L11a202

L11a201

L11a203

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a202 at Knotilus!

Link Presentations

[edit Notes on L11a202's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {t(1)^{2}t(2)^{6}-t(1)t(2)^{6}-t(1)^{2}t(2)^{5}+t(1)t(2)^{5}-t(2)^{5}+t(1)^{2}t(2)^{4}-t(1)t(2)^{4}+t(2)^{4}-t(1)^{2}t(2)^{3}+t(1)t(2)^{3}-t(2)^{3}+t(1)^{2}t(2)^{2}-t(1)t(2)^{2}+t(2)^{2}-t(1)^{2}t(2)+t(1)t(2)-t(2)-t(1)+1}{t(1)t(2)^{3}}}$ (db)
Jones polynomial	$-{\frac {1}{q^{3/2}}}+{\frac {1}{q^{5/2}}}-{\frac {3}{q^{7/2}}}+{\frac {3}{q^{9/2}}}-{\frac {5}{q^{11/2}}}+{\frac {5}{q^{13/2}}}-{\frac {5}{q^{15/2}}}+{\frac {5}{q^{17/2}}}-{\frac {4}{q^{19/2}}}+{\frac {3}{q^{21/2}}}-{\frac {2}{q^{23/2}}}+{\frac {1}{q^{25/2}}}$ (db)
Signature	-7 (db)
HOMFLY-PT polynomial	$-z^{7}a^{9}-6z^{5}a^{9}-11z^{3}a^{9}-7za^{9}-2a^{9}z^{-1}+z^{9}a^{7}+8z^{7}a^{7}+23z^{5}a^{7}+30z^{3}a^{7}+19za^{7}+5a^{7}z^{-1}-z^{7}a^{5}-7z^{5}a^{5}-16z^{3}a^{5}-13za^{5}-3a^{5}z^{-1}$ (db)
Kauffman polynomial	$a^{16}z^{2}+2a^{15}z^{3}+3a^{14}z^{4}-2a^{14}z^{2}+4a^{13}z^{5}-6a^{13}z^{3}+5a^{12}z^{6}-13a^{12}z^{4}+6a^{12}z^{2}-a^{12}+5a^{11}z^{7}-17a^{11}z^{5}+13a^{11}z^{3}-3a^{11}z+4a^{10}z^{8}-15a^{10}z^{6}+11a^{10}z^{4}+a^{10}z^{2}+3a^{9}z^{9}-14a^{9}z^{7}+18a^{9}z^{5}-9a^{9}z^{3}+6a^{9}z-2a^{9}z^{-1}+a^{8}z^{10}-2a^{8}z^{8}-11a^{8}z^{6}+31a^{8}z^{4}-22a^{8}z^{2}+5a^{8}+4a^{7}z^{9}-27a^{7}z^{7}+62a^{7}z^{5}-59a^{7}z^{3}+25a^{7}z-5a^{7}z^{-1}+a^{6}z^{10}-6a^{6}z^{8}+9a^{6}z^{6}+4a^{6}z^{4}-14a^{6}z^{2}+5a^{6}+a^{5}z^{9}-8a^{5}z^{7}+23a^{5}z^{5}-29a^{5}z^{3}+16a^{5}z-3a^{5}z^{-1}$ (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   \ -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 χ -2                       1 1 -4                         0 -6                   3 1   2 -8                 1 1     0 -10               4 2       2 -12             2 2         0 -14           3 3           0 -16         2 2             0 -18       2 3               -1 -20     1 2                 1 -22   1 2                   -1 -24   1                     1 -26 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-8$ $i=-6$ $r=-9$ ${\mathbb {Z} }$ $r=-8$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-7$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-6$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-5$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-4$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-3$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-2$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{4}$ $r=-1$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=0$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{3}$ $r=1$ ${\mathbb {Z} }$ $r=2$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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