L9a40

L9a39

L9a41

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9a40 at Knotilus!
	L9a40 is $9_{4}^{2}$ in the Rolfsen table of links.

National swiss museum

Povray depiction

Link Presentations

[edit Notes on L9a40's Link Presentations]

Planar diagram presentation	X10,1,11,2 X2,11,3,12 X12,3,13,4 X8,9,1,10 X18,13,9,14 X14,8,15,7 X16,6,17,5 X6,16,7,15 X4,18,5,17
Gauss code	{1, -2, 3, -9, 7, -8, 6, -4}, {4, -1, 2, -3, 5, -6, 8, -7, 9, -5}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(u-1)(v-1)\left(u^{2}v+uv^{2}+uv+u+v\right)}{u^{3/2}v^{3/2}}}$ (db)
Jones polynomial	$-q^{5/2}+q^{3/2}-2{\sqrt {q}}+{\frac {3}{\sqrt {q}}}-{\frac {4}{q^{3/2}}}+{\frac {3}{q^{5/2}}}-{\frac {4}{q^{7/2}}}+{\frac {3}{q^{9/2}}}-{\frac {2}{q^{11/2}}}+{\frac {1}{q^{13/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$-z^{3}a^{5}-2za^{5}+z^{5}a^{3}+3z^{3}a^{3}+za^{3}+z^{5}a+4z^{3}a+4za+az^{-1}-z^{3}a^{-1}-3za^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$a^{8}z^{2}+2a^{7}z^{3}+3a^{6}z^{4}-2a^{6}z^{2}+4a^{5}z^{5}-8a^{5}z^{3}+4a^{5}z+3a^{4}z^{6}-6a^{4}z^{4}+a^{4}z^{2}+2a^{3}z^{7}-4a^{3}z^{5}-2a^{3}z^{3}+2a^{3}z+a^{2}z^{8}-2a^{2}z^{6}-2a^{2}z^{4}+2a^{2}z^{2}+3az^{7}+z^{7}a^{-1}-14az^{5}-6z^{5}a^{-1}+19az^{3}+11z^{3}a^{-1}-8az+az^{-1}-6za^{-1}+a^{-1}z^{-1}+z^{8}-5z^{6}+7z^{4}-2z^{2}-1$ (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   \ -5 -4 -3 -2 -1 0 1 2 3 4 χ 6                   1 1 4                     0 2               2 1   1 0             1       -1 -2           3 2       1 -4         2 3         1 -6       2 1           1 -8     1 2             1 -10   1 2               -1 -12   1                 1 -14 1                   -1

Integral Khovanov Homology (db, data source)

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

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.

L9a39 L9a41