L11n386

L11n385

L11n387

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n386 at Knotilus!

Link Presentations

[edit Notes on L11n386's Link Presentations]

Planar diagram presentation	X6172 X14,7,15,8 X15,1,16,4 X5,12,6,13 X3849 X9,16,10,17 X17,19,18,22 X11,20,12,21 X19,10,20,11 X21,5,22,18 X2,14,3,13
Gauss code	{1, -11, -5, 3}, {-9, 8, -10, 7}, {-4, -1, 2, 5, -6, 9, -8, 4, 11, -2, -3, 6, -7, 10}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(u-1)(v-1)(w-1)\left(w^{2}-w+1\right)}{{\sqrt {u}}{\sqrt {v}}w^{3/2}}}$ (db)
Jones polynomial	$-q^{-8}+2q^{-7}-4q^{-6}+6q^{-5}-8q^{-4}+9q^{-3}-6q^{-2}+2q+7q^{-1}-3$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$-z^{4}a^{6}-3z^{2}a^{6}-a^{6}z^{-2}-3a^{6}+z^{6}a^{4}+5z^{4}a^{4}+11z^{2}a^{4}+4a^{4}z^{-2}+10a^{4}-3z^{4}a^{2}-10z^{2}a^{2}-5a^{2}z^{-2}-11a^{2}+2z^{2}+2z^{-2}+4$ (db)
Kauffman polynomial	$a^{9}z^{5}-3a^{9}z^{3}+a^{9}z+2a^{8}z^{6}-5a^{8}z^{4}+a^{8}z^{2}+3a^{7}z^{7}-9a^{7}z^{5}+9a^{7}z^{3}-5a^{7}z+a^{7}z^{-1}+3a^{6}z^{8}-10a^{6}z^{6}+14a^{6}z^{4}-9a^{6}z^{2}-a^{6}z^{-2}+5a^{6}+a^{5}z^{9}+2a^{5}z^{7}-17a^{5}z^{5}+35a^{5}z^{3}-22a^{5}z+5a^{5}z^{-1}+5a^{4}z^{8}-20a^{4}z^{6}+39a^{4}z^{4}-34a^{4}z^{2}-4a^{4}z^{-2}+16a^{4}+a^{3}z^{9}-8a^{3}z^{5}+26a^{3}z^{3}-26a^{3}z+9a^{3}z^{-1}+2a^{2}z^{8}-8a^{2}z^{6}+23a^{2}z^{4}-32a^{2}z^{2}-5a^{2}z^{-2}+17a^{2}+az^{7}-az^{5}+3az^{3}-10az+5az^{-1}+3z^{4}-8z^{2}-2z^{-2}+7$ (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 3                   2 2 1                 1   -1 -1               6 2   4 -3             4 5     1 -5           5 2       3 -7         3 4         1 -9       3 5           -2 -11     1 3             2 -13   1 3               -2 -15   1                 1 -17 1                   -1

Integral Khovanov Homology (db, data source)

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

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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