L9a54

L9a53

L9a55

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9a54 at Knotilus!
	L9a54 is $9_{9}^{3}$ in the Rolfsen table of links.

Link Presentations

[edit Notes on L9a54's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(t(1)-1)(t(2)-1)(t(3)-1)\left(t(3)^{2}-t(3)+1\right)}{{\sqrt {t(1)}}{\sqrt {t(2)}}t(3)^{3/2}}}$ (db)
Jones polynomial	$-q^{6}+3q^{5}-6q^{4}+8q^{3}+q^{-3}-7q^{2}-2q^{-2}+9q+5q^{-1}-6$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$z^{6}a^{-2}+4z^{4}a^{-2}-z^{4}a^{-4}-2z^{4}+a^{2}z^{2}+7z^{2}a^{-2}-2z^{2}a^{-4}-6z^{2}+2a^{2}+6a^{-2}-2a^{-4}-6+a^{2}z^{-2}+a^{-2}z^{-2}-2z^{-2}$ (db)
Kauffman polynomial	$z^{8}a^{-2}+z^{8}+2az^{7}+7z^{7}a^{-1}+5z^{7}a^{-3}+a^{2}z^{6}+11z^{6}a^{-2}+8z^{6}a^{-4}+4z^{6}-6az^{5}-16z^{5}a^{-1}-4z^{5}a^{-3}+6z^{5}a^{-5}-4a^{2}z^{4}-35z^{4}a^{-2}-14z^{4}a^{-4}+3z^{4}a^{-6}-22z^{4}+4az^{3}+3z^{3}a^{-1}-7z^{3}a^{-3}-5z^{3}a^{-5}+z^{3}a^{-7}+6a^{2}z^{2}+30z^{2}a^{-2}+10z^{2}a^{-4}+26z^{2}+2az+6za^{-1}+6za^{-3}+2za^{-5}-4a^{2}-12a^{-2}-4a^{-4}-11-2az^{-1}-2a^{-1}z^{-1}+a^{2}z^{-2}+a^{-2}z^{-2}+2z^{-2}$ (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   \ -4 -3 -2 -1 0 1 2 3 4 5 χ 13                   1 -1 11                 2   2 9               4 1   -3 7             4 2     2 5           3 4       1 3         6 4         2 1       4 7           3 -1     1 2             -1 -3   1 4               3 -5   1                 -1 -7 1                   1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=1$ $i=3$ $r=-4$ ${\mathbb {Z} }$ $r=-3$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-2$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-1$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=0$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{6}$ $r=1$ 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 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3}} ${\mathbb {Z} }^{3}$ $r=2$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=3$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=4$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=5$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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