L9n16

L9n15

L9n17

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9n16 at Knotilus!
	L9n16 is $9_{51}^{2}$ in the Rolfsen table of links.

Link Presentations

[edit Notes on L9n16's Link Presentations]

Planar diagram presentation	X8192 X11,17,12,16 X3,10,4,11 X2,15,3,16 X12,5,13,6 X6718 X14,10,15,9 X18,14,7,13 X17,4,18,5
Gauss code	{1, -4, -3, 9, 5, -6}, {6, -1, 7, 3, -2, -5, 8, -7, 4, 2, -9, -8}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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