L11n187

L11n186

L11n188

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n187 at Knotilus!

Link Presentations

[edit Notes on L11n187's Link Presentations]

Planar diagram presentation	X8192 X12,4,13,3 X22,12,7,11 X15,20,16,21 X18,10,19,9 X10,20,11,19 X21,14,22,15 X16,6,17,5 X2738 X4,14,5,13 X6,18,1,17
Gauss code	{1, -9, 2, -10, 8, -11}, {9, -1, 5, -6, 3, -2, 10, 7, -4, -8, 11, -5, 6, 4, -7, -3}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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