L9n14

L9n13

L9n15

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9n14 at Knotilus!
	L9n14 is $9_{50}^{2}$ in the Rolfsen table of links.

Link Presentations

[edit Notes on L9n14's Link Presentations]

Planar diagram presentation	X8192 X11,17,12,16 X3,10,4,11 X15,3,16,2 X5,13,6,12 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 $u$, $v$, $w$, ...)	$-{\frac {u^{2}v^{2}-u^{2}v+uv-v+1}{uv}}$ (db)
Jones polynomial	$q^{9/2}-q^{7/2}+q^{5/2}-{\frac {1}{q^{5/2}}}-2q^{3/2}+{\frac {1}{q^{3/2}}}+{\sqrt {q}}-{\frac {2}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$z^{3}a^{-3}+3za^{-3}+a^{-3}z^{-1}-z^{5}a^{-1}+az^{3}-5z^{3}a^{-1}+3az-7za^{-1}+2az^{-1}-3a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-z^{7}a^{-1}-z^{7}a^{-3}-2z^{6}a^{-2}-z^{6}a^{-4}-z^{6}+5z^{5}a^{-1}+5z^{5}a^{-3}+9z^{4}a^{-2}+5z^{4}a^{-4}+4z^{4}-2az^{3}-9z^{3}a^{-1}-7z^{3}a^{-3}-a^{2}z^{2}-10z^{2}a^{-2}-6z^{2}a^{-4}-5z^{2}-a^{3}z+4az+9za^{-1}+4za^{-3}+3a^{-2}+a^{-4}+3-2az^{-1}-3a^{-1}z^{-1}-a^{-3}z^{-1}$ (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   \ -2 -1 0 1 2 3 4 5 χ 10               1 -1 8                 0 6           1 1   0 4         1       1 2         1       1 0     2 1         1 -2     1           1 -4 1 1             0 -6 1               1

Integral Khovanov Homology (db, data source)

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

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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