L11n390

L11n389

L11n391

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n390 at Knotilus!

Link Presentations

[edit Notes on L11n390's Link Presentations]

Planar diagram presentation	X6172 X16,7,17,8 X4,17,1,18 X5,12,6,13 X3849 X22,14,19,13 X20,10,21,9 X10,20,11,19 X14,22,15,21 X11,18,12,5 X15,2,16,3
Gauss code	{1, 11, -5, -3}, {8, -7, 9, -6}, {-4, -1, 2, 5, 7, -8, -10, 4, 6, -9, -11, -2, 3, 10}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(t(3)-1)\left(t(1)t(3)^{4}+t(1)t(3)^{3}-t(1)t(3)^{2}+t(1)t(2)t(3)^{2}-t(2)t(3)^{2}+t(3)^{2}+t(2)t(3)+t(2)\right)}{{\sqrt {t(1)}}{\sqrt {t(2)}}t(3)^{5/2}}}$ (db)
Jones polynomial	$q^{-7}-3q^{-6}+3q^{-5}-3q^{-4}+4q^{-3}+q^{2}-2q^{-2}+2q^{-1}+1$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$a^{6}z^{2}-a^{6}z^{-2}-a^{6}+4a^{4}z^{2}+4a^{4}z^{-2}+8a^{4}-a^{2}z^{6}-7a^{2}z^{4}-14a^{2}z^{2}-5a^{2}z^{-2}-13a^{2}+z^{4}+5z^{2}+2z^{-2}+6$ (db)
Kauffman polynomial	$z^{4}a^{8}-z^{2}a^{8}+3z^{5}a^{7}-6z^{3}a^{7}+a^{7}z^{-1}+2z^{6}a^{6}-3z^{4}a^{6}-a^{6}z^{-2}+a^{6}+3z^{5}a^{5}-9za^{5}+5a^{5}z^{-1}+6z^{4}a^{4}-13z^{2}a^{4}-4a^{4}z^{-2}+12a^{4}+z^{7}a^{3}-9z^{5}a^{3}+28z^{3}a^{3}-27za^{3}+9a^{3}z^{-1}+z^{8}a^{2}-10z^{6}a^{2}+31z^{4}a^{2}-37z^{2}a^{2}-5a^{2}z^{-2}+21a^{2}+z^{7}a-9z^{5}a+22z^{3}a-18za+5az^{-1}+z^{8}-8z^{6}+21z^{4}-23z^{2}-2z^{-2}+11$ (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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 χ 5                     1 1 3                     1 1 1                 1     1 -1             3         3 -3           2 3 1       0 -5         3 1           2 -7         2 1           1 -9     3 3               0 -11   1 1                 0 -13   2                   -2 -15 1                     1

Integral Khovanov Homology (db, data source)

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

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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