L7a7

L7a6

L7n1

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L7a7 at Knotilus!
	L7a7 is $7_{1}^{3}$ in the Rolfsen table of links.

Link Presentations

[edit Notes on L7a7's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X14,12,9,11 X8,14,5,13 X12,8,13,7 X2536 X4,9,1,10
Gauss code	{1, -6, 2, -7}, {6, -1, 5, -4}, {7, -2, 3, -5, 4, -3}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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.

L7a6 L7n1