L11n83

L11n82

L11n84

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n83 at Knotilus!

Link Presentations

[edit Notes on L11n83's Link Presentations]

Planar diagram presentation	X6172 X20,7,21,8 X4,21,1,22 X5,14,6,15 X10,4,11,3 X11,16,12,17 X15,12,16,13 X13,22,14,5 X18,9,19,10 X2,18,3,17 X8,19,9,20
Gauss code	{1, -10, 5, -3}, {-4, -1, 2, -11, 9, -5, -6, 7, -8, 4, -7, 6, 10, -9, 11, -2, 3, 8}

A Braid Representative	{{{braid_table}}}
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {2(t(1)-1)(t(2)-1)\left(t(2)^{2}-t(2)+1\right)}{{\sqrt {t(1)}}t(2)^{3/2}}}$ (db)
Jones polynomial	$-{\sqrt {q}}+{\frac {2}{\sqrt {q}}}-{\frac {5}{q^{3/2}}}+{\frac {6}{q^{5/2}}}-{\frac {8}{q^{7/2}}}+{\frac {8}{q^{9/2}}}-{\frac {8}{q^{11/2}}}+{\frac {5}{q^{13/2}}}-{\frac {3}{q^{15/2}}}+{\frac {2}{q^{17/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$-a^{9}z^{-1}-a^{7}z^{3}+2a^{7}z^{-1}+a^{5}z^{5}+2a^{5}z^{3}+a^{5}z-a^{5}z^{-1}+a^{3}z^{5}+2a^{3}z^{3}+a^{3}z+a^{3}z^{-1}-az^{3}-2az-az^{-1}$ (db)
Kauffman polynomial	$3a^{10}z^{4}-8a^{10}z^{2}+3a^{10}+a^{9}z^{7}-a^{9}z^{5}-a^{9}z^{3}+a^{9}z-a^{9}z^{-1}+2a^{8}z^{8}-8a^{8}z^{6}+19a^{8}z^{4}-20a^{8}z^{2}+7a^{8}+a^{7}z^{9}-a^{7}z^{7}-a^{7}z^{5}+5a^{7}z^{3}+2a^{7}z-2a^{7}z^{-1}+4a^{6}z^{8}-13a^{6}z^{6}+22a^{6}z^{4}-14a^{6}z^{2}+4a^{6}+a^{5}z^{9}-2a^{5}z^{5}+3a^{5}z^{3}+a^{5}z-a^{5}z^{-1}+2a^{4}z^{8}-3a^{4}z^{6}+2a^{4}z^{4}-2a^{4}z^{2}+2a^{3}z^{7}-a^{3}z^{5}-6a^{3}z^{3}+3a^{3}z-a^{3}z^{-1}+2a^{2}z^{6}-4a^{2}z^{4}+a^{2}+az^{5}-3az^{3}+3az-az^{-1}$ (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   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 χ 2                   1 1 0                 1   -1 -2               4 1   3 -4             4 3     -1 -6           4 2       2 -8         4 4         0 -10       4 4           0 -12     1 4             3 -14   2 4               -2 -16   1                 1 -18 2                   -2

Integral Khovanov Homology (db, data source)

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

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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