L11a87

L11a86

L11a88

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a87 at Knotilus!

Link Presentations

[edit Notes on L11a87's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {uv^{5}-5uv^{4}+10uv^{3}-12uv^{2}+7uv-2u-2v^{5}+7v^{4}-12v^{3}+10v^{2}-5v+1}{{\sqrt {u}}v^{5/2}}}$ (db)
Jones polynomial	$-{\frac {14}{q^{9/2}}}-q^{7/2}+{\frac {19}{q^{7/2}}}+5q^{5/2}-{\frac {24}{q^{5/2}}}-11q^{3/2}+{\frac {24}{q^{3/2}}}+{\frac {1}{q^{15/2}}}-{\frac {3}{q^{13/2}}}+{\frac {7}{q^{11/2}}}+17{\sqrt {q}}-{\frac {22}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{7}(-z)-a^{7}z^{-1}+3a^{5}z^{3}+6a^{5}z+4a^{5}z^{-1}-3a^{3}z^{5}-8a^{3}z^{3}-9a^{3}z-4a^{3}z^{-1}+az^{7}+3az^{5}-z^{5}a^{-1}+4az^{3}-z^{3}a^{-1}+2az+az^{-1}$ (db)
Kauffman polynomial	$-2a^{4}z^{10}-2a^{2}z^{10}-4a^{5}z^{9}-12a^{3}z^{9}-8az^{9}-4a^{6}z^{8}-9a^{4}z^{8}-18a^{2}z^{8}-13z^{8}-3a^{7}z^{7}-a^{5}z^{7}+12a^{3}z^{7}-az^{7}-11z^{7}a^{-1}-a^{8}z^{6}+4a^{6}z^{6}+19a^{4}z^{6}+37a^{2}z^{6}-5z^{6}a^{-2}+18z^{6}+8a^{7}z^{5}+18a^{5}z^{5}+14a^{3}z^{5}+20az^{5}+15z^{5}a^{-1}-z^{5}a^{-3}+3a^{8}z^{4}+7a^{6}z^{4}-a^{4}z^{4}-13a^{2}z^{4}+4z^{4}a^{-2}-4z^{4}-8a^{7}z^{3}-25a^{5}z^{3}-25a^{3}z^{3}-12az^{3}-4z^{3}a^{-1}-3a^{8}z^{2}-11a^{6}z^{2}-15a^{4}z^{2}-8a^{2}z^{2}-z^{2}+4a^{7}z+16a^{5}z+15a^{3}z+3az+a^{8}+4a^{6}+7a^{4}+4a^{2}+1-a^{7}z^{-1}-4a^{5}z^{-1}-4a^{3}z^{-1}-az^{-1}$ (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                     4   -4 4                   7 1   6 2                 10 4     -6 0               12 7       5 -2             13 11         -2 -4           11 11           0 -6         8 13             5 -8       6 11               -5 -10     2 9                 7 -12   1 5                   -4 -14   2                     2 -16 1                       -1

Integral Khovanov Homology (db, data source)

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

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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