L11n284

L11n283

L11n285

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n284 at Knotilus!

Link Presentations

[edit Notes on L11n284's Link Presentations]

Planar diagram presentation	X6172 X12,6,13,5 X8493 X2,14,3,13 X14,7,15,8 X18,10,19,9 X17,11,18,22 X11,21,12,20 X21,17,22,16 X4,15,1,16 X10,20,5,19
Gauss code	{1, -4, 3, -10}, {2, -1, 5, -3, 6, -11}, {-8, -2, 4, -5, 10, 9, -7, -6, 11, 8, -9, 7}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(t(1)-1)(t(2)-1)(t(3)-1)\left(t(2)t(3)^{3}+1\right)}{{\sqrt {t(1)}}t(2)t(3)^{2}}}$ (db)
Jones polynomial	$-2q^{7}+3q^{6}-4q^{5}+6q^{4}-4q^{3}+6q^{2}-3q-q^{-1}+3$ (db)
Signature	4 (db)
HOMFLY-PT polynomial	$z^{2}a^{-8}-a^{-8}z^{-2}-z^{6}a^{-6}-5z^{4}a^{-6}-5z^{2}a^{-6}+4a^{-6}z^{-2}+2a^{-6}+z^{8}a^{-4}+6z^{6}a^{-4}+11z^{4}a^{-4}+6z^{2}a^{-4}-5a^{-4}z^{-2}-5a^{-4}-z^{6}a^{-2}-4z^{4}a^{-2}-2z^{2}a^{-2}+2a^{-2}z^{-2}+3a^{-2}$ (db)
Kauffman polynomial	$2z^{9}a^{-3}+2z^{9}a^{-5}+3z^{8}a^{-2}+7z^{8}a^{-4}+4z^{8}a^{-6}+z^{7}a^{-1}-7z^{7}a^{-3}-6z^{7}a^{-5}+2z^{7}a^{-7}-15z^{6}a^{-2}-35z^{6}a^{-4}-20z^{6}a^{-6}-4z^{5}a^{-1}+z^{5}a^{-3}-3z^{5}a^{-5}-8z^{5}a^{-7}+21z^{4}a^{-2}+49z^{4}a^{-4}+29z^{4}a^{-6}+z^{4}a^{-8}+3z^{3}a^{-1}+4z^{3}a^{-3}+7z^{3}a^{-5}+6z^{3}a^{-7}-9z^{2}a^{-2}-21z^{2}a^{-4}-14z^{2}a^{-6}-2z^{2}a^{-8}+5za^{-3}+9za^{-5}+5za^{-7}+za^{-9}-3a^{-2}-4a^{-4}-2a^{-6}-5a^{-3}z^{-1}-9a^{-5}z^{-1}-5a^{-7}z^{-1}-a^{-9}z^{-1}+2a^{-2}z^{-2}+5a^{-4}z^{-2}+4a^{-6}z^{-2}+a^{-8}z^{-2}$ (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   \ -3 -2 -1 0 1 2 3 4 5 χ 15                 2 -2 13               2 1 1 11             3 2   -1 9           3 2 1   2 7         3 5       2 5       3 2 1       2 3     2 5           3 1   1 1             0 -1   2               2 -3 1                 -1

Integral Khovanov Homology (db, data source)

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

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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