L9a55

L9a54

L9n1

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9a55 at Knotilus!
	L9a55 is $9_{1}^{4}$ in the Rolfsen table of links.

Link Presentations

[edit Notes on L9a55's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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.

L9a54 L9n1