L11a213

L11a212

L11a214

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a213 at Knotilus!

Link Presentations

[edit Notes on L11a213's Link Presentations]

Planar diagram presentation	X8192 X12,3,13,4 X18,10,19,9 X22,20,7,19 X20,14,21,13 X14,22,15,21 X10,16,11,15 X16,6,17,5 X2738 X4,11,5,12 X6,18,1,17
Gauss code	{1, -9, 2, -10, 8, -11}, {9, -1, 3, -7, 10, -2, 5, -6, 7, -8, 11, -3, 4, -5, 6, -4}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {t(1)^{2}t(2)^{4}-3t(1)t(2)^{4}+2t(2)^{4}-4t(1)^{2}t(2)^{3}+7t(1)t(2)^{3}-5t(2)^{3}+6t(1)^{2}t(2)^{2}-9t(1)t(2)^{2}+6t(2)^{2}-5t(1)^{2}t(2)+7t(1)t(2)-4t(2)+2t(1)^{2}-3t(1)+1}{t(1)t(2)^{2}}}$ (db)
Jones polynomial	$q^{17/2}-4q^{15/2}+9q^{13/2}-14q^{11/2}+19q^{9/2}-21q^{7/2}+20q^{5/2}-18q^{3/2}+12{\sqrt {q}}-{\frac {8}{\sqrt {q}}}+{\frac {3}{q^{3/2}}}-{\frac {1}{q^{5/2}}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$z^{3}a^{-7}+za^{-7}-2z^{5}a^{-5}-4z^{3}a^{-5}-za^{-5}+a^{-5}z^{-1}+z^{7}a^{-3}+3z^{5}a^{-3}+3z^{3}a^{-3}-2a^{-3}z^{-1}-2z^{5}a^{-1}+az^{3}-5z^{3}a^{-1}+2az-3za^{-1}+az^{-1}$ (db)
Kauffman polynomial	$-2z^{10}a^{-2}-2z^{10}a^{-4}-4z^{9}a^{-1}-12z^{9}a^{-3}-8z^{9}a^{-5}-6z^{8}a^{-2}-16z^{8}a^{-4}-13z^{8}a^{-6}-3z^{8}-az^{7}+9z^{7}a^{-1}+27z^{7}a^{-3}+4z^{7}a^{-5}-13z^{7}a^{-7}+33z^{6}a^{-2}+51z^{6}a^{-4}+19z^{6}a^{-6}-9z^{6}a^{-8}+10z^{6}+4az^{5}+z^{5}a^{-1}-9z^{5}a^{-3}+15z^{5}a^{-5}+17z^{5}a^{-7}-4z^{5}a^{-9}-35z^{4}a^{-2}-42z^{4}a^{-4}-8z^{4}a^{-6}+8z^{4}a^{-8}-z^{4}a^{-10}-10z^{4}-6az^{3}-10z^{3}a^{-1}-3z^{3}a^{-3}-8z^{3}a^{-5}-8z^{3}a^{-7}+z^{3}a^{-9}+13z^{2}a^{-2}+18z^{2}a^{-4}+4z^{2}a^{-6}-3z^{2}a^{-8}+2z^{2}+4az+3za^{-1}-2za^{-3}+za^{-7}-3a^{-2}-5a^{-4}-2a^{-6}+1-az^{-1}+2a^{-3}z^{-1}+a^{-5}z^{-1}$ (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 18                       1 -1 16                     3   3 14                   6 1   -5 12                 8 3     5 10               11 6       -5 8             10 8         2 6           10 11           1 4         8 10             -2 2       5 11               6 0     3 7                 -4 -2   1 6                   5 -4   2                     -2 -6 1                       1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=2$ $i=4$ $r=-4$ ${\mathbb {Z} }$ $r=-3$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-2$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{3}$ $r=-1$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=0$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{8}$ $r=1$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=2$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=3$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{11}$ $r=4$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=5$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=6$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=7$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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