L8a10

L8a9

L8a11

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L8a10 at Knotilus!
	L8a10 is $8_{5}^{2}$ in the Rolfsen table of links.

Symmetric version

Mongolian ornament

Link Presentations

[edit Notes on L8a10's Link Presentations]

Planar diagram presentation	X8192 X10,3,11,4 X12,15,13,16 X14,5,15,6 X4,13,5,14 X16,11,7,12 X2738 X6,9,1,10
Gauss code	{1, -7, 2, -5, 4, -8}, {7, -1, 8, -2, 6, -3, 5, -4, 3, -6}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {2t(2)t(1)^{2}-t(1)^{2}+2t(2)^{2}t(1)-3t(2)t(1)+2t(1)-t(2)^{2}+2t(2)}{t(1)t(2)}}$ (db)
Jones polynomial	$-{\frac {1}{q^{3/2}}}+{\frac {2}{q^{5/2}}}-{\frac {4}{q^{7/2}}}+{\frac {4}{q^{9/2}}}-{\frac {5}{q^{11/2}}}+{\frac {4}{q^{13/2}}}-{\frac {3}{q^{15/2}}}+{\frac {2}{q^{17/2}}}-{\frac {1}{q^{19/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a^{9}z-a^{7}z^{3}+a^{7}z^{-1}-2a^{5}z^{3}-3a^{5}z-a^{5}z^{-1}-a^{3}z^{3}-a^{3}z$ (db)
Kauffman polynomial	$-z^{5}a^{11}+3z^{3}a^{11}-2za^{11}-2z^{6}a^{10}+6z^{4}a^{10}-4z^{2}a^{10}-z^{7}a^{9}+4z^{3}a^{9}-za^{9}-4z^{6}a^{8}+8z^{4}a^{8}-3z^{2}a^{8}-z^{7}a^{7}-2z^{5}a^{7}+6z^{3}a^{7}-4za^{7}+a^{7}z^{-1}-2z^{6}a^{6}+2z^{2}a^{6}-a^{6}-3z^{5}a^{5}+4z^{3}a^{5}-4za^{5}+a^{5}z^{-1}-2z^{4}a^{4}+z^{2}a^{4}-z^{3}a^{3}+za^{3}$ (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 -2                 1 1 -4               2 1 -1 -6             2     2 -8           2 2     0 -10         3 2       1 -12       1 2         1 -14     2 3           -1 -16   1 2             1 -18   1               -1 -20 1                 1

Integral Khovanov Homology (db, data source)

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

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.

L8a9 L8a11