L7a3

L7a2

L7a4

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L7a3 at Knotilus!
	L7a3 is $7_{4}^{2}$ in the Rolfsen table of links.

Link Presentations

[edit Notes on L7a3's Link Presentations]

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

A Braid Representative	{{{braid_table}}}
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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.

L7a2 L7a4