L9a35

L9a34

L9a36

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9a35 at Knotilus!
	L9a35 is $9_{9}^{2}$ in the Rolfsen table of links.

More symmetric depiction

Link Presentations

[edit Notes on L9a35's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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