L11n90

L11n89

L11n91

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n90 at Knotilus!

Link Presentations

[edit Notes on L11n90's Link Presentations]

Planar diagram presentation	X6172 X12,3,13,4 X7,16,8,17 X17,22,18,5 X14,9,15,10 X10,20,11,19 X21,9,22,8 X18,14,19,13 X20,15,21,16 X2536 X4,11,1,12
Gauss code	{1, -10, 2, -11}, {10, -1, -3, 7, 5, -6, 11, -2, 8, -5, 9, 3, -4, -8, 6, -9, -7, 4}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(t(2)-2)(t(1)+t(2))(2t(2)-1)}{{\sqrt {t(1)}}t(2)^{3/2}}}$ (db)
Jones polynomial	$-{\frac {3}{\sqrt {q}}}+{\frac {4}{q^{3/2}}}-{\frac {5}{q^{5/2}}}+{\frac {6}{q^{7/2}}}-{\frac {6}{q^{9/2}}}+{\frac {5}{q^{11/2}}}-{\frac {4}{q^{13/2}}}+{\frac {2}{q^{15/2}}}-{\frac {1}{q^{17/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{9}z^{-1}-3za^{7}-2a^{7}z^{-1}+2z^{3}a^{5}+2za^{5}+a^{5}z^{-1}+2z^{3}a^{3}+2za^{3}+a^{3}z^{-1}-3za-az^{-1}$ (db)
Kauffman polynomial	$-z^{7}a^{9}+5z^{5}a^{9}-8z^{3}a^{9}+5za^{9}-a^{9}z^{-1}-2z^{8}a^{8}+9z^{6}a^{8}-11z^{4}a^{8}+4z^{2}a^{8}-a^{8}-z^{9}a^{7}+15z^{5}a^{7}-25z^{3}a^{7}+13za^{7}-2a^{7}z^{-1}-5z^{8}a^{6}+21z^{6}a^{6}-25z^{4}a^{6}+13z^{2}a^{6}-3a^{6}-z^{9}a^{5}-z^{7}a^{5}+15z^{5}a^{5}-19z^{3}a^{5}+9za^{5}-a^{5}z^{-1}-3z^{8}a^{4}+12z^{6}a^{4}-17z^{4}a^{4}+12z^{2}a^{4}-2a^{4}-2z^{7}a^{3}+5z^{5}a^{3}-2z^{3}a^{3}-3za^{3}+a^{3}z^{-1}-3z^{4}a^{2}+3z^{2}a^{2}-a^{2}-4za+az^{-1}$ (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   \ -8 -7 -6 -5 -4 -3 -2 -1 0 χ 0                 3 3 -2               2 1 -1 -4             3 2   1 -6           3 2     -1 -8         3 3       0 -10       3 4         1 -12     1 2           -1 -14   1 3             2 -16   1               -1 -18 1                 1

Integral Khovanov Homology (db, data source)

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

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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