L9a21

L9a20

L9a22

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9a21 at Knotilus!
	L9a21 is $9_{34}^{2}$ in the Rolfsen table of links.

Link Presentations

[edit Notes on L9a21's Link Presentations]

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {\left(t(1)t(2)^{2}-t(1)t(2)+t(2)+t(1)-1\right)\left(t(1)t(2)^{2}-t(2)^{2}-t(1)t(2)+t(2)-1\right)}{t(1)t(2)^{2}}}$ (db)
Jones polynomial	$-q^{7/2}+3q^{5/2}-5q^{3/2}+7{\sqrt {q}}-{\frac {9}{\sqrt {q}}}+{\frac {8}{q^{3/2}}}-{\frac {8}{q^{5/2}}}+{\frac {5}{q^{7/2}}}-{\frac {3}{q^{9/2}}}+{\frac {1}{q^{11/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$-a^{3}z^{5}-3a^{3}z^{3}-2a^{3}z+a^{3}z^{-1}+az^{7}+5az^{5}-z^{5}a^{-1}+8az^{3}-3z^{3}a^{-1}+3az-2za^{-1}-az^{-1}$ (db)
Kauffman polynomial	$-2a^{2}z^{8}-2z^{8}-4a^{3}z^{7}-8az^{7}-4z^{7}a^{-1}-4a^{4}z^{6}-a^{2}z^{6}-3z^{6}a^{-2}-3a^{5}z^{5}+5a^{3}z^{5}+19az^{5}+10z^{5}a^{-1}-z^{5}a^{-3}-a^{6}z^{4}+4a^{4}z^{4}+4a^{2}z^{4}+7z^{4}a^{-2}+6z^{4}+4a^{5}z^{3}-3a^{3}z^{3}-17az^{3}-8z^{3}a^{-1}+2z^{3}a^{-3}+a^{6}z^{2}-2a^{2}z^{2}-3z^{2}a^{-2}-4z^{2}-a^{5}z+4az+3za^{-1}-a^{2}+a^{3}z^{-1}+az^{-1}$ (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 8                   1 1 6                 2   -2 4               3 1   2 2             4 2     -2 0           5 3       2 -2         4 5         1 -4       4 4           0 -6     2 5             3 -8   1 3               -2 -10   2                 2 -12 1                   -1

Integral Khovanov Homology (db, data source)

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

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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