L11n184

L11n183

L11n185

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n184 at Knotilus!

Link Presentations

[edit Notes on L11n184's Link Presentations]

Planar diagram presentation	X8192 X9,21,10,20 X6,21,1,22 X18,8,19,7 X3,10,4,11 X15,12,16,13 X5,14,6,15 X13,4,14,5 X11,16,12,17 X22,18,7,17 X19,2,20,3
Gauss code	{1, 11, -5, 8, -7, -3}, {4, -1, -2, 5, -9, 6, -8, 7, -6, 9, 10, -4, -11, 2, 3, -10}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-2$ $i=0$ $r=-8$ ${\mathbb {Z} }$ $r=-7$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-6$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }^{2}$ $r=-5$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-3$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-2$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-1$ ${\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=0$ ${\mathbb {Z} }^{2}$ ${\mathbb {Z} }$ $r=1$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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