L8a5

L8a4

L8a6

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L8a5 at Knotilus!
	L8a5 is $8_{11}^{2}$ in the Rolfsen table of links.

Link Presentations

[edit Notes on L8a5's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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