L9n7

L9n6

L9n8

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9n7 at Knotilus!
	L9n7 is $9_{48}^{2}$ in the Rolfsen table of links.

Link Presentations

[edit Notes on L9n7's Link Presentations]

Planar diagram presentation	X6172 X16,7,17,8 X4,17,1,18 X5,12,6,13 X3849 X9,14,10,15 X13,10,14,11 X11,18,12,5 X15,2,16,3
Gauss code	{1, 9, -5, -3}, {-4, -1, 2, 5, -6, 7, -8, 4, -7, 6, -9, -2, 3, 8}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {2uv^{3}-2uv^{2}+uv+v^{2}-2v+2}{{\sqrt {u}}v^{3/2}}}$ (db)
Jones polynomial	$-{\frac {1}{q^{5/2}}}+{\frac {1}{q^{7/2}}}-{\frac {3}{q^{9/2}}}+{\frac {3}{q^{11/2}}}-{\frac {4}{q^{13/2}}}+{\frac {3}{q^{15/2}}}-{\frac {3}{q^{17/2}}}+{\frac {2}{q^{19/2}}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$-a^{11}z^{-1}+z^{3}a^{9}+3za^{9}+2a^{9}z^{-1}-z^{5}a^{7}-3z^{3}a^{7}-za^{7}-z^{5}a^{5}-4z^{3}a^{5}-4za^{5}-a^{5}z^{-1}$ (db)
Kauffman polynomial	$3a^{12}z^{2}-2a^{12}+a^{11}z^{5}+a^{11}z^{3}-2a^{11}z+a^{11}z^{-1}+2a^{10}z^{6}-6a^{10}z^{4}+11a^{10}z^{2}-5a^{10}+a^{9}z^{7}-2a^{9}z^{5}+5a^{9}z^{3}-6a^{9}z+2a^{9}z^{-1}+3a^{8}z^{6}-8a^{8}z^{4}+7a^{8}z^{2}-3a^{8}+a^{7}z^{7}-2a^{7}z^{5}+a^{6}z^{6}-2a^{6}z^{4}-a^{6}z^{2}+a^{6}+a^{5}z^{5}-4a^{5}z^{3}+4a^{5}z-a^{5}z^{-1}$ (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 -4               1 1 -6             1 1 0 -8           2     2 -10         1 1     0 -12       3 2       1 -14     1 2         1 -16   2 2           0 -18   1             1 -20 2               -2

Integral Khovanov Homology (db, data source)

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

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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