L9a44

L9a43

L9a45

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9a44 at Knotilus!
	L9a44 is $9_{3}^{3}$ in the Rolfsen table of links.

Link Presentations

[edit Notes on L9a44's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X18,14,9,13 X16,12,17,11 X12,18,13,17 X8,16,5,15 X14,8,15,7 X2536 X4,9,1,10
Gauss code	{1, -8, 2, -9}, {8, -1, 7, -6}, {9, -2, 4, -5, 3, -7, 6, -4, 5, -3}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {-t(1)t(3)^{3}+t(1)t(2)t(3)^{3}-2t(2)t(3)^{3}+t(3)^{3}+2t(1)t(3)^{2}-t(1)t(2)t(3)^{2}+2t(2)t(3)^{2}-t(3)^{2}-2t(1)t(3)+t(1)t(2)t(3)-2t(2)t(3)+t(3)+2t(1)-t(1)t(2)+t(2)-1}{{\sqrt {t(1)}}{\sqrt {t(2)}}t(3)^{3/2}}}$ (db)
Jones polynomial	$-q^{6}+3q^{5}-6q^{4}+7q^{3}-7q^{2}+8q-5+5q^{-1}-q^{-2}+q^{-3}$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$z^{6}a^{-2}+4z^{4}a^{-2}-z^{4}a^{-4}-2z^{4}+a^{2}z^{2}+7z^{2}a^{-2}-2z^{2}a^{-4}-7z^{2}+3a^{2}+8a^{-2}-2a^{-4}-9+2a^{2}z^{-2}+4a^{-2}z^{-2}-a^{-4}z^{-2}-5z^{-2}$ (db)
Kauffman polynomial	$z^{3}a^{-7}+3z^{4}a^{-6}+6z^{5}a^{-5}-6z^{3}a^{-5}+3za^{-5}-a^{-5}z^{-1}+7z^{6}a^{-4}-11z^{4}a^{-4}+6z^{2}a^{-4}+a^{-4}z^{-2}-2a^{-4}+4z^{7}a^{-3}-z^{5}a^{-3}-12z^{3}a^{-3}+13za^{-3}-5a^{-3}z^{-1}+z^{8}a^{-2}+a^{2}z^{6}+7z^{6}a^{-2}-5a^{2}z^{4}-21z^{4}a^{-2}+9a^{2}z^{2}+16z^{2}a^{-2}+2a^{2}z^{-2}+4a^{-2}z^{-2}-7a^{2}-10a^{-2}+az^{7}+5z^{7}a^{-1}-az^{5}-8z^{5}a^{-1}-6az^{3}-11z^{3}a^{-1}+11az+21za^{-1}-5az^{-1}-9a^{-1}z^{-1}+z^{8}+z^{6}-12z^{4}+19z^{2}+5z^{-2}-14$ (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   \ -4 -3 -2 -1 0 1 2 3 4 5 χ 13                   1 -1 11                 2   2 9               4 1   -3 7             3 2     1 5           4 4       0 3         4 3         1 1       4 7           3 -1     1 1             0 -3     4               4 -5 1 1                 0 -7 1                   1

Integral Khovanov Homology (db, data source)

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

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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