L11n50

L11n49

L11n51

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n50 at Knotilus!

Link Presentations

[edit Notes on L11n50's Link Presentations]

Planar diagram presentation	X6172 X10,4,11,3 X12,8,13,7 X13,18,14,19 X9,17,10,16 X17,9,18,8 X19,22,20,5 X15,21,16,20 X21,15,22,14 X2536 X4,12,1,11
Gauss code	{1, -10, 2, -11}, {10, -1, 3, 6, -5, -2, 11, -3, -4, 9, -8, 5, -6, 4, -7, 8, -9, 7}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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