L9a34

L9a33

L9a35

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9a34 at Knotilus!
	L9a34 is $9_{6}^{2}$ in the Rolfsen table of links.

Link Presentations

[edit Notes on L9a34's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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