L11a168

L11a167

L11a169

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a168 at Knotilus!

Link Presentations

[edit Notes on L11a168's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {u^{2}v^{6}-3u^{2}v^{5}+4u^{2}v^{4}-4u^{2}v^{3}+3u^{2}v^{2}-u^{2}v-uv^{6}+4uv^{5}-7uv^{4}+7uv^{3}-7uv^{2}+4uv-u-v^{5}+3v^{4}-4v^{3}+4v^{2}-3v+1}{uv^{3}}}$ (db)
Jones polynomial	$-q^{5/2}+4q^{3/2}-8{\sqrt {q}}+{\frac {12}{\sqrt {q}}}-{\frac {18}{q^{3/2}}}+{\frac {19}{q^{5/2}}}-{\frac {20}{q^{7/2}}}+{\frac {18}{q^{9/2}}}-{\frac {13}{q^{11/2}}}+{\frac {8}{q^{13/2}}}-{\frac {4}{q^{15/2}}}+{\frac {1}{q^{17/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$-a^{5}z^{7}-4a^{5}z^{5}-4a^{5}z^{3}+a^{3}z^{9}+6a^{3}z^{7}+12a^{3}z^{5}+8a^{3}z^{3}+a^{3}z^{-1}-az^{7}-4az^{5}-4az^{3}-az-az^{-1}$ (db)
Kauffman polynomial	$a^{10}z^{4}+4a^{9}z^{5}-2a^{9}z^{3}+8a^{8}z^{6}-7a^{8}z^{4}+2a^{8}z^{2}+11a^{7}z^{7}-13a^{7}z^{5}+4a^{7}z^{3}+11a^{6}z^{8}-14a^{6}z^{6}+4a^{6}z^{2}+8a^{5}z^{9}-9a^{5}z^{7}-4a^{5}z^{5}+2a^{5}z^{3}+3a^{4}z^{10}+7a^{4}z^{8}-30a^{4}z^{6}+18a^{4}z^{4}+14a^{3}z^{9}-42a^{3}z^{7}+38a^{3}z^{5}-14a^{3}z^{3}+a^{3}z-a^{3}z^{-1}+3a^{2}z^{10}-22a^{2}z^{6}+23a^{2}z^{4}-4a^{2}z^{2}+a^{2}+6az^{9}-21az^{7}+z^{7}a^{-1}+22az^{5}-3z^{5}a^{-1}-8az^{3}+2z^{3}a^{-1}+az-az^{-1}+4z^{8}-14z^{6}+13z^{4}-2z^{2}$ (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                     3   -3 2                   5 1   4 0                 7 3     -4 -2               11 5       6 -4             9 8         -1 -6           11 10           1 -8         8 10             2 -10       5 10               -5 -12     3 8                 5 -14   1 5                   -4 -16   3                     3 -18 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-4$ $i=-2$ $r=-7$ ${\mathbb {Z} }$ $r=-6$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-4$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=-3$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=-2$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{11}$ $r=-1$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=0$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{11}$ $r=1$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=2$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=3$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=4$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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