L11n371

L11n370

L11n372

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n371 at Knotilus!

Link Presentations

[edit Notes on L11n371's Link Presentations]

Planar diagram presentation	X6172 X3,10,4,11 X7,14,8,15 X15,22,16,17 X17,16,18,5 X9,19,10,18 X13,21,14,20 X19,13,20,12 X21,9,22,8 X2536 X11,4,12,1
Gauss code	{1, -10, -2, 11}, {-5, 6, -8, 7, -9, 4}, {10, -1, -3, 9, -6, 2, -11, 8, -7, 3, -4, 5}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(t(3)-1)(t(1)t(3)+1)(t(3)t(2)-t(2)+1)(t(2)t(3)-t(3)+1)}{{\sqrt {t(1)}}t(2)t(3)^{2}}}$ (db)
Jones polynomial	$q^{3}-3q^{2}+7q-9+12q^{-1}-12q^{-2}+12q^{-3}-8q^{-4}+6q^{-5}-2q^{-6}$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$-a^{6}z^{2}-2a^{6}+a^{4}z^{6}+5a^{4}z^{4}+10a^{4}z^{2}+a^{4}z^{-2}+8a^{4}-a^{2}z^{8}-6a^{2}z^{6}-14a^{2}z^{4}-17a^{2}z^{2}-2a^{2}z^{-2}-11a^{2}+z^{6}+4z^{4}+6z^{2}+z^{-2}+5$ (db)
Kauffman polynomial	$3a^{7}z^{3}-2a^{7}z+a^{6}z^{6}+6a^{6}z^{4}-8a^{6}z^{2}+2a^{6}+5a^{5}z^{7}-11a^{5}z^{5}+17a^{5}z^{3}-7a^{5}z+6a^{4}z^{8}-18a^{4}z^{6}+33a^{4}z^{4}-28a^{4}z^{2}-a^{4}z^{-2}+10a^{4}+2a^{3}z^{9}+5a^{3}z^{7}-24a^{3}z^{5}+30a^{3}z^{3}-14a^{3}z+2a^{3}z^{-1}+10a^{2}z^{8}-29a^{2}z^{6}+z^{6}a^{-2}+33a^{2}z^{4}-3z^{4}a^{-2}-26a^{2}z^{2}+3z^{2}a^{-2}-2a^{2}z^{-2}+12a^{2}-a^{-2}+2az^{9}+3az^{7}+3z^{7}a^{-1}-21az^{5}-8z^{5}a^{-1}+21az^{3}+5z^{3}a^{-1}-10az-za^{-1}+2az^{-1}+4z^{8}-9z^{6}+3z^{4}-3z^{2}-z^{-2}+4$ (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   \ -5 -4 -3 -2 -1 0 1 2 3 4 χ 7                   1 1 5                 2   -2 3               5 1   4 1             5 3     -2 -1           7 4       3 -3         6 6         0 -5       6 6           0 -7     3 7             4 -9   3 5               -2 -11   4                 4 -13 2                   -2

Integral Khovanov Homology (db, data source)

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

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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