L11a178

L11a177

L11a179

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a178 at Knotilus!

Link Presentations

[edit Notes on L11a178's Link Presentations]

Planar diagram presentation	X8192 X10,4,11,3 X16,7,17,8 X20,10,21,9 X22,15,7,16 X14,6,15,5 X18,14,19,13 X12,20,13,19 X4,12,5,11 X6,18,1,17 X2,21,3,22
Gauss code	{1, -11, 2, -9, 6, -10}, {3, -1, 4, -2, 9, -8, 7, -6, 5, -3, 10, -7, 8, -4, 11, -5}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {2u^{2}v^{4}-5u^{2}v^{3}+6u^{2}v^{2}-4u^{2}v+u^{2}-3uv^{4}+10uv^{3}-15uv^{2}+10uv-3u+v^{4}-4v^{3}+6v^{2}-5v+2}{uv^{2}}}$ (db)
Jones polynomial	$-16q^{9/2}+21q^{7/2}-{\frac {1}{q^{7/2}}}-25q^{5/2}+{\frac {4}{q^{5/2}}}+25q^{3/2}-{\frac {10}{q^{3/2}}}+q^{15/2}-4q^{13/2}+9q^{11/2}-22{\sqrt {q}}+{\frac {16}{\sqrt {q}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$-z^{7}a^{-1}-z^{7}a^{-3}+az^{5}-3z^{5}a^{-1}-3z^{5}a^{-3}+z^{5}a^{-5}+2az^{3}-4z^{3}a^{-1}-3z^{3}a^{-3}+2z^{3}a^{-5}+2az-3za^{-1}+za^{-3}+za^{-5}+az^{-1}-2a^{-1}z^{-1}+2a^{-3}z^{-1}-a^{-5}z^{-1}$ (db)
Kauffman polynomial	$-3z^{10}a^{-2}-3z^{10}a^{-4}-9z^{9}a^{-1}-16z^{9}a^{-3}-7z^{9}a^{-5}-15z^{8}a^{-2}-10z^{8}a^{-4}-7z^{8}a^{-6}-12z^{8}-9az^{7}+7z^{7}a^{-1}+29z^{7}a^{-3}+9z^{7}a^{-5}-4z^{7}a^{-7}-4a^{2}z^{6}+42z^{6}a^{-2}+32z^{6}a^{-4}+14z^{6}a^{-6}-z^{6}a^{-8}+21z^{6}-a^{3}z^{5}+14az^{5}+9z^{5}a^{-1}-12z^{5}a^{-3}+3z^{5}a^{-5}+9z^{5}a^{-7}+4a^{2}z^{4}-34z^{4}a^{-2}-23z^{4}a^{-4}-7z^{4}a^{-6}+2z^{4}a^{-8}-16z^{4}+a^{3}z^{3}-9az^{3}-16z^{3}a^{-1}-5z^{3}a^{-3}-5z^{3}a^{-5}-6z^{3}a^{-7}+9z^{2}a^{-2}+6z^{2}a^{-4}+z^{2}a^{-6}-z^{2}a^{-8}+5z^{2}+4az+9za^{-1}+7za^{-3}+3za^{-5}+za^{-7}-a^{-2}-az^{-1}-2a^{-1}z^{-1}-2a^{-3}z^{-1}-a^{-5}z^{-1}$ (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   \ -4 -3 -2 -1 0 1 2 3 4 5 6 7 χ 16                       1 -1 14                     3   3 12                   6 1   -5 10                 10 3     7 8               11 6       -5 6             14 10         4 4           12 12           0 2         10 13             -3 0       7 13               6 -2     3 9                 -6 -4   1 7                   6 -6   3                     -3 -8 1                       1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=0$ $i=2$ $r=-4$ ${\mathbb {Z} }$ $r=-3$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-2$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-1$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=0$ ${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{10}$ $r=1$ ${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{12}$ ${\mathbb {Z} }^{12}$ $r=2$ ${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{13}$ ${\mathbb {Z} }^{14}$ $r=3$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{11}$ $r=4$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=5$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=6$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=7$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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