L7n1

L7a7

L7n2

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L7n1 at Knotilus!
	L7n1 is $7_{7}^{2}$ in the Rolfsen table of links.

Link Presentations

[edit Notes on L7n1's Link Presentations]

Planar diagram presentation	X6172 X12,7,13,8 X4,13,1,14 X5,10,6,11 X3849 X9,14,10,5 X11,2,12,3
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 $u$, $v$, $w$, ...)	${\frac {-uv^{3}-1}{{\sqrt {u}}v^{3/2}}}$ (db)
Jones polynomial	$-{\frac {1}{q^{5/2}}}-{\frac {1}{q^{9/2}}}-{\frac {1}{q^{13/2}}}+{\frac {1}{q^{15/2}}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$-a^{9}z^{-1}+a^{7}z^{3}+4a^{7}z+3a^{7}z^{-1}-a^{5}z^{5}-5a^{5}z^{3}-6a^{5}z-2a^{5}z^{-1}$ (db)
Kauffman polynomial	$a^{10}+a^{9}z-a^{9}z^{-1}+a^{8}z^{4}-4a^{8}z^{2}+3a^{8}+a^{7}z^{5}-5a^{7}z^{3}+7a^{7}z-3a^{7}z^{-1}+a^{6}z^{4}-4a^{6}z^{2}+3a^{6}+a^{5}z^{5}-5a^{5}z^{3}+6a^{5}z-2a^{5}z^{-1}$ (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   \ -5 -4 -3 -2 -1 0 χ -4           1 1 -6           1 1 -8       1     1 -10   1         1 -12   2 1       1 -14             0 -16 1           -1

Integral Khovanov Homology (db, data source)

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

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.

L7a7 L7n2