L11a164

L11a163

L11a165

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a164 at Knotilus!

Link Presentations

[edit Notes on L11a164's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {u^{2}v^{6}-2u^{2}v^{5}+3u^{2}v^{4}-3u^{2}v^{3}+2u^{2}v^{2}+2uv^{5}-4uv^{4}+5uv^{3}-4uv^{2}+2uv+2v^{4}-3v^{3}+3v^{2}-2v+1}{uv^{3}}}$ (db)
Jones polynomial	$q^{9/2}-2q^{7/2}+5q^{5/2}-8q^{3/2}+10{\sqrt {q}}-{\frac {13}{\sqrt {q}}}+{\frac {12}{q^{3/2}}}-{\frac {11}{q^{5/2}}}+{\frac {8}{q^{7/2}}}-{\frac {5}{q^{9/2}}}+{\frac {2}{q^{11/2}}}-{\frac {1}{q^{13/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$-az^{9}+a^{3}z^{7}-8az^{7}+z^{7}a^{-1}+6a^{3}z^{5}-25az^{5}+6z^{5}a^{-1}+13a^{3}z^{3}-37az^{3}+13z^{3}a^{-1}+11a^{3}z-25az+11za^{-1}+3a^{3}z^{-1}-5az^{-1}+2a^{-1}z^{-1}$ (db)
Kauffman polynomial	$a^{7}z^{5}-3a^{7}z^{3}+2a^{7}z+2a^{6}z^{6}-4a^{6}z^{4}+a^{6}z^{2}+3a^{5}z^{7}-6a^{5}z^{5}+4a^{5}z^{3}-2a^{5}z+3a^{4}z^{8}-5a^{4}z^{6}+z^{6}a^{-4}+4a^{4}z^{4}-4z^{4}a^{-4}-2a^{4}z^{2}+4z^{2}a^{-4}-a^{-4}+3a^{3}z^{9}-9a^{3}z^{7}+2z^{7}a^{-3}+20a^{3}z^{5}-6z^{5}a^{-3}-22a^{3}z^{3}+3z^{3}a^{-3}+13a^{3}z-3a^{3}z^{-1}+a^{2}z^{10}+3a^{2}z^{8}+3z^{8}a^{-2}-17a^{2}z^{6}-9z^{6}a^{-2}+34a^{2}z^{4}+7z^{4}a^{-2}-23a^{2}z^{2}-3z^{2}a^{-2}+5a^{2}+6az^{9}+3z^{9}a^{-1}-24az^{7}-10z^{7}a^{-1}+49az^{5}+16z^{5}a^{-1}-51az^{3}-19z^{3}a^{-1}+28az+11za^{-1}-5az^{-1}-2a^{-1}z^{-1}+z^{10}+3z^{8}-20z^{6}+37z^{4}-27z^{2}+5$ (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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 10                       1 -1 8                     1   1 6                   4 1   -3 4                 4 1     3 2               6 4       -2 0             7 4         3 -2           6 7           1 -4         5 6             -1 -6       3 6               3 -8     2 5                 -3 -10     3                   3 -12 1 2                     -1 -14 1                       1

Integral Khovanov Homology (db, data source)

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

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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