L11a182

L11a181

L11a183

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a182 at Knotilus!

Link Presentations

[edit Notes on L11a182's Link Presentations]

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {2u^{2}v^{4}-5u^{2}v^{3}+6u^{2}v^{2}-4u^{2}v+u^{2}-3uv^{4}+9uv^{3}-11uv^{2}+9uv-3u+v^{4}-4v^{3}+6v^{2}-5v+2}{uv^{2}}}$ (db)
Jones polynomial	$23q^{9/2}-23q^{7/2}+19q^{5/2}-15q^{3/2}+{\frac {1}{q^{3/2}}}-q^{19/2}+4q^{17/2}-9q^{15/2}+15q^{13/2}-20q^{11/2}+8{\sqrt {q}}-{\frac {4}{\sqrt {q}}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$z^{7}a^{-3}+z^{7}a^{-5}-z^{5}a^{-1}+3z^{5}a^{-3}+3z^{5}a^{-5}-z^{5}a^{-7}-2z^{3}a^{-1}+3z^{3}a^{-3}+3z^{3}a^{-5}-2z^{3}a^{-7}+za^{-3}+za^{-5}-za^{-7}+a^{-1}z^{-1}-a^{-3}z^{-1}$ (db)
Kauffman polynomial	$z^{5}a^{-11}-z^{3}a^{-11}+4z^{6}a^{-10}-5z^{4}a^{-10}+z^{2}a^{-10}+8z^{7}a^{-9}-12z^{5}a^{-9}+6z^{3}a^{-9}-za^{-9}+10z^{8}a^{-8}-16z^{6}a^{-8}+11z^{4}a^{-8}-3z^{2}a^{-8}+7z^{9}a^{-7}-3z^{7}a^{-7}-8z^{5}a^{-7}+7z^{3}a^{-7}-za^{-7}+2z^{10}a^{-6}+15z^{8}a^{-6}-36z^{6}a^{-6}+26z^{4}a^{-6}-6z^{2}a^{-6}+12z^{9}a^{-5}-16z^{7}a^{-5}-z^{5}a^{-5}+5z^{3}a^{-5}-za^{-5}+2z^{10}a^{-4}+11z^{8}a^{-4}-28z^{6}a^{-4}+15z^{4}a^{-4}-2z^{2}a^{-4}+5z^{9}a^{-3}-z^{7}a^{-3}-16z^{5}a^{-3}+13z^{3}a^{-3}-2za^{-3}-a^{-3}z^{-1}+6z^{8}a^{-2}-11z^{6}a^{-2}+3z^{4}a^{-2}+z^{2}a^{-2}+a^{-2}+4z^{7}a^{-1}-10z^{5}a^{-1}+8z^{3}a^{-1}-za^{-1}-a^{-1}z^{-1}+z^{6}-2z^{4}+z^{2}$ (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   \ -3 -2 -1 0 1 2 3 4 5 6 7 8 χ 20                       1 1 18                     3   -3 16                   6 1   5 14                 9 3     -6 12               11 6       5 10             12 9         -3 8           11 11           0 6         9 13             4 4       6 10               -4 2     3 10                 7 0   1 5                   -4 -2   3                     3 -4 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=2$ $i=4$ $r=-3$ ${\mathbb {Z} }$ $r=-2$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-1$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=0$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{6}$ $r=1$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=2$ ${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{11}$ $r=3$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{12}$ ${\mathbb {Z} }^{12}$ $r=4$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{11}$ $r=5$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=6$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=7$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=8$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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