L11n282

L11n281

L11n283

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n282 at Knotilus!

Link Presentations

[edit Notes on L11n282's Link Presentations]

Planar diagram presentation	X6172 X5,12,6,13 X8493 X2,14,3,13 X14,7,15,8 X18,10,19,9 X17,11,18,22 X11,21,12,20 X21,17,22,16 X4,15,1,16 X10,20,5,19
Gauss code	{1, -4, 3, -10}, {-2, -1, 5, -3, 6, -11}, {-8, 2, 4, -5, 10, 9, -7, -6, 11, 8, -9, 7}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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