L9n5

L9n4

L9n6

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9n5 at Knotilus!
	L9n5 is $9_{44}^{2}$ in the Rolfsen table of links.

Link Presentations

[edit Notes on L9n5's Link Presentations]

Planar diagram presentation	X6172 X14,7,15,8 X15,1,16,4 X5,12,6,13 X3849 X9,16,10,17 X11,18,12,5 X17,10,18,11 X2,14,3,13
Gauss code	{1, -9, -5, 3}, {-4, -1, 2, 5, -6, 8, -7, 4, 9, -2, -3, 6, -8, 7}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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