L11a410

L11a409

L11a411

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a410 at Knotilus!

Link Presentations

[edit Notes on L11a410's Link Presentations]

Planar diagram presentation	X6172 X12,4,13,3 X20,16,21,15 X14,8,15,7 X10,12,5,11 X22,18,11,17 X16,22,17,21 X8,19,9,20 X18,9,19,10 X2536 X4,14,1,13
Gauss code	{1, -10, 2, -11}, {10, -1, 4, -8, 9, -5}, {5, -2, 11, -4, 3, -7, 6, -9, 8, -3, 7, -6}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {t(1)t(2)^{2}t(3)^{4}-t(2)^{2}t(3)^{4}+t(1)t(3)^{4}-2t(1)t(2)t(3)^{4}+t(2)t(3)^{4}-t(1)t(2)^{2}t(3)^{3}+2t(2)^{2}t(3)^{3}-2t(1)t(3)^{3}+2t(1)t(2)t(3)^{3}-2t(2)t(3)^{3}+t(3)^{3}+t(1)t(2)^{2}t(3)^{2}-2t(2)^{2}t(3)^{2}+2t(1)t(3)^{2}-2t(1)t(2)t(3)^{2}+2t(2)t(3)^{2}-t(3)^{2}-t(1)t(2)^{2}t(3)+2t(2)^{2}t(3)-2t(1)t(3)+2t(1)t(2)t(3)-2t(2)t(3)+t(3)-t(2)^{2}+t(1)-t(1)t(2)+2t(2)-1}{{\sqrt {t(1)}}t(2)t(3)^{2}}}$ (db)
Jones polynomial	$-q^{9}+3q^{8}-6q^{7}+9q^{6}-12q^{5}+13q^{4}-12q^{3}+12q^{2}-7q+6-2q^{-1}+q^{-2}$ (db)
Signature	4 (db)
HOMFLY-PT polynomial	$z^{8}a^{-4}-2z^{6}a^{-2}+6z^{6}a^{-4}-z^{6}a^{-6}-10z^{4}a^{-2}+14z^{4}a^{-4}-4z^{4}a^{-6}+z^{4}-17z^{2}a^{-2}+17z^{2}a^{-4}-5z^{2}a^{-6}+4z^{2}-13a^{-2}+11a^{-4}-3a^{-6}+5-5a^{-2}z^{-2}+4a^{-4}z^{-2}-a^{-6}z^{-2}+2z^{-2}$ (db)
Kauffman polynomial	$z^{3}a^{-11}+3z^{4}a^{-10}+6z^{5}a^{-9}-4z^{3}a^{-9}+za^{-9}+9z^{6}a^{-8}-12z^{4}a^{-8}+3z^{2}a^{-8}+11z^{7}a^{-7}-23z^{5}a^{-7}+12z^{3}a^{-7}-4za^{-7}+a^{-7}z^{-1}+10z^{8}a^{-6}-26z^{6}a^{-6}+19z^{4}a^{-6}-12z^{2}a^{-6}-a^{-6}z^{-2}+5a^{-6}+5z^{9}a^{-5}-4z^{7}a^{-5}-30z^{5}a^{-5}+43z^{3}a^{-5}-21za^{-5}+5a^{-5}z^{-1}+z^{10}a^{-4}+12z^{8}a^{-4}-62z^{6}a^{-4}+87z^{4}a^{-4}-52z^{2}a^{-4}-4a^{-4}z^{-2}+18a^{-4}+7z^{9}a^{-3}-24z^{7}a^{-3}+9z^{5}a^{-3}+31z^{3}a^{-3}-29za^{-3}+9a^{-3}z^{-1}+z^{10}a^{-2}+3z^{8}a^{-2}-33z^{6}a^{-2}+67z^{4}a^{-2}-53z^{2}a^{-2}-5a^{-2}z^{-2}+21a^{-2}+2z^{9}a^{-1}-9z^{7}a^{-1}+10z^{5}a^{-1}+5z^{3}a^{-1}-13za^{-1}+5a^{-1}z^{-1}+z^{8}-6z^{6}+14z^{4}-16z^{2}-2z^{-2}+9$ (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   \ -4 -3 -2 -1 0 1 2 3 4 5 6 7 χ 19                       1 -1 17                     2   2 15                   4 1   -3 13                 5 2     3 11               7 4       -3 9             6 5         1 7           7 8           1 5         5 5             0 3       4 9               5 1     2 3                 -1 -1   1 5                   4 -3   1                     -1 -5 1                       1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=3$ $i=5$ $r=-4$ ${\mathbb {Z} }$ $r=-3$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}} $r=-2$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }^{2}$ $r=-1$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=0$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{5}$ $r=1$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=2$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{6}$ $r=3$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=4$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=5$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=6$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=7$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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