L7a6

L7a5

L7a7

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L7a6 at Knotilus!
	L7a6 is $7_{1}^{2}$ in the Rolfsen table of links.

Link Presentations

[edit Notes on L7a6's Link Presentations]

Planar diagram presentation	X8192 X10,4,11,3 X14,10,7,9 X12,6,13,5 X2738 X4,12,5,11 X6,14,1,13
Gauss code	{1, -5, 2, -6, 4, -7}, {5, -1, 3, -2, 6, -4, 7, -3}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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