L9n18

L9n17

L9n19

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9n18 at Knotilus!
	L9n18 is $9_{53}^{2}$ in the Rolfsen table of links.

Link Presentations

[edit Notes on L9n18's Link Presentations]

Planar diagram presentation	X10,1,11,2 X2,11,3,12 X12,3,13,4 X18,5,9,6 X7,14,8,15 X13,16,14,17 X15,8,16,1 X6,9,7,10 X4,17,5,18
Gauss code	{1, -2, 3, -9, 4, -8, -5, 7}, {8, -1, 2, -3, -6, 5, -7, 6, 9, -4}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {\left(t(2)t(1)^{2}+1\right)\left(t(1)t(2)^{2}+1\right)}{t(1)^{3/2}t(2)^{3/2}}}$ (db)
Jones polynomial	$-{\frac {1}{q^{7/2}}}-{\frac {1}{q^{11/2}}}+{\frac {1}{q^{15/2}}}-{\frac {1}{q^{21/2}}}$ (db)
Signature	-6 (db)
HOMFLY-PT polynomial	$a^{11}(-z)+a^{9}z^{5}+6a^{9}z^{3}+8a^{9}z+a^{9}z^{-1}-a^{7}z^{7}-7a^{7}z^{5}-15a^{7}z^{3}-11a^{7}z-a^{7}z^{-1}$ (db)
Kauffman polynomial	$-z^{3}a^{13}+3za^{13}+za^{11}-z^{6}a^{10}+6z^{4}a^{10}-8z^{2}a^{10}-z^{7}a^{9}+7z^{5}a^{9}-14z^{3}a^{9}+9za^{9}-a^{9}z^{-1}-z^{6}a^{8}+6z^{4}a^{8}-8z^{2}a^{8}+a^{8}-z^{7}a^{7}+7z^{5}a^{7}-15z^{3}a^{7}+11za^{7}-a^{7}z^{-1}$ (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   \ -8 -7 -6 -5 -4 -3 -2 -1 0 χ -6                 1 1 -8                 1 1 -10             1     1 -12         1         1 -14       1 1 1       -1 -16       1           -1 -18     1 1           0 -20 1                 1 -22 1                 1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-8$ $i=-6$ $i=-4$ $r=-8$ ${\mathbb {Z} }$ ${\mathbb {Z} }$ $r=-7$ $r=-6$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }$ ${\mathbb {Z} }$ $r=-3$ ${\mathbb {Z} }$ $r=-2$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-1$ $r=0$ ${\mathbb {Z} }$ ${\mathbb {Z} }$

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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