L9n12

L9n11

L9n13

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9n12 at Knotilus!
	L9n12 is $9_{59}^{2}$ in the Rolfsen table of links.

Link Presentations

[edit Notes on L9n12's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {-t(2)^{5}-t(1)t(2)^{3}+t(2)^{3}+t(1)t(2)^{2}-t(2)^{2}-t(1)}{{\sqrt {t(1)}}t(2)^{5/2}}}$ (db)
Jones polynomial	$q^{11/2}-q^{9/2}+q^{7/2}-q^{5/2}+q^{3/2}-{\sqrt {q}}-{\frac {1}{\sqrt {q}}}-{\frac {1}{q^{5/2}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$za^{-5}+2za^{-3}+2a^{-3}z^{-1}-z^{5}a^{-1}+az^{3}-6z^{3}a^{-1}+4az-9za^{-1}+3az^{-1}-5a^{-1}z^{-1}$ (db)
Kauffman polynomial	$z^{4}a^{-6}-3z^{2}a^{-6}+a^{-6}+z^{5}a^{-5}-3z^{3}a^{-5}+za^{-5}+z^{4}a^{-4}-2z^{2}a^{-4}+z^{3}a^{-3}-3za^{-3}+2a^{-3}z^{-1}+z^{6}a^{-2}-7z^{4}a^{-2}+13z^{2}a^{-2}-5a^{-2}+az^{7}+z^{7}a^{-1}-7az^{5}-8z^{5}a^{-1}+15az^{3}+19z^{3}a^{-1}-12az-16za^{-1}+3az^{-1}+5a^{-1}z^{-1}+z^{6}-7z^{4}+12z^{2}-5$ (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   \ -4 -3 -2 -1 0 1 2 3 4 5 χ 12                   1 -1 10                     0 8               1 1   0 6           1 1       0 4           1 1       0 2       1 2 1         0 0         2           2 -2     1               1 -4 1                   1 -6 1                   1

Integral Khovanov Homology (db, data source)

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

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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