L9n9

L9n8

L9n10

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9n9 at Knotilus!
	L9n9 is $9_{60}^{2}$ in the Rolfsen table of links.

Link Presentations

[edit Notes on L9n9's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X7,14,8,15 X15,18,16,5 X11,16,12,17 X17,12,18,13 X13,8,14,9 X2536 X4,9,1,10
Gauss code	{1, -8, 2, -9}, {8, -1, -3, 7, 9, -2, -5, 6, -7, 3, -4, 5, -6, 4}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {t(1)t(2)^{3}+t(2)^{3}-t(2)^{2}-t(1)t(2)+t(1)+1}{{\sqrt {t(1)}}t(2)^{3/2}}}$ (db)
Jones polynomial	$-{\frac {1}{q^{5/2}}}-{\frac {1}{q^{11/2}}}-{\frac {1}{q^{15/2}}}+{\frac {1}{q^{17/2}}}-{\frac {1}{q^{19/2}}}+{\frac {1}{q^{21/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$-a^{11}z^{-1}+2za^{9}+2a^{9}z^{-1}+za^{7}-z^{5}a^{5}-5z^{3}a^{5}-5za^{5}-a^{5}z^{-1}$ (db)
Kauffman polynomial	$-z^{6}a^{12}+5z^{4}a^{12}-6z^{2}a^{12}+2a^{12}-z^{7}a^{11}+5z^{5}a^{11}-6z^{3}a^{11}+3za^{11}-a^{11}z^{-1}-2z^{6}a^{10}+11z^{4}a^{10}-15z^{2}a^{10}+5a^{10}-z^{7}a^{9}+6z^{5}a^{9}-10z^{3}a^{9}+7za^{9}-2a^{9}z^{-1}-z^{6}a^{8}+6z^{4}a^{8}-8z^{2}a^{8}+3a^{8}+z^{3}a^{7}-za^{7}+z^{2}a^{6}-a^{6}-z^{5}a^{5}+5z^{3}a^{5}-5za^{5}+a^{5}z^{-1}$ (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   \ -9 -8 -7 -6 -5 -4 -3 -2 -1 0 χ -4                   1 1 -6                   1 1 -8             1 1     0 -10           1         1 -12         1 3 1       1 -14       1             1 -16       1 1           0 -18   1 1               0 -20                     0 -22 1                   -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-6$ $i=-4$ $i=-2$ $r=-9$ ${\mathbb {Z} }$ $r=-8$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-7$ ${\mathbb {Z} }$ $r=-6$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }^{3}$ ${\mathbb {Z} }$ $r=-3$ ${\mathbb {Z} }$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-2$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-1$ $r=0$ ${\mathbb {Z} }$ ${\mathbb {Z} }$

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.

L9n8 L9n10