L11a82

L11a81

L11a83

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a82 at Knotilus!

Link Presentations

[edit Notes on L11a82's Link Presentations]

Planar diagram presentation	X6172 X12,3,13,4 X16,8,17,7 X22,13,5,14 X14,17,15,18 X20,10,21,9 X18,21,19,22 X8,16,9,15 X10,20,11,19 X2536 X4,11,1,12
Gauss code	{1, -10, 2, -11}, {10, -1, 3, -8, 6, -9, 11, -2, 4, -5, 8, -3, 5, -7, 9, -6, 7, -4}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {2uv^{4}-10uv^{3}+14uv^{2}-6uv+u+v^{5}-6v^{4}+14v^{3}-10v^{2}+2v}{{\sqrt {u}}v^{5/2}}}$ (db)
Jones polynomial	$-{\frac {14}{q^{9/2}}}-q^{7/2}+{\frac {18}{q^{7/2}}}+3q^{5/2}-{\frac {22}{q^{5/2}}}-8q^{3/2}+{\frac {21}{q^{3/2}}}+{\frac {1}{q^{15/2}}}-{\frac {4}{q^{13/2}}}+{\frac {8}{q^{11/2}}}+14{\sqrt {q}}-{\frac {18}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$-za^{7}+3z^{3}a^{5}+3za^{5}+2a^{5}z^{-1}-2z^{5}a^{3}-4z^{3}a^{3}-8za^{3}-4a^{3}z^{-1}-z^{5}a+3z^{3}a+6za+3az^{-1}+2z^{3}a^{-1}-za^{-1}-a^{-1}z^{-1}-za^{-3}$ (db)
Kauffman polynomial	$a^{8}z^{6}-2a^{8}z^{4}+a^{8}z^{2}+4a^{7}z^{7}-10a^{7}z^{5}+8a^{7}z^{3}-3a^{7}z+6a^{6}z^{8}-12a^{6}z^{6}+5a^{6}z^{4}-a^{6}z^{2}+a^{6}+4a^{5}z^{9}+5a^{5}z^{7}-34a^{5}z^{5}+36a^{5}z^{3}-15a^{5}z+2a^{5}z^{-1}+a^{4}z^{10}+17a^{4}z^{8}-47a^{4}z^{6}+40a^{4}z^{4}-14a^{4}z^{2}+2a^{4}+9a^{3}z^{9}-a^{3}z^{7}-40a^{3}z^{5}+z^{5}a^{-3}+53a^{3}z^{3}-2z^{3}a^{-3}-24a^{3}z+za^{-3}+4a^{3}z^{-1}+a^{2}z^{10}+19a^{2}z^{8}-51a^{2}z^{6}+3z^{6}a^{-2}+53a^{2}z^{4}-4z^{4}a^{-2}-24a^{2}z^{2}+z^{2}a^{-2}+3a^{2}+5az^{9}+4az^{7}+6z^{7}a^{-1}-26az^{5}-9z^{5}a^{-1}+34az^{3}+7z^{3}a^{-1}-17az-4za^{-1}+3az^{-1}+a^{-1}z^{-1}+8z^{8}-14z^{6}+16z^{4}-11z^{2}+1$ (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                   6 1   5 2                 8 2     -6 0               10 6       4 -2             12 9         -3 -4           10 9           1 -6         8 12             4 -8       6 10               -4 -10     3 9                 6 -12   1 5                   -4 -14   3                     3 -16 1                       -1

Integral Khovanov Homology (db, data source)

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

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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