integral Khovanov Homology (db, data source)

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

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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