L11a199

L11a198

L11a200

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a199 at Knotilus!

Link Presentations

[edit Notes on L11a199's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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