L9a51

L9a50

L9a52

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9a51 at Knotilus!
	L9a51 is $9_{11}^{3}$ in the Rolfsen table of links.

Link Presentations

[edit Notes on L9a51's Link Presentations]

Planar diagram presentation	X6172 X12,4,13,3 X8,12,9,11 X18,8,11,7 X16,13,17,14 X14,6,15,5 X10,16,5,15 X2,9,3,10 X4,18,1,17
Gauss code	{1, -8, 2, -9}, {6, -1, 4, -3, 8, -7}, {3, -2, 5, -6, 7, -5, 9, -4}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {uv^{2}w^{2}-2uv^{2}w-2uvw^{2}+4uvw-2uv+uw^{2}-2uw+u-v^{2}w^{2}+2v^{2}w-v^{2}+2vw^{2}-4vw+2v+2w-1}{{\sqrt {u}}vw}}$ (db)
Jones polynomial	$q^{7}-3q^{6}+7q^{5}-8q^{4}+11q^{3}-10q^{2}-q^{-2}+9q+4q^{-1}-6$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$z^{2}a^{-6}+a^{-6}z^{-2}+a^{-6}-2z^{4}a^{-4}-4z^{2}a^{-4}-2a^{-4}z^{-2}-3a^{-4}+z^{6}a^{-2}+3z^{4}a^{-2}+3z^{2}a^{-2}+a^{-2}z^{-2}+a^{-2}-z^{4}-z^{2}+1$ (db)
Kauffman polynomial	$z^{4}a^{-8}-z^{2}a^{-8}+3z^{5}a^{-7}-2z^{3}a^{-7}+6z^{6}a^{-6}-10z^{4}a^{-6}+10z^{2}a^{-6}+a^{-6}z^{-2}-5a^{-6}+5z^{7}a^{-5}-3z^{5}a^{-5}-4z^{3}a^{-5}+6za^{-5}-2a^{-5}z^{-1}+2z^{8}a^{-4}+8z^{6}a^{-4}-21z^{4}a^{-4}+17z^{2}a^{-4}+2a^{-4}z^{-2}-8a^{-4}+10z^{7}a^{-3}-16z^{5}a^{-3}+2z^{3}a^{-3}+6za^{-3}-2a^{-3}z^{-1}+2z^{8}a^{-2}+6z^{6}a^{-2}-18z^{4}a^{-2}+9z^{2}a^{-2}+a^{-2}z^{-2}-3a^{-2}+5z^{7}a^{-1}+az^{5}-9z^{5}a^{-1}-az^{3}+3z^{3}a^{-1}+4z^{6}-8z^{4}+3z^{2}+1$ (db)

Khovanov Homology

The coefficients of the monomials Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t^rq^j} are shown, along with their alternating sums $\chi$ (fixed $j$, alternation over $r$).

\   r    \    j   \ -3 -2 -1 0 1 2 3 4 5 6 χ 15                   1 1 13                 3 1 -2 11               4     4 9             4 3     -1 7           7 4       3 5         4 5         1 3       5 6           -1 1     3 6             3 -1   1 3               -2 -3   3                 3 -5 1                   -1

Integral Khovanov Homology (db, data source)

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

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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