L9n22

L9n21

L9n23

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9n22 at Knotilus!
	L9n22 is $9_{15}^{3}$ in the Rolfsen table of links.

Link Presentations

[edit Notes on L9n22's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X14,7,15,8 X8,13,5,14 X11,17,12,16 X15,9,16,18 X17,13,18,12 X2536 X4,9,1,10
Gauss code	{1, -8, 2, -9}, {8, -1, 3, -4}, {9, -2, -5, 7, 4, -3, -6, 5, -7, 6}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {uv+uw^{3}-uw^{2}+uw-u+vw^{3}-vw^{2}+vw-v-w^{3}}{{\sqrt {u}}{\sqrt {v}}w^{3/2}}}$ (db)
Jones polynomial	$-2q+3-3q^{-1}+4q^{-2}-2q^{-3}+4q^{-4}-q^{-5}+q^{-6}$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$2a^{6}z^{-2}+a^{6}-3z^{2}a^{4}-5a^{4}z^{-2}-8a^{4}+2z^{4}a^{2}+8z^{2}a^{2}+4a^{2}z^{-2}+10a^{2}-2z^{2}-z^{-2}-3$ (db)
Kauffman polynomial	$a^{6}z^{6}-5a^{6}z^{4}+9a^{6}z^{2}+2a^{6}z^{-2}-7a^{6}+a^{5}z^{7}-2a^{5}z^{5}-5a^{5}z^{3}+11a^{5}z-5a^{5}z^{-1}+5a^{4}z^{6}-20a^{4}z^{4}+23a^{4}z^{2}+5a^{4}z^{-2}-14a^{4}+a^{3}z^{7}+2a^{3}z^{5}-17a^{3}z^{3}+21a^{3}z-9a^{3}z^{-1}+4a^{2}z^{6}-14a^{2}z^{4}+16a^{2}z^{2}+4a^{2}z^{-2}-10a^{2}+4az^{5}-12az^{3}+13az+3za^{-1}-5az^{-1}-a^{-1}z^{-1}+z^{4}+2z^{2}+z^{-2}-2$ (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 3               2 -2 1             1   1 -1           3 3   0 -3         1 1 1   1 -5       1 3       2 -7     3 1         2 -9   1 4           3 -11                 0 -13 1               1

Integral Khovanov Homology (db, data source)

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

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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