L9n21

L9n20

L9n22

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9n21 at Knotilus!
	L9n21 is $9_{17}^{3}$ in the Rolfsen table of links.

Link Presentations

[edit Notes on L9n21's Link Presentations]

Planar diagram presentation	X6172 X3,11,4,10 X7,15,8,14 X13,5,14,8 X16,11,17,12 X18,15,9,16 X12,17,13,18 X2536 X9,1,10,4
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 $u$, $v$, $w$, ...)	${\frac {-uv-uw^{2}+uw-vw^{2}+vw+w^{3}}{{\sqrt {u}}{\sqrt {v}}w^{3/2}}}$ (db)
Jones polynomial	$-q^{-5}+q^{-4}+q^{3}-q^{-3}+q^{-2}+q+2$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$-z^{2}a^{4}-a^{4}z^{-2}-2a^{4}+z^{4}a^{2}+5z^{2}a^{2}+4a^{2}z^{-2}+8a^{2}-z^{4}-6z^{2}-5z^{-2}-9+z^{2}a^{-2}+2a^{-2}z^{-2}+3a^{-2}$ (db)
Kauffman polynomial	$a^{4}z^{6}+a^{2}z^{6}+z^{6}a^{-2}+z^{6}+a^{5}z^{5}+2a^{3}z^{5}+2az^{5}+z^{5}a^{-1}-4a^{4}z^{4}-6a^{2}z^{4}-6z^{4}a^{-2}-8z^{4}-4a^{5}z^{3}-10a^{3}z^{3}-13az^{3}-7z^{3}a^{-1}+3a^{4}z^{2}+11a^{2}z^{2}+10z^{2}a^{-2}+18z^{2}+3a^{5}z+13a^{3}z+21az+11za^{-1}-2a^{4}-10a^{2}-7a^{-2}-14-a^{5}z^{-1}-5a^{3}z^{-1}-9az^{-1}-5a^{-1}z^{-1}+a^{4}z^{-2}+4a^{2}z^{-2}+2a^{-2}z^{-2}+5z^{-2}$ (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   \ -5 -4 -3 -2 -1 0 1 2 3 4 χ 7                   1 1 5                   1 1 3               1     1 1           3         3 -1         1 4 1       2 -3       1             1 -5       1 1           0 -7   1 1               0 -9                     0 -11 1                   -1

Integral Khovanov Homology (db, data source)

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

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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