L11a205

L11a204

L11a206

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a205 at Knotilus!

Link Presentations

[edit Notes on L11a205's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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