L11n444

L11n443

L11n445

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n444 at Knotilus!

Link Presentations

[edit Notes on L11n444's Link Presentations]

Planar diagram presentation	X6172 X2536 X11,19,12,18 X10,3,11,4 X4,9,1,10 X16,7,17,8 X8,15,5,16 X13,20,14,21 X19,15,20,22 X21,12,22,13 X17,9,18,14
Gauss code	{1, -2, 4, -5}, {2, -1, 6, -7}, {5, -4, -3, 10, -8, 11}, {7, -6, -11, 3, -9, 8, -10, 9}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {-t(1)t(4)^{2}t(3)^{2}-t(2)t(4)^{2}t(3)^{2}+t(4)^{2}t(3)^{2}-t(2)t(3)^{2}+t(1)t(4)t(3)^{2}+2t(2)t(4)t(3)^{2}-t(4)t(3)^{2}+2t(1)t(4)^{2}t(3)+t(2)t(4)^{2}t(3)-t(4)^{2}t(3)+t(1)t(3)-t(1)t(2)t(3)+2t(2)t(3)-2t(1)t(4)t(3)-2t(2)t(4)t(3)-t(1)t(4)^{2}-t(1)+t(1)t(2)-t(2)+2t(1)t(4)-t(1)t(2)t(4)+t(2)t(4)}{{\sqrt {t(1)}}{\sqrt {t(2)}}t(3)t(4)}}$ (db)
Jones polynomial	$2{\sqrt {q}}-{\frac {5}{\sqrt {q}}}+{\frac {6}{q^{3/2}}}-{\frac {10}{q^{5/2}}}+{\frac {8}{q^{7/2}}}-{\frac {10}{q^{9/2}}}+{\frac {6}{q^{11/2}}}-{\frac {6}{q^{13/2}}}+{\frac {2}{q^{15/2}}}-{\frac {1}{q^{17/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{9}z^{-1}+a^{9}z^{-3}-4za^{7}-7a^{7}z^{-1}-3a^{7}z^{-3}+5z^{3}a^{5}+14za^{5}+12a^{5}z^{-1}+3a^{5}z^{-3}-2z^{5}a^{3}-8z^{3}a^{3}-13za^{3}-7a^{3}z^{-1}-a^{3}z^{-3}+2z^{3}a+3za+az^{-1}$ (db)
Kauffman polynomial	$-z^{7}a^{9}+5z^{5}a^{9}-10z^{3}a^{9}+10za^{9}-5a^{9}z^{-1}+a^{9}z^{-3}-2z^{8}a^{8}+7z^{6}a^{8}-4z^{4}a^{8}-7z^{2}a^{8}-3a^{8}z^{-2}+9a^{8}-z^{9}a^{7}-4z^{7}a^{7}+31z^{5}a^{7}-50z^{3}a^{7}+35za^{7}-14a^{7}z^{-1}+3a^{7}z^{-3}-7z^{8}a^{6}+21z^{6}a^{6}-4z^{4}a^{6}-24z^{2}a^{6}-6a^{6}z^{-2}+21a^{6}-z^{9}a^{5}-11z^{7}a^{5}+54z^{5}a^{5}-74z^{3}a^{5}+49za^{5}-18a^{5}z^{-1}+3a^{5}z^{-3}-5z^{8}a^{4}+9z^{6}a^{4}+11z^{4}a^{4}-28z^{2}a^{4}-3a^{4}z^{-2}+18a^{4}-8z^{7}a^{3}+27z^{5}a^{3}-39z^{3}a^{3}+30za^{3}-11a^{3}z^{-1}+a^{3}z^{-3}-5z^{6}a^{2}+11z^{4}a^{2}-14z^{2}a^{2}+6a^{2}-z^{5}a-5z^{3}a+6za-2az^{-1}-3z^{2}+1$ (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 1 χ 2                   2 -2 0                 3   3 -2               4 3   -1 -4             6 3 1   4 -6           5 7       2 -8         5 4 1       2 -10       4 8           4 -12     2 2             0 -14   1 5               4 -16   1                 -1 -18 1                   1

Integral Khovanov Homology (db, data source)

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

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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