L9n13

L9n12

L9n14

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9n13 at Knotilus!
	L9n13 is $9_{54}^{2}$ in the Rolfsen table of links.

Link Presentations

[edit Notes on L9n13's Link Presentations]

Planar diagram presentation	X8192 X11,17,12,16 X3,10,4,11 X15,3,16,2 X5,13,6,12 X6718 X9,14,10,15 X13,18,14,7 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 $u$, $v$, $w$, ...)	$-{\frac {(t(2)t(1)-t(1)+1)(t(1)t(2)-t(2)+1)}{t(1)t(2)}}$ (db)
Jones polynomial	$-{\frac {2}{q^{9/2}}}+{\frac {3}{q^{7/2}}}-{\frac {4}{q^{5/2}}}+{\frac {2}{q^{3/2}}}-{\frac {1}{q^{13/2}}}+{\frac {2}{q^{11/2}}}+{\sqrt {q}}-{\frac {3}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$z^{3}a^{5}+2za^{5}-z^{5}a^{3}-4z^{3}a^{3}-4za^{3}+a^{3}z^{-1}+z^{3}a+za-az^{-1}$ (db)
Kauffman polynomial	$-z^{5}a^{7}+3z^{3}a^{7}-za^{7}-2z^{6}a^{6}+7z^{4}a^{6}-5z^{2}a^{6}-z^{7}a^{5}+2z^{5}a^{5}+za^{5}-3z^{6}a^{4}+9z^{4}a^{4}-7z^{2}a^{4}-z^{7}a^{3}+3z^{5}a^{3}-6z^{3}a^{3}+4za^{3}+a^{3}z^{-1}-z^{6}a^{2}+2z^{4}a^{2}-3z^{2}a^{2}-a^{2}-3z^{3}a+2za+az^{-1}-z^{2}$ (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 χ 2               1 -1 0             2   2 -2           1 2   1 -4         3 1     2 -6       1 2       1 -8     1 2         -1 -10   1 1           0 -12   1             -1 -14 1               1

Integral Khovanov Homology (db, data source)

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

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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