L7a5

L7a4

L7a6

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L7a5 at Knotilus!
	L7a5 is $7^2_2$ in the Rolfsen table of links.

Link Presentations

[edit Notes on L7a5's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Khovanov Homology

The coefficients of the monomials $t^rq^j$ are shown, along with their alternating sums $\chi$ (fixed $j$, alternation over $r$).

\   r    \    j   \ -5 -4 -3 -2 -1 0 1 2 χ 4               1 1 2             1   -1 0           2 1   1 -2         2 2     0 -4       2 1       1 -6     1 3         2 -8   1 1           0 -10   1             1 -12 1               -1

Integral Khovanov Homology (db, data source)

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-2$ $i=0$ $r=-5$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}$ $r=-1$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=0$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}$ $r=1$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=2$ ${\mathbb Z}_2$ ${\mathbb Z}$

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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