L8a11

L8a10

L8a12

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L8a11 at Knotilus!
	L8a11 is $8_{3}^{2}$ in the Rolfsen table of links.

Link Presentations

[edit Notes on L8a11's Link Presentations]

Planar diagram presentation	X8192 X2,9,3,10 X10,3,11,4 X14,5,15,6 X16,11,7,12 X12,15,13,16 X6718 X4,13,5,14
Gauss code	{1, -2, 3, -8, 4, -7}, {7, -1, 2, -3, 5, -6, 8, -4, 6, -5}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {2t(2)^{2}t(1)^{2}-t(2)t(1)^{2}-t(2)^{2}t(1)+3t(2)t(1)-t(1)-t(2)+2}{t(1)t(2)}}$ (db)
Jones polynomial	$-{\frac {1}{q^{5/2}}}+{\frac {1}{q^{7/2}}}-{\frac {3}{q^{9/2}}}+{\frac {3}{q^{11/2}}}-{\frac {4}{q^{13/2}}}+{\frac {4}{q^{15/2}}}-{\frac {3}{q^{17/2}}}+{\frac {2}{q^{19/2}}}-{\frac {1}{q^{21/2}}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$z^{3}a^{9}+2za^{9}-z^{5}a^{7}-3z^{3}a^{7}-za^{7}+a^{7}z^{-1}-z^{5}a^{5}-4z^{3}a^{5}-4za^{5}-a^{5}z^{-1}$ (db)
Kauffman polynomial	$-z^{3}a^{13}+za^{13}-2z^{4}a^{12}+2z^{2}a^{12}-2z^{5}a^{11}+z^{3}a^{11}-2z^{6}a^{10}+3z^{4}a^{10}-3z^{2}a^{10}-z^{7}a^{9}+z^{5}a^{9}-2z^{3}a^{9}+2za^{9}-3z^{6}a^{8}+7z^{4}a^{8}-4z^{2}a^{8}-z^{7}a^{7}+2z^{5}a^{7}-za^{7}+a^{7}z^{-1}-z^{6}a^{6}+2z^{4}a^{6}+z^{2}a^{6}-a^{6}-z^{5}a^{5}+4z^{3}a^{5}-4za^{5}+a^{5}z^{-1}$ (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   \ -8 -7 -6 -5 -4 -3 -2 -1 0 χ -4                 1 1 -6               1 1 0 -8             2     2 -10           1 1     0 -12         3 2       1 -14       1 1         0 -16     2 3           -1 -18   1 2             1 -20   1               -1 -22 1                 1

Integral Khovanov Homology (db, data source)

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

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.

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