L9a10

L9a9

L9a11

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9a10 at Knotilus!
	L9a10 is $9_{36}^{2}$ in the Rolfsen table of links.

Link Presentations

[edit Notes on L9a10's Link Presentations]

Planar diagram presentation	X6172 X16,7,17,8 X4,17,1,18 X12,6,13,5 X10,4,11,3 X18,12,5,11 X14,9,15,10 X2,14,3,13 X8,15,9,16
Gauss code	{1, -8, 5, -3}, {4, -1, 2, -9, 7, -5, 6, -4, 8, -7, 9, -2, 3, -6}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {2(u-1)(v-1)\left(v^{2}-v+1\right)}{{\sqrt {u}}v^{3/2}}}$ (db)
Jones polynomial	$-q^{9/2}+{\frac {1}{q^{9/2}}}+3q^{7/2}-{\frac {3}{q^{7/2}}}-6q^{5/2}+{\frac {4}{q^{5/2}}}+7q^{3/2}-{\frac {7}{q^{3/2}}}-8{\sqrt {q}}+{\frac {8}{\sqrt {q}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$az^{5}+z^{5}a^{-1}-a^{3}z^{3}+2az^{3}+2z^{3}a^{-1}-z^{3}a^{-3}-a^{3}z+2za^{-1}-za^{-3}+a^{3}z^{-1}-2az^{-1}+2a^{-1}z^{-1}-a^{-3}z^{-1}$ (db)
Kauffman polynomial	$z^{3}a^{-5}+a^{4}z^{6}-3a^{4}z^{4}+3z^{4}a^{-4}+2a^{4}z^{2}+3a^{3}z^{7}-11a^{3}z^{5}+6z^{5}a^{-3}+11a^{3}z^{3}-6z^{3}a^{-3}-a^{3}z+3za^{-3}-a^{3}z^{-1}-a^{-3}z^{-1}+2a^{2}z^{8}-3a^{2}z^{6}+7z^{6}a^{-2}-4a^{2}z^{4}-10z^{4}a^{-2}+4a^{2}z^{2}+4z^{2}a^{-2}+8az^{7}+5z^{7}a^{-1}-22az^{5}-5z^{5}a^{-1}+13az^{3}-5z^{3}a^{-1}+2az+6za^{-1}-2az^{-1}-2a^{-1}z^{-1}+2z^{8}+3z^{6}-14z^{4}+6z^{2}-1$ (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   \ -5 -4 -3 -2 -1 0 1 2 3 4 χ 10                   1 1 8                 2   -2 6               4 1   3 4             3 2     -1 2           5 4       1 0         5 5         0 -2       2 3           -1 -4     2 5             3 -6   1 2               -1 -8   2                 2 -10 1                   -1

Integral Khovanov Homology (db, data source)

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

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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