L7a1

L6n1

L7a2

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L7a1 at Knotilus!
	L7a1 is $7_{6}^{2}$ in the Rolfsen table of links.

Link Presentations

[edit Notes on L7a1's Link Presentations]

Planar diagram presentation	X6172 X12,7,13,8 X4,13,1,14 X10,6,11,5 X8493 X14,10,5,9 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 $u$, $v$, $w$, ...)	${\frac {(u-1)(v-1)\left(v^{2}-v+1\right)}{{\sqrt {u}}v^{3/2}}}$ (db)
Jones polynomial	$-q^{9/2}+3q^{7/2}-4q^{5/2}+{\frac {1}{q^{5/2}}}+4q^{3/2}-{\frac {3}{q^{3/2}}}-5{\sqrt {q}}+{\frac {3}{\sqrt {q}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$-z^{3}a^{-3}-za^{-3}+z^{5}a^{-1}-az^{3}+3z^{3}a^{-1}-az+2za^{-1}+az^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$z^{3}a^{-5}+3z^{4}a^{-4}-2z^{2}a^{-4}+4z^{5}a^{-3}-5z^{3}a^{-3}+2za^{-3}+2z^{6}a^{-2}+a^{2}z^{4}+z^{4}a^{-2}-a^{2}z^{2}-3z^{2}a^{-2}+3az^{5}+7z^{5}a^{-1}-6az^{3}-12z^{3}a^{-1}+2az+4za^{-1}+az^{-1}+a^{-1}z^{-1}+2z^{6}-z^{4}-2z^{2}-1$ (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   \ -3 -2 -1 0 1 2 3 4 χ 10               1 1 8             2   -2 6           2 1   1 4         2 2     0 2       3 2       1 0     2 4         2 -2   1 1           0 -4   2             2 -6 1               -1

Integral Khovanov Homology (db, data source)

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

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.

L6n1 L7a2