L9a30

L9a29

L9a31

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9a30 at Knotilus!
	L9a30 is $9_{3}^{2}$ in the Rolfsen table of links.

Link Presentations

[edit Notes on L9a30's Link Presentations]

Planar diagram presentation	X8192 X2,9,3,10 X10,3,11,4 X6718 X16,13,17,14 X14,6,15,5 X4,16,5,15 X18,11,7,12 X12,17,13,18
Gauss code	{1, -2, 3, -7, 6, -4}, {4, -1, 2, -3, 8, -9, 5, -6, 7, -5, 9, -8}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {2u^{2}v^{2}-2u^{2}v-2uv^{2}+3uv-2u-2v+2}{uv}}$ (db)
Jones polynomial	$-{\sqrt {q}}+{\frac {1}{\sqrt {q}}}-{\frac {3}{q^{3/2}}}+{\frac {4}{q^{5/2}}}-{\frac {5}{q^{7/2}}}+{\frac {5}{q^{9/2}}}-{\frac {5}{q^{11/2}}}+{\frac {3}{q^{13/2}}}-{\frac {2}{q^{15/2}}}+{\frac {1}{q^{17/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a^{7}\left(-z^{3}\right)-2a^{7}z+a^{5}z^{5}+3a^{5}z^{3}+2a^{5}z+a^{3}z^{5}+3a^{3}z^{3}+2a^{3}z+a^{3}z^{-1}-az^{3}-3az-az^{-1}$ (db)
Kauffman polynomial	$a^{10}z^{4}-2a^{10}z^{2}+2a^{9}z^{5}-4a^{9}z^{3}+a^{9}z+2a^{8}z^{6}-3a^{8}z^{4}+a^{8}z^{2}+2a^{7}z^{7}-5a^{7}z^{5}+7a^{7}z^{3}-2a^{7}z+a^{6}z^{8}-2a^{6}z^{6}+4a^{6}z^{4}-a^{6}z^{2}+3a^{5}z^{7}-9a^{5}z^{5}+12a^{5}z^{3}-4a^{5}z+a^{4}z^{8}-3a^{4}z^{6}+6a^{4}z^{4}-5a^{4}z^{2}+a^{3}z^{7}-a^{3}z^{5}-3a^{3}z^{3}+3a^{3}z-a^{3}z^{-1}+a^{2}z^{6}-2a^{2}z^{4}-a^{2}z^{2}+a^{2}+az^{5}-4az^{3}+4az-az^{-1}$ (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   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 χ 2                   1 1 0                     0 -2               3 1   2 -4             2 1     -1 -6           3 2       1 -8         3 3         0 -10       2 2           0 -12     1 3             2 -14   1 2               -1 -16   1                 1 -18 1                   -1

Integral Khovanov Homology (db, data source)

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

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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