L11n206

L11n205

L11n207

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n206 at Knotilus!

Link Presentations

[edit Notes on L11n206's Link Presentations]

Planar diagram presentation	X10,1,11,2 X16,8,17,7 X18,12,19,11 X19,3,20,2 X3,12,4,13 X13,21,14,20 X14,5,15,6 X6,9,7,10 X22,16,9,15 X8,18,1,17 X4,22,5,21
Gauss code	{1, 4, -5, -11, 7, -8, 2, -10}, {8, -1, 3, 5, -6, -7, 9, -2, 10, -3, -4, 6, 11, -9}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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