L11a385

L11a384

L11a386

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a385 at Knotilus!

Link Presentations

[edit Notes on L11a385's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {-2t(1)t(3)^{3}+2t(1)t(2)t(3)^{3}-4t(2)t(3)^{3}+2t(3)^{3}+6t(1)t(3)^{2}-4t(1)t(2)t(3)^{2}+7t(2)t(3)^{2}-4t(3)^{2}-7t(1)t(3)+4t(1)t(2)t(3)-6t(2)t(3)+4t(3)+4t(1)-2t(1)t(2)+2t(2)-2}{{\sqrt {t(1)}}{\sqrt {t(2)}}t(3)^{3/2}}}$ (db)
Jones polynomial	$-q^{8}+4q^{7}-10q^{6}+14q^{5}-18q^{4}+21q^{3}+q^{-3}-19q^{2}-2q^{-2}+17q+7q^{-1}-10$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$z^{6}a^{-2}+z^{6}a^{-4}+z^{4}a^{-2}+2z^{4}a^{-4}-z^{4}a^{-6}-2z^{4}+a^{2}z^{2}-z^{2}a^{-2}+4z^{2}a^{-4}-z^{2}a^{-6}-4z^{2}+2a^{2}-3a^{-2}+6a^{-4}-2a^{-6}-3+a^{2}z^{-2}-2a^{-2}z^{-2}+3a^{-4}z^{-2}-a^{-6}z^{-2}-z^{-2}$ (db)
Kauffman polynomial	$z^{10}a^{-2}+z^{10}a^{-4}+2z^{9}a^{-1}+7z^{9}a^{-3}+5z^{9}a^{-5}+8z^{8}a^{-2}+15z^{8}a^{-4}+10z^{8}a^{-6}+3z^{8}+2az^{7}+6z^{7}a^{-1}+z^{7}a^{-3}+6z^{7}a^{-5}+9z^{7}a^{-7}+a^{2}z^{6}-15z^{6}a^{-2}-31z^{6}a^{-4}-17z^{6}a^{-6}+4z^{6}a^{-8}-4z^{6}-4az^{5}-23z^{5}a^{-1}-33z^{5}a^{-3}-31z^{5}a^{-5}-16z^{5}a^{-7}+z^{5}a^{-9}-4a^{2}z^{4}+6z^{4}a^{-2}+18z^{4}a^{-4}+10z^{4}a^{-6}-4z^{4}a^{-8}-2z^{4}+23z^{3}a^{-1}+50z^{3}a^{-3}+39z^{3}a^{-5}+11z^{3}a^{-7}-z^{3}a^{-9}+6a^{2}z^{2}-3z^{2}a^{-2}-4z^{2}a^{-4}-2z^{2}a^{-6}+5z^{2}+4az-10za^{-1}-34za^{-3}-27za^{-5}-7za^{-7}-4a^{2}+4a^{-2}+5a^{-4}+a^{-6}-3-2az^{-1}+2a^{-1}z^{-1}+10a^{-3}z^{-1}+8a^{-5}z^{-1}+2a^{-7}z^{-1}+a^{2}z^{-2}-2a^{-2}z^{-2}-3a^{-4}z^{-2}-a^{-6}z^{-2}+z^{-2}$ (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   \ -4 -3 -2 -1 0 1 2 3 4 5 6 7 χ 17                       1 -1 15                     3   3 13                   7 1   -6 11                 7 3     4 9               11 7       -4 7             10 7         3 5           9 11           2 3         8 10             -2 1       5 12               7 -1     2 5                 -3 -3     5                   5 -5 1 2                     -1 -7 1                       1

Integral Khovanov Homology (db, data source)

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

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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