L9n1

L9a55

L9n2

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9n1 at Knotilus!
	L9n1 is $9_{45}^{2}$ in the Rolfsen table of links.

Link Presentations

[edit Notes on L9n1's Link Presentations]

Planar diagram presentation	X6172 X14,7,15,8 X4,15,1,16 X5,10,6,11 X3849 X11,18,12,5 X17,12,18,13 X9,16,10,17 X13,2,14,3
Gauss code	{1, 9, -5, -3}, {-4, -1, 2, 5, -8, 4, -6, 7, -9, -2, 3, 8, -7, 6}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {2t(1)t(2)^{3}-t(1)t(2)^{2}-t(2)+2}{{\sqrt {t(1)}}t(2)^{3/2}}}$ (db)
Jones polynomial	$-{\frac {1}{q^{5/2}}}+{\frac {1}{q^{7/2}}}-{\frac {2}{q^{9/2}}}+{\frac {1}{q^{11/2}}}-{\frac {3}{q^{13/2}}}+{\frac {2}{q^{15/2}}}-{\frac {1}{q^{17/2}}}+{\frac {1}{q^{19/2}}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$-a^{11}z^{-1}+z^{3}a^{9}+4za^{9}+2a^{9}z^{-1}-z^{5}a^{7}-4z^{3}a^{7}-3za^{7}-z^{5}a^{5}-4z^{3}a^{5}-3za^{5}-a^{5}z^{-1}$ (db)
Kauffman polynomial	$a^{12}z^{2}-2a^{12}+a^{11}z^{3}-a^{11}z+a^{11}z^{-1}+a^{10}z^{6}-5a^{10}z^{4}+10a^{10}z^{2}-5a^{10}+a^{9}z^{7}-5a^{9}z^{5}+9a^{9}z^{3}-5a^{9}z+2a^{9}z^{-1}+2a^{8}z^{6}-8a^{8}z^{4}+9a^{8}z^{2}-3a^{8}+a^{7}z^{7}-4a^{7}z^{5}+4a^{7}z^{3}-a^{7}z+a^{6}z^{6}-3a^{6}z^{4}+a^{6}+a^{5}z^{5}-4a^{5}z^{3}+3a^{5}z-a^{5}z^{-1}$ (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   \ -7 -6 -5 -4 -3 -2 -1 0 χ -4               1 1 -6             1 1 0 -8           1     1 -10       1 1 1     1 -12       3 1       2 -14       1         1 -16   1 2           -1 -18                 0 -20 1               -1

Integral Khovanov Homology (db, data source)

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

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.

L9a55 L9n2