L7a4

L7a3

L7a5

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L7a4 at Knotilus!
	L7a4 is $7_{3}^{2}$ in the Rolfsen table of links.

Link Presentations

[edit Notes on L7a4's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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