L11n216

L11n215

L11n217

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n216 at Knotilus!

Link Presentations

[edit Notes on L11n216's Link Presentations]

Planar diagram presentation	X10,1,11,2 X12,3,13,4 X18,8,19,7 X14,6,15,5 X17,9,18,22 X21,17,22,16 X20,13,21,14 X6,16,7,15 X4,20,5,19 X2,9,3,10 X8,11,1,12
Gauss code	{1, -10, 2, -9, 4, -8, 3, -11}, {10, -1, 11, -2, 7, -4, 8, 6, -5, -3, 9, -7, -6, 5}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(t(1)-1)(t(2)-1)\left(t(2)t(1)^{2}+t(2)^{2}t(1)-2t(2)t(1)+t(1)+t(2)\right)}{t(1)^{3/2}t(2)^{3/2}}}$ (db)
Jones polynomial	$-q^{13/2}+3q^{11/2}-5q^{9/2}+7q^{7/2}-8q^{5/2}+8q^{3/2}-8{\sqrt {q}}+{\frac {4}{\sqrt {q}}}-{\frac {3}{q^{3/2}}}+{\frac {1}{q^{5/2}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$-z^{3}a^{-5}-za^{-5}+z^{5}a^{-3}+2z^{3}a^{-3}+za^{-3}+z^{5}a^{-1}-az^{3}+2z^{3}a^{-1}-az+za^{-1}+az^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$z^{5}a^{-7}-2z^{3}a^{-7}+3z^{6}a^{-6}-7z^{4}a^{-6}+2z^{2}a^{-6}+4z^{7}a^{-5}-10z^{5}a^{-5}+6z^{3}a^{-5}-2za^{-5}+3z^{8}a^{-4}-7z^{6}a^{-4}+6z^{4}a^{-4}-2z^{2}a^{-4}+z^{9}a^{-3}+z^{7}a^{-3}-6z^{5}a^{-3}+8z^{3}a^{-3}-2za^{-3}+4z^{8}a^{-2}-13z^{6}a^{-2}+a^{2}z^{4}+20z^{4}a^{-2}-2a^{2}z^{2}-8z^{2}a^{-2}+z^{9}a^{-1}-3z^{7}a^{-1}+3az^{5}+8z^{5}a^{-1}-6az^{3}-6z^{3}a^{-1}+2az+2za^{-1}+az^{-1}+a^{-1}z^{-1}+z^{8}-3z^{6}+8z^{4}-6z^{2}-1$ (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   \ -3 -2 -1 0 1 2 3 4 5 6 χ 14                   1 1 12                 2   -2 10               3 1   2 8             4 2     -2 6           4 3       1 4         4 4         0 2       4 4           0 0     2 6             4 -2   1 2               -1 -4   2                 2 -6 1                   -1

Integral Khovanov Homology (db, data source)

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

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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