L11a187

L11a186

L11a188

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a187 at Knotilus!

Link Presentations

[edit Notes on L11a187's Link Presentations]

Planar diagram presentation	X8192 X10,3,11,4 X20,14,21,13 X16,5,17,6 X4,15,5,16 X18,12,19,11 X22,18,7,17 X14,20,15,19 X12,22,13,21 X2738 X6,9,1,10
Gauss code	{1, -10, 2, -5, 4, -11}, {10, -1, 11, -2, 6, -9, 3, -8, 5, -4, 7, -6, 8, -3, 9, -7}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {2t(1)t(2)^{4}+2t(1)^{2}t(2)^{3}-7t(1)t(2)^{3}+4t(2)^{3}-5t(1)^{2}t(2)^{2}+11t(1)t(2)^{2}-5t(2)^{2}+4t(1)^{2}t(2)-7t(1)t(2)+2t(2)+2t(1)}{t(1)t(2)^{2}}}$ (db)
Jones polynomial	$q^{9/2}-{\frac {7}{q^{9/2}}}-3q^{7/2}+{\frac {10}{q^{7/2}}}+6q^{5/2}-{\frac {15}{q^{5/2}}}-10q^{3/2}+{\frac {16}{q^{3/2}}}-{\frac {1}{q^{13/2}}}+{\frac {3}{q^{11/2}}}+14{\sqrt {q}}-{\frac {16}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{5}z^{3}+a^{5}z+a^{5}z^{-1}-a^{3}z^{5}-a^{3}z^{3}+z^{3}a^{-3}-a^{3}z+za^{-3}-2az^{5}-z^{5}a^{-1}-4az^{3}-z^{3}a^{-1}-4az-2az^{-1}+a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-a^{2}z^{10}-z^{10}-4a^{3}z^{9}-7az^{9}-3z^{9}a^{-1}-6a^{4}z^{8}-9a^{2}z^{8}-4z^{8}a^{-2}-7z^{8}-6a^{5}z^{7}+4a^{3}z^{7}+16az^{7}+3z^{7}a^{-1}-3z^{7}a^{-3}-3a^{6}z^{6}+12a^{4}z^{6}+29a^{2}z^{6}+10z^{6}a^{-2}-z^{6}a^{-4}+25z^{6}-a^{7}z^{5}+15a^{5}z^{5}+a^{3}z^{5}-20az^{5}+4z^{5}a^{-1}+9z^{5}a^{-3}+5a^{6}z^{4}-9a^{4}z^{4}-38a^{2}z^{4}-7z^{4}a^{-2}+3z^{4}a^{-4}-34z^{4}+2a^{7}z^{3}-15a^{5}z^{3}-5a^{3}z^{3}+17az^{3}-2z^{3}a^{-1}-7z^{3}a^{-3}+19a^{2}z^{2}+4z^{2}a^{-2}-2z^{2}a^{-4}+25z^{2}+7a^{5}z-9az-za^{-1}+za^{-3}+a^{4}-3a^{2}-2a^{-2}-5-a^{5}z^{-1}+2az^{-1}+a^{-1}z^{-1}$ (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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 10                       1 -1 8                     2   2 6                   4 1   -3 4                 6 2     4 2               8 4       -4 0             8 6         2 -2           9 9           0 -4         6 7             -1 -6       4 9               5 -8     3 6                 -3 -10     4                   4 -12 1 3                     -2 -14 1                       1

Integral Khovanov Homology (db, data source)

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

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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