L7a2

L7a1

L7a3

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L7a2 at Knotilus!
	L7a2 is $7_{5}^{2}$ in the Rolfsen table of links.

Link Presentations

[edit Notes on L7a2's Link Presentations]

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {-2uv^{2}+2uv-u-v^{3}+2v^{2}-2v}{{\sqrt {u}}v^{3/2}}}$ (db)
Jones polynomial	$-{\frac {1}{q^{3/2}}}+{\frac {2}{q^{5/2}}}-{\frac {4}{q^{7/2}}}+{\frac {3}{q^{9/2}}}-{\frac {4}{q^{11/2}}}+{\frac {3}{q^{13/2}}}-{\frac {2}{q^{15/2}}}+{\frac {1}{q^{17/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$-a^{9}z^{-1}+3a^{7}z+3a^{7}z^{-1}-2a^{5}z^{3}-4a^{5}z-2a^{5}z^{-1}-a^{3}z^{3}-a^{3}z$ (db)
Kauffman polynomial	$a^{10}z^{4}-2a^{10}z^{2}+a^{10}+2a^{9}z^{5}-4a^{9}z^{3}+2a^{9}z-a^{9}z^{-1}+a^{8}z^{6}+2a^{8}z^{4}-7a^{8}z^{2}+3a^{8}+5a^{7}z^{5}-10a^{7}z^{3}+8a^{7}z-3a^{7}z^{-1}+a^{6}z^{6}+3a^{6}z^{4}-6a^{6}z^{2}+3a^{6}+3a^{5}z^{5}-5a^{5}z^{3}+5a^{5}z-2a^{5}z^{-1}+2a^{4}z^{4}-a^{4}z^{2}+a^{3}z^{3}-a^{3}z$ (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   \ -7 -6 -5 -4 -3 -2 -1 0 χ -2               1 1 -4             2 1 -1 -6           2     2 -8         1 2     1 -10       3 2       1 -12     1 2         1 -14   1 2           -1 -16   1             1 -18 1               -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-4$ $i=-2$ $r=-7$ ${\mathbb {Z} }$ $r=-6$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{3}$ $r=-3$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-2$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-1$ ${\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=0$ ${\mathbb {Z} }$ ${\mathbb {Z} }$

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.

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