L9a42

L9a41

L9a43

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9a42 at Knotilus!
	L9a42 is $9_{41}^{2}$ in the Rolfsen table of links.

Link Presentations

[edit Notes on L9a42's Link Presentations]

Planar diagram presentation	X10,1,11,2 X12,4,13,3 X18,5,9,6 X6,9,7,10 X16,12,17,11 X14,8,15,7 X4,14,5,13 X8,16,1,15 X2,17,3,18
Gauss code	{1, -9, 2, -7, 3, -4, 6, -8}, {4, -1, 5, -2, 7, -6, 8, -5, 9, -3}

A Braid Representative	{{{braid_table}}}
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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