L9n4

L9n3

L9n5

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9n4 at Knotilus!
	L9n4 is $9_{43}^{2}$ in the Rolfsen table of links.

Link Presentations

[edit Notes on L9n4's Link Presentations]

Planar diagram presentation	X6172 X14,7,15,8 X4,15,1,16 X5,12,6,13 X3849 X9,16,10,17 X11,18,12,5 X17,10,18,11 X13,2,14,3
Gauss code	{1, 9, -5, -3}, {-4, -1, 2, 5, -6, 8, -7, 4, -9, -2, 3, 6, -8, 7}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {-uv^{5}-1}{{\sqrt {u}}v^{5/2}}}$ (db)
Jones polynomial	$-{\frac {1}{q^{7/2}}}+{\frac {1}{q^{21/2}}}-{\frac {1}{q^{15/2}}}-{\frac {1}{q^{11/2}}}$ (db)
Signature	-7 (db)
HOMFLY-PT polynomial	$a^{11}(-z)-2a^{11}z^{-1}+a^{9}z^{5}+6a^{9}z^{3}+10a^{9}z+5a^{9}z^{-1}-a^{7}z^{7}-7a^{7}z^{5}-15a^{7}z^{3}-11a^{7}z-3a^{7}z^{-1}$ (db)
Kauffman polynomial	$-a^{14}-z^{3}a^{11}+4za^{11}-2a^{11}z^{-1}-z^{6}a^{10}+6z^{4}a^{10}-10z^{2}a^{10}+5a^{10}-z^{7}a^{9}+7z^{5}a^{9}-16z^{3}a^{9}+15za^{9}-5a^{9}z^{-1}-z^{6}a^{8}+6z^{4}a^{8}-10z^{2}a^{8}+5a^{8}-z^{7}a^{7}+7z^{5}a^{7}-15z^{3}a^{7}+11za^{7}-3a^{7}z^{-1}$ (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   \ -7 -6 -5 -4 -3 -2 -1 0 χ -6               1 1 -8               1 1 -10           1     1 -12       1         1 -14       2 1       1 -16   1             1 -18   1 1           0 -20 1               -1 -22 1               -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-8$ $i=-6$ $i=-4$ $r=-7$ ${\mathbb {Z} }$ ${\mathbb {Z} }$ $r=-6$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }^{2}$ ${\mathbb {Z} }$ $r=-3$ ${\mathbb {Z} }$ $r=-2$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-1$ $r=0$ ${\mathbb {Z} }$ ${\mathbb {Z} }$

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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