L9a22

L9a21

L9a23

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9a22 at Knotilus!
	L9a22 is $9_{35}^{2}$ in the Rolfsen table of links.

Link Presentations

[edit Notes on L9a22's Link Presentations]

Planar diagram presentation	X8192 X2,9,3,10 X10,3,11,4 X16,12,17,11 X12,6,13,5 X4,17,5,18 X14,7,15,8 X18,13,7,14 X6,16,1,15
Gauss code	{1, -2, 3, -6, 5, -9}, {7, -1, 2, -3, 4, -5, 8, -7, 9, -4, 6, -8}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {u^{2}v^{4}-2u^{2}v^{3}+2u^{2}v^{2}-u^{2}v-uv^{4}+3uv^{3}-3uv^{2}+3uv-u-v^{3}+2v^{2}-2v+1}{uv^{2}}}$ (db)
Jones polynomial	$q^{3/2}-3{\sqrt {q}}+{\frac {4}{\sqrt {q}}}-{\frac {7}{q^{3/2}}}+{\frac {7}{q^{5/2}}}-{\frac {8}{q^{7/2}}}+{\frac {7}{q^{9/2}}}-{\frac {5}{q^{11/2}}}+{\frac {3}{q^{13/2}}}-{\frac {1}{q^{15/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$-a^{3}z^{7}+a^{5}z^{5}-5a^{3}z^{5}+az^{5}+3a^{5}z^{3}-8a^{3}z^{3}+3az^{3}+2a^{5}z-4a^{3}z+az+a^{3}z^{-1}-az^{-1}$ (db)
Kauffman polynomial	$-z^{3}a^{9}-3z^{4}a^{8}+z^{2}a^{8}-5z^{5}a^{7}+4z^{3}a^{7}-za^{7}-6z^{6}a^{6}+8z^{4}a^{6}-3z^{2}a^{6}-5z^{7}a^{5}+8z^{5}a^{5}-3z^{3}a^{5}+za^{5}-2z^{8}a^{4}-2z^{6}a^{4}+13z^{4}a^{4}-7z^{2}a^{4}-8z^{7}a^{3}+24z^{5}a^{3}-19z^{3}a^{3}+4za^{3}+a^{3}z^{-1}-2z^{8}a^{2}+3z^{6}a^{2}+5z^{4}a^{2}-5z^{2}a^{2}-a^{2}-3z^{7}a+11z^{5}a-11z^{3}a+2za+az^{-1}-z^{6}+3z^{4}-2z^{2}$ (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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 χ 4                   1 -1 2                 2   2 0               2 1   -1 -2             5 2     3 -4           3 3       0 -6         5 4         1 -8       3 4           1 -10     2 4             -2 -12   1 3               2 -14   2                 -2 -16 1                   1

Integral Khovanov Homology (db, data source)

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

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.

L9a21 L9a23