L9a39

L9a38

L9a40

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9a39 at Knotilus!
	L9a39 is $9_{2}^{2}$ in the Rolfsen table of links.

Link Presentations

[edit Notes on L9a39's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {t(2)^{3}t(1)^{3}-t(2)^{2}t(1)^{3}-t(2)^{3}t(1)^{2}+3t(2)^{2}t(1)^{2}-t(2)t(1)^{2}-t(2)^{2}t(1)+3t(2)t(1)-t(1)-t(2)+1}{t(1)^{3/2}t(2)^{3/2}}}$ (db)
Jones polynomial	${\frac {4}{q^{9/2}}}-{\frac {5}{q^{7/2}}}+{\frac {4}{q^{5/2}}}+q^{3/2}-{\frac {4}{q^{3/2}}}-{\frac {1}{q^{15/2}}}+{\frac {2}{q^{13/2}}}-{\frac {3}{q^{11/2}}}-2{\sqrt {q}}+{\frac {2}{\sqrt {q}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$-a^{3}z^{7}+a^{5}z^{5}-6a^{3}z^{5}+az^{5}+4a^{5}z^{3}-12a^{3}z^{3}+4az^{3}+4a^{5}z-9a^{3}z+3az+a^{5}z^{-1}-a^{3}z^{-1}$ (db)
Kauffman polynomial	$-z^{3}a^{9}+za^{9}-2z^{4}a^{8}+2z^{2}a^{8}-2z^{5}a^{7}+z^{3}a^{7}-2z^{6}a^{6}+2z^{4}a^{6}-z^{2}a^{6}-2z^{7}a^{5}+5z^{5}a^{5}-7z^{3}a^{5}+5za^{5}-a^{5}z^{-1}-z^{8}a^{4}+z^{6}a^{4}+3z^{4}a^{4}-4z^{2}a^{4}+a^{4}-4z^{7}a^{3}+16z^{5}a^{3}-20z^{3}a^{3}+10za^{3}-a^{3}z^{-1}-z^{8}a^{2}+2z^{6}a^{2}+3z^{4}a^{2}-4z^{2}a^{2}-2z^{7}a+9z^{5}a-11z^{3}a+4za-z^{6}+4z^{4}-3z^{2}$ (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                 1   1 0               1 1   0 -2             3 1     2 -4           2 2       0 -6         3 2         1 -8       1 2           1 -10     2 3             -1 -12   1 2               1 -14   1                 -1 -16 1                   1

Integral Khovanov Homology (db, data source)

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

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.

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