L11n360

L11n359

L11n361

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n360 at Knotilus!

Link Presentations

[edit Notes on L11n360's Link Presentations]

Planar diagram presentation	X6172 X12,7,13,8 X4,13,1,14 X5,18,6,19 X8493 X9,21,10,20 X19,11,20,10 X17,14,18,15 X15,22,16,17 X21,16,22,5 X2,12,3,11
Gauss code	{1, -11, 5, -3}, {-8, 4, -7, 6, -10, 9}, {-4, -1, 2, -5, -6, 7, 11, -2, 3, 8, -9, 10}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(t(1)-1)(t(3)-1)^{2}}{{\sqrt {t(1)}}t(3)}}$ (db)
Jones polynomial	$q^{3}-2q^{2}+3q-3+4q^{-1}-2q^{-2}+3q^{-3}+q^{-6}-q^{-7}$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$-z^{2}a^{6}-2a^{6}+z^{4}a^{4}+5z^{2}a^{4}+a^{4}z^{-2}+5a^{4}-z^{4}a^{2}-3z^{2}a^{2}-2a^{2}z^{-2}-4a^{2}-z^{4}-2z^{2}+z^{-2}+z^{2}a^{-2}+a^{-2}$ (db)
Kauffman polynomial	$a^{7}z^{7}-6a^{7}z^{5}+9a^{7}z^{3}-3a^{7}z+a^{6}z^{8}-7a^{6}z^{6}+14a^{6}z^{4}-12a^{6}z^{2}+5a^{6}+a^{5}z^{7}-9a^{5}z^{5}+18a^{5}z^{3}-10a^{5}z+2a^{4}z^{8}-16a^{4}z^{6}+39a^{4}z^{4}-38a^{4}z^{2}-a^{4}z^{-2}+15a^{4}+a^{3}z^{9}-5a^{3}z^{7}+2a^{3}z^{5}+13a^{3}z^{3}-13a^{3}z+2a^{3}z^{-1}+3a^{2}z^{8}-18a^{2}z^{6}+z^{6}a^{-2}+35a^{2}z^{4}-4z^{4}a^{-2}-31a^{2}z^{2}+3z^{2}a^{-2}-2a^{2}z^{-2}+13a^{2}-a^{-2}+az^{9}-3az^{7}+2z^{7}a^{-1}-3az^{5}-8z^{5}a^{-1}+10az^{3}+6z^{3}a^{-1}-7az-za^{-1}+2az^{-1}+2z^{8}-8z^{6}+6z^{4}-2z^{2}-z^{-2}+3$ (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 3 4 χ 7                       1 1 5                     1   -1 3                   2 1   1 1               1 2 1     0 -1             1 4 2       1 -3           1 2 3         2 -5         1 4 2           1 -7       1 1 3             3 -9       1 1               0 -11   1 1 1                 1 -13                         0 -15 1                       -1

Integral Khovanov Homology (db, data source)

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

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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