L11a400

L11a399

L11a401

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a400 at Knotilus!

Link Presentations

[edit Notes on L11a400's Link Presentations]

Planar diagram presentation	X6172 X12,6,13,5 X8493 X2,14,3,13 X14,7,15,8 X18,12,19,11 X20,9,21,10 X10,19,5,20 X4,15,1,16 X22,18,11,17 X16,22,17,21
Gauss code	{1, -4, 3, -9}, {2, -1, 5, -3, 7, -8}, {6, -2, 4, -5, 9, -11, 10, -6, 8, -7, 11, -10}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(u-1)(v-1)^{2}(w-1)^{2}\left(w^{2}+1\right)}{{\sqrt {u}}vw^{2}}}$ (db)
Jones polynomial	$q^{7}-4q^{6}+8q^{5}-14q^{4}+18q^{3}-20q^{2}+21q-16+14q^{-1}-7q^{-2}+4q^{-3}-q^{-4}$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$-z^{8}a^{-2}-5z^{6}a^{-2}+z^{6}a^{-4}+2z^{6}-a^{2}z^{4}-9z^{4}a^{-2}+3z^{4}a^{-4}+7z^{4}-2a^{2}z^{2}-5z^{2}a^{-2}+2z^{2}a^{-4}+5z^{2}+a^{2}+4a^{-2}-a^{-4}-4+2a^{2}z^{-2}+4a^{-2}z^{-2}-a^{-4}z^{-2}-5z^{-2}$ (db)
Kauffman polynomial	$2z^{10}a^{-2}+2z^{10}+5az^{9}+13z^{9}a^{-1}+8z^{9}a^{-3}+4a^{2}z^{8}+18z^{8}a^{-2}+13z^{8}a^{-4}+9z^{8}+a^{3}z^{7}-14az^{7}-32z^{7}a^{-1}-5z^{7}a^{-3}+12z^{7}a^{-5}-15a^{2}z^{6}-67z^{6}a^{-2}-24z^{6}a^{-4}+8z^{6}a^{-6}-50z^{6}-3a^{3}z^{5}+7az^{5}+11z^{5}a^{-1}-19z^{5}a^{-3}-16z^{5}a^{-5}+4z^{5}a^{-7}+18a^{2}z^{4}+68z^{4}a^{-2}+18z^{4}a^{-4}-6z^{4}a^{-6}+z^{4}a^{-8}+61z^{4}+2a^{3}z^{3}+az^{3}+6z^{3}a^{-1}+15z^{3}a^{-3}+6z^{3}a^{-5}-2z^{3}a^{-7}-7a^{2}z^{2}-24z^{2}a^{-2}-8z^{2}a^{-4}-23z^{2}+5az+9za^{-1}+5za^{-3}+za^{-5}-3a^{2}-2a^{-2}-4-5az^{-1}-9a^{-1}z^{-1}-5a^{-3}z^{-1}-a^{-5}z^{-1}+2a^{2}z^{-2}+4a^{-2}z^{-2}+a^{-4}z^{-2}+5z^{-2}$ (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   \ -5 -4 -3 -2 -1 0 1 2 3 4 5 6 χ 15                       1 1 13                     3   -3 11                   5 1   4 9                 9 3     -6 7               9 5       4 5             11 9         -2 3           10 9           1 1         8 13             5 -1       6 8               -2 -3     3 10                 7 -5   1 4                   -3 -7   3                     3 -9 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=1$ $i=3$ $r=-5$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-3$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-2$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{6}$ $r=-1$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=0$ ${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{10}$ $r=1$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{11}$ $r=2$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=3$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=4$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=5$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=6$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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