L11a166

L11a165

L11a167

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a166 at Knotilus!

Link Presentations

[edit Notes on L11a166's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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