L7n2

L7n1

L8a1

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L7n2 at Knotilus!
	L7n2 is $7_{8}^{2}$ in the Rolfsen table of links.

Link Presentations

[edit Notes on L7n2's Link Presentations]

Planar diagram presentation	X6172 X12,7,13,8 X13,1,14,4 X5,10,6,11 X3849 X9,14,10,5 X2,12,3,11
Gauss code	{1, -7, -5, 3}, {-4, -1, 2, 5, -6, 4, 7, -2, -3, 6}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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.

L7n1 L8a1