L11a167

L11a166

L11a168

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a167 at Knotilus!

Link Presentations

[edit Notes on L11a167's Link Presentations]

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

A Braid Representative	{{{braid_table}}}
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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