L8a7

L8a6

L8a8

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L8a7 at Knotilus!
	L8a7 is $8_{14}^{2}$ in the Rolfsen table of links.

Link Presentations

[edit Notes on L8a7's Link Presentations]

Planar diagram presentation	X6172 X14,7,15,8 X4,15,1,16 X10,5,11,6 X12,3,13,4 X16,11,5,12 X2,9,3,10 X8,13,9,14
Gauss code	{1, -7, 5, -3}, {4, -1, 2, -8, 7, -4, 6, -5, 8, -2, 3, -6}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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