L11a210

L11a209

L11a211

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a210 at Knotilus!

Link Presentations

[edit Notes on L11a210's Link Presentations]

Planar diagram presentation	X8192 X20,9,21,10 X6718 X22,15,7,16 X4,14,5,13 X16,6,17,5 X12,18,13,17 X10,4,11,3 X18,12,19,11 X14,21,15,22 X2,20,3,19
Gauss code	{1, -11, 8, -5, 6, -3}, {3, -1, 2, -8, 9, -7, 5, -10, 4, -6, 7, -9, 11, -2, 10, -4}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {2t(1)^{2}t(2)^{4}-3t(1)t(2)^{4}+t(2)^{4}-5t(1)^{2}t(2)^{3}+9t(1)t(2)^{3}-4t(2)^{3}+6t(1)^{2}t(2)^{2}-11t(1)t(2)^{2}+6t(2)^{2}-4t(1)^{2}t(2)+9t(1)t(2)-5t(2)+t(1)^{2}-3t(1)+2}{t(1)t(2)^{2}}}$ (db)
Jones polynomial	$-q^{11/2}+4q^{9/2}-9q^{7/2}+14q^{5/2}-20q^{3/2}+22{\sqrt {q}}-{\frac {23}{\sqrt {q}}}+{\frac {20}{q^{3/2}}}-{\frac {15}{q^{5/2}}}+{\frac {9}{q^{7/2}}}-{\frac {4}{q^{9/2}}}+{\frac {1}{q^{11/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$az^{7}+z^{7}a^{-1}-a^{3}z^{5}+3az^{5}+3z^{5}a^{-1}-z^{5}a^{-3}-2a^{3}z^{3}+3az^{3}+3z^{3}a^{-1}-2z^{3}a^{-3}-a^{3}z+az+2za^{-1}-za^{-3}+a^{-1}z^{-1}-a^{-3}z^{-1}$ (db)
Kauffman polynomial	$-3z^{10}a^{-2}-3z^{10}-10az^{9}-16z^{9}a^{-1}-6z^{9}a^{-3}-15a^{2}z^{8}-5z^{8}a^{-2}-4z^{8}a^{-4}-16z^{8}-14a^{3}z^{7}+8az^{7}+40z^{7}a^{-1}+17z^{7}a^{-3}-z^{7}a^{-5}-9a^{4}z^{6}+24a^{2}z^{6}+37z^{6}a^{-2}+13z^{6}a^{-4}+57z^{6}-4a^{5}z^{5}+19a^{3}z^{5}+17az^{5}-20z^{5}a^{-1}-11z^{5}a^{-3}+3z^{5}a^{-5}-a^{6}z^{4}+7a^{4}z^{4}-11a^{2}z^{4}-38z^{4}a^{-2}-13z^{4}a^{-4}-44z^{4}+a^{5}z^{3}-10a^{3}z^{3}-15az^{3}-2z^{3}a^{-1}-z^{3}a^{-3}-3z^{3}a^{-5}-2a^{4}z^{2}+2a^{2}z^{2}+10z^{2}a^{-2}+4z^{2}a^{-4}+10z^{2}+2a^{3}z+3az+za^{-5}-a^{-2}+a^{-1}z^{-1}+a^{-3}z^{-1}$ (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   \ -5 -4 -3 -2 -1 0 1 2 3 4 5 6 χ 12                       1 1 10                     3   -3 8                   6 1   5 6                 8 3     -5 4               12 6       6 2             11 9         -2 0           12 11           1 -2         9 12             3 -4       6 11               -5 -6     3 9                 6 -8   1 6                   -5 -10   3                     3 -12 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-2$ $i=0$ $r=-5$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-3$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-2$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=-1$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=0$ ${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{12}$ $r=1$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{11}$ $r=2$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{12}$ $r=3$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=4$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=5$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=6$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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