L9a52

L9a51

L9a53

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9a52 at Knotilus!
	L9a52 is $9_{8}^{3}$ in the Rolfsen table of links.

Link Presentations

[edit Notes on L9a52's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X16,11,17,12 X14,8,15,7 X8,14,9,13 X18,15,13,16 X12,17,5,18 X2536 X4,9,1,10
Gauss code	{1, -8, 2, -9}, {5, -4, 6, -3, 7, -6}, {8, -1, 4, -5, 9, -2, 3, -7}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(v-1)(w-1)\left(2uw-u+w^{2}-2w\right)}{{\sqrt {u}}{\sqrt {v}}w^{3/2}}}$ (db)
Jones polynomial	$-q^{-8}+3q^{-7}-5q^{-6}+8q^{-5}-7q^{-4}+9q^{-3}-7q^{-2}+q+5q^{-1}-2$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$-a^{8}+3a^{6}z^{2}+a^{6}z^{-2}+3a^{6}-2a^{4}z^{4}-4a^{4}z^{2}-2a^{4}z^{-2}-4a^{4}-a^{2}z^{4}+a^{2}z^{-2}+a^{2}+z^{2}+1$ (db)
Kauffman polynomial	$a^{9}z^{5}-2a^{9}z^{3}+a^{9}z+3a^{8}z^{6}-7a^{8}z^{4}+5a^{8}z^{2}-2a^{8}+3a^{7}z^{7}-3a^{7}z^{5}-5a^{7}z^{3}+3a^{7}z+a^{6}z^{8}+7a^{6}z^{6}-22a^{6}z^{4}+20a^{6}z^{2}+a^{6}z^{-2}-9a^{6}+6a^{5}z^{7}-8a^{5}z^{5}-2a^{5}z^{3}+7a^{5}z-2a^{5}z^{-1}+a^{4}z^{8}+7a^{4}z^{6}-19a^{4}z^{4}+20a^{4}z^{2}+2a^{4}z^{-2}-10a^{4}+3a^{3}z^{7}-2a^{3}z^{5}-a^{3}z^{3}+5a^{3}z-2a^{3}z^{-1}+3a^{2}z^{6}-3a^{2}z^{4}+3a^{2}z^{2}+a^{2}z^{-2}-3a^{2}+2az^{5}-2az^{3}+z^{4}-2z^{2}+1$ (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   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 χ 3                   1 1 1                 1   -1 -1               4 1   3 -3             5 3     -2 -5           4 2       2 -7         3 5         2 -9       5 4           1 -11     2 5             3 -13   1 3               -2 -15   2                 2 -17 1                   -1

Integral Khovanov Homology (db, data source)

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

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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