L9a7

L9a6

L9a8

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9a7 at Knotilus!
	L9a7 is $9_{17}^{2}$ in the Rolfsen table of links.

Link Presentations

[edit Notes on L9a7's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X16,13,17,14 X14,7,15,8 X8,15,9,16 X18,11,5,12 X12,17,13,18 X2536 X4,9,1,10
Gauss code	{1, -8, 2, -9}, {8, -1, 4, -5, 9, -2, 6, -7, 3, -4, 5, -3, 7, -6}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {3uv^{2}-4uv+2u+2v^{3}-4v^{2}+3v}{{\sqrt {u}}v^{3/2}}}$ (db)
Jones polynomial	${\frac {5}{q^{9/2}}}-{\frac {4}{q^{7/2}}}+{\frac {2}{q^{5/2}}}-{\frac {1}{q^{3/2}}}+{\frac {1}{q^{21/2}}}-{\frac {2}{q^{19/2}}}+{\frac {4}{q^{17/2}}}-{\frac {5}{q^{15/2}}}+{\frac {5}{q^{13/2}}}-{\frac {7}{q^{11/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$-a^{11}z^{-1}+3a^{9}z+2a^{9}z^{-1}-2a^{7}z^{3}-2a^{7}z-2a^{5}z^{3}-2a^{5}z-a^{5}z^{-1}-a^{3}z^{3}-a^{3}z$ (db)
Kauffman polynomial	$-z^{6}a^{12}+4z^{4}a^{12}-5z^{2}a^{12}+2a^{12}-2z^{7}a^{11}+7z^{5}a^{11}-6z^{3}a^{11}+2za^{11}-a^{11}z^{-1}-z^{8}a^{10}-z^{6}a^{10}+13z^{4}a^{10}-15z^{2}a^{10}+5a^{10}-5z^{7}a^{9}+15z^{5}a^{9}-12z^{3}a^{9}+5za^{9}-2a^{9}z^{-1}-z^{8}a^{8}-3z^{6}a^{8}+13z^{4}a^{8}-10z^{2}a^{8}+3a^{8}-3z^{7}a^{7}+5z^{5}a^{7}-2z^{3}a^{7}-za^{7}-3z^{6}a^{6}+2z^{4}a^{6}+z^{2}a^{6}-a^{6}-3z^{5}a^{5}+3z^{3}a^{5}-3za^{5}+a^{5}z^{-1}-2z^{4}a^{4}+z^{2}a^{4}-z^{3}a^{3}+za^{3}$ (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   \ -9 -8 -7 -6 -5 -4 -3 -2 -1 0 χ -2                   1 1 -4                 2 1 -1 -6               2     2 -8             3 2     -1 -10           4 2       2 -12         2 4         2 -14       3 3           0 -16     1 2             1 -18   1 3               -2 -20   1                 1 -22 1                   -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-4$ $i=-2$ $r=-9$ ${\mathbb {Z} }$ $r=-8$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-7$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-6$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-5$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-4$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{4}$ $r=-3$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-2$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-1$ ${\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=0$ ${\mathbb {Z} }$ ${\mathbb {Z} }$

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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