L9a20

L9a19

L9a21

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9a20 at Knotilus!
	L9a20 is $9_{42}^{2}$ in the Rolfsen table of links.

Link Presentations

[edit Notes on L9a20's Link Presentations]

Planar diagram presentation	X8192 X16,11,17,12 X10,4,11,3 X2,15,3,16 X12,5,13,6 X6718 X14,10,15,9 X18,14,7,13 X4,18,5,17
Gauss code	{1, -4, 3, -9, 5, -6}, {6, -1, 7, -3, 2, -5, 8, -7, 4, -2, 9, -8}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {t(1)^{2}t(2)^{4}-t(1)t(2)^{4}-3t(1)^{2}t(2)^{3}+4t(1)t(2)^{3}-t(2)^{3}+3t(1)^{2}t(2)^{2}-7t(1)t(2)^{2}+3t(2)^{2}-t(1)^{2}t(2)+4t(1)t(2)-3t(2)-t(1)+1}{t(1)t(2)^{2}}}$ (db)
Jones polynomial	$-q^{7/2}+4q^{5/2}-7q^{3/2}+9{\sqrt {q}}-{\frac {12}{\sqrt {q}}}+{\frac {11}{q^{3/2}}}-{\frac {10}{q^{5/2}}}+{\frac {7}{q^{7/2}}}-{\frac {4}{q^{9/2}}}+{\frac {1}{q^{11/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$-a^{3}z^{5}-2a^{3}z^{3}+a^{3}z^{-1}+az^{7}+4az^{5}-z^{5}a^{-1}+4az^{3}-2z^{3}a^{-1}-az-az^{-1}$ (db)
Kauffman polynomial	$-3a^{2}z^{8}-3z^{8}-7a^{3}z^{7}-13az^{7}-6z^{7}a^{-1}-7a^{4}z^{6}-6a^{2}z^{6}-4z^{6}a^{-2}-3z^{6}-4a^{5}z^{5}+8a^{3}z^{5}+25az^{5}+12z^{5}a^{-1}-z^{5}a^{-3}-a^{6}z^{4}+8a^{4}z^{4}+16a^{2}z^{4}+7z^{4}a^{-2}+14z^{4}+3a^{5}z^{3}-2a^{3}z^{3}-12az^{3}-6z^{3}a^{-1}+z^{3}a^{-3}-2a^{4}z^{2}-6a^{2}z^{2}-2z^{2}a^{-2}-6z^{2}-a^{3}z-az-a^{2}+a^{3}z^{-1}+az^{-1}$ (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   \ -5 -4 -3 -2 -1 0 1 2 3 4 χ 8                   1 1 6                 3   -3 4               4 1   3 2             5 3     -2 0           7 4       3 -2         5 6         1 -4       5 6           -1 -6     3 6             3 -8   1 4               -3 -10   3                 3 -12 1                   -1

Integral Khovanov Homology (db, data source)

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

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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