L9n20

L9n19

L9n21

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9n20 at Knotilus!
	L9n20 is $9_{16}^{3}$ in the Rolfsen table of links.

Link Presentations

[edit Notes on L9n20's Link Presentations]

Planar diagram presentation	X6172 X3,11,4,10 X7,15,8,14 X13,5,14,8 X11,17,12,16 X15,9,16,18 X17,13,18,12 X2536 X9,1,10,4
Gauss code	{1, -8, -2, 9}, {8, -1, -3, 4}, {-9, 2, -5, 7, -4, 3, -6, 5, -7, 6}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {-2uvw^{2}+2uvw-uv+uw^{2}-uw+vw^{2}-vw+w^{3}-2w^{2}+2w}{{\sqrt {u}}{\sqrt {v}}w^{3/2}}}$ (db)
Jones polynomial	$-q^{8}+2q^{7}-4q^{6}+5q^{5}-4q^{4}+6q^{3}-3q^{2}+3q$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$-2z^{4}a^{-4}+3z^{2}a^{-2}-7z^{2}a^{-4}+3z^{2}a^{-6}+5a^{-2}-10a^{-4}+6a^{-6}-a^{-8}+2a^{-2}z^{-2}-5a^{-4}z^{-2}+4a^{-6}z^{-2}-a^{-8}z^{-2}$ (db)
Kauffman polynomial	$z^{7}a^{-5}+z^{7}a^{-7}+4z^{6}a^{-4}+6z^{6}a^{-6}+2z^{6}a^{-8}+3z^{5}a^{-3}+5z^{5}a^{-5}+3z^{5}a^{-7}+z^{5}a^{-9}-11z^{4}a^{-4}-16z^{4}a^{-6}-5z^{4}a^{-8}-6z^{3}a^{-3}-18z^{3}a^{-5}-15z^{3}a^{-7}-3z^{3}a^{-9}+6z^{2}a^{-2}+18z^{2}a^{-4}+15z^{2}a^{-6}+3z^{2}a^{-8}+11za^{-3}+21za^{-5}+13za^{-7}+3za^{-9}-7a^{-2}-14a^{-4}-10a^{-6}-2a^{-8}-5a^{-3}z^{-1}-9a^{-5}z^{-1}-5a^{-7}z^{-1}-a^{-9}z^{-1}+2a^{-2}z^{-2}+5a^{-4}z^{-2}+4a^{-6}z^{-2}+a^{-8}z^{-2}$ (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   \ 0 1 2 3 4 5 6 7 χ 17               1 -1 15             1   1 13           3 1   -2 11         2 1     1 9       3 4       1 7     3 1         2 5     3           3 3 3 3             0 1 3               3

Integral Khovanov Homology (db, data source)

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

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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