L11n286

L11n285

L11n287

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n286 at Knotilus!

Link Presentations

[edit Notes on L11n286's Link Presentations]

Planar diagram presentation	X6172 X5,12,6,13 X3849 X2,14,3,13 X14,7,15,8 X15,1,16,4 X21,18,22,19 X9,21,10,20 X19,5,20,10 X11,16,12,17 X17,22,18,11
Gauss code	{1, -4, -3, 6}, {-2, -1, 5, 3, -8, 9}, {-10, 2, 4, -5, -6, 10, -11, 7, -9, 8, -7, 11}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {2uvw^{2}-3uvw-2uw^{2}+2uw-2v^{2}w+2v^{2}+3vw-2v}{{\sqrt {u}}vw}}$ (db)
Jones polynomial	$-q^{-7}+2q^{-6}-3q^{-5}+5q^{-4}-5q^{-3}+2q^{2}+6q^{-2}-2q-5q^{-1}+5$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$-z^{2}a^{6}-a^{6}+z^{4}a^{4}+2z^{2}a^{4}+2a^{4}+z^{4}a^{2}+z^{2}a^{2}+a^{2}z^{-2}+a^{2}-3z^{2}-2z^{-2}-4+a^{-2}z^{-2}+2a^{-2}$ (db)
Kauffman polynomial	$a^{5}z^{9}+a^{3}z^{9}+2a^{6}z^{8}+4a^{4}z^{8}+2a^{2}z^{8}+a^{7}z^{7}-2a^{5}z^{7}-a^{3}z^{7}+2az^{7}-10a^{6}z^{6}-17a^{4}z^{6}-5a^{2}z^{6}+2z^{6}-5a^{7}z^{5}-5a^{5}z^{5}-4a^{3}z^{5}-3az^{5}+z^{5}a^{-1}+15a^{6}z^{4}+23a^{4}z^{4}+3a^{2}z^{4}-5z^{4}+7a^{7}z^{3}+10a^{5}z^{3}+2a^{3}z^{3}-2az^{3}-z^{3}a^{-1}-8a^{6}z^{2}-16a^{4}z^{2}-a^{2}z^{2}+3z^{2}a^{-2}+10z^{2}-2a^{7}z-4a^{5}z+6az+4za^{-1}+2a^{6}+4a^{4}-2a^{2}-4a^{-2}-7-2az^{-1}-2a^{-1}z^{-1}+a^{2}z^{-2}+a^{-2}z^{-2}+2z^{-2}$ (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   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 χ 5                   2 2 3                 1 1 0 1               4 1   3 -1             3 3     0 -3           3 2       1 -5         3 4         1 -7       2 2           0 -9     1 3             2 -11   1 2               -1 -13   1                 1 -15 1                   -1

Integral Khovanov Homology (db, data source)

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

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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