L11n369

L11n368

L11n370

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n369 at Knotilus!

Link Presentations

[edit Notes on L11n369's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X7,14,8,15 X20,16,21,15 X18,11,19,12 X12,17,13,18 X22,20,17,19 X16,22,5,21 X13,8,14,9 X2536 X4,9,1,10
Gauss code	{1, -10, 2, -11}, {6, -5, 7, -4, 8, -7}, {10, -1, -3, 9, 11, -2, 5, -6, -9, 3, 4, -8}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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