L11n80

L11n79

L11n81

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n80 at Knotilus!

Link Presentations

[edit Notes on L11n80's Link Presentations]

Planar diagram presentation	X6172 X16,7,17,8 X18,9,19,10 X8,17,9,18 X19,1,20,4 X5,14,6,15 X3,10,4,11 X11,20,12,21 X13,22,14,5 X21,12,22,13 X2,16,3,15
Gauss code	{1, -11, -7, 5}, {-6, -1, 2, -4, 3, 7, -8, 10, -9, 6, 11, -2, 4, -3, -5, 8, -10, 9}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(t(1)-1)(t(2)-1)\left(t(2)^{2}+1\right)\left(t(2)^{2}-t(2)+1\right)}{{\sqrt {t(1)}}t(2)^{5/2}}}$ (db)
Jones polynomial	$-{\frac {2}{q^{5/2}}}+{\frac {2}{q^{7/2}}}-{\frac {6}{q^{9/2}}}+{\frac {7}{q^{11/2}}}-{\frac {8}{q^{13/2}}}+{\frac {8}{q^{15/2}}}-{\frac {7}{q^{17/2}}}+{\frac {5}{q^{19/2}}}-{\frac {2}{q^{21/2}}}+{\frac {1}{q^{23/2}}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$-z^{5}a^{9}-4z^{3}a^{9}-6za^{9}-3a^{9}z^{-1}+z^{7}a^{7}+6z^{5}a^{7}+15z^{3}a^{7}+18za^{7}+7a^{7}z^{-1}-2z^{5}a^{5}-9z^{3}a^{5}-12za^{5}-4a^{5}z^{-1}$ (db)
Kauffman polynomial	$a^{14}z^{4}-2a^{14}z^{2}+a^{14}+2a^{13}z^{5}-2a^{13}z^{3}+3a^{12}z^{6}-3a^{12}z^{4}+a^{12}z^{2}+3a^{11}z^{7}-3a^{11}z^{5}+2a^{11}z^{3}+2a^{10}z^{8}-a^{10}z^{6}+a^{10}z^{2}+a^{9}z^{9}-a^{9}z^{7}+6a^{9}z^{5}-11a^{9}z^{3}+8a^{9}z-3a^{9}z^{-1}+3a^{8}z^{8}-8a^{8}z^{6}+15a^{8}z^{4}-19a^{8}z^{2}+7a^{8}+a^{7}z^{9}-4a^{7}z^{7}+14a^{7}z^{5}-27a^{7}z^{3}+22a^{7}z-7a^{7}z^{-1}+a^{6}z^{8}-4a^{6}z^{6}+11a^{6}z^{4}-17a^{6}z^{2}+7a^{6}+3a^{5}z^{5}-12a^{5}z^{3}+14a^{5}z-4a^{5}z^{-1}$ (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   \ -9 -8 -7 -6 -5 -4 -3 -2 -1 0 χ -4                   2 2 -6                 2 2 0 -8               4     4 -10             3 2     -1 -12           5 4       1 -14         3 3         0 -16       4 5           -1 -18     1 3             2 -20   1 4               -3 -22   1                 1 -24 1                   -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-6$ $i=-4$ $r=-9$ ${\mathbb {Z} }$ $r=-8$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-7$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-6$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=-5$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-4$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=-3$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-2$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=-1$ ${\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=0$ ${\mathbb {Z} }^{2}$ ${\mathbb {Z} }^{2}$

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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