L9a41

L9a40

L9a42

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9a41 at Knotilus!
	L9a41 is $9_{23}^{2}$ in the Rolfsen table of links.

Link Presentations

[edit Notes on L9a41's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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