L8n7

L8n6

L8n8

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L8n7 at Knotilus!
	L8n7 is $8_{2}^{4}$ in the Rolfsen table of links.

Link Presentations

[edit Notes on L8n7's Link Presentations]

Planar diagram presentation	X6172 X2536 X11,13,12,16 X3,11,4,10 X9,1,10,4 X7,15,8,14 X13,5,14,8 X15,9,16,12
Gauss code	{1, -2, -4, 5}, {2, -1, -6, 7}, {-5, 4, -3, 8}, {-7, 6, -8, 3}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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