L8a20

L8a19

L8a21

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L8a20 at Knotilus!
	L8a20 is $8_{4}^{3}$ in the Rolfsen table of links.

Depiction obtained with knotilus

Link Presentations

[edit Notes on L8a20's Link Presentations]

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

A Braid Representative	{{{braid_table}}}
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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.

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