L11a389

L11a388

L11a390

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a389 at Knotilus!

Link Presentations

[edit Notes on L11a389's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {-t(1)t(3)^{3}+t(1)t(2)t(3)^{3}-2t(2)t(3)^{3}+t(3)^{3}+6t(1)t(3)^{2}-5t(1)t(2)t(3)^{2}+8t(2)t(3)^{2}-5t(3)^{2}-8t(1)t(3)+5t(1)t(2)t(3)-6t(2)t(3)+5t(3)+2t(1)-t(1)t(2)+t(2)-1}{{\sqrt {t(1)}}{\sqrt {t(2)}}t(3)^{3/2}}}$ (db)
Jones polynomial	$-q^{5}+4q^{4}-8q^{3}+14q^{2}-17q+19-18q^{-1}+16q^{-2}-10q^{-3}+6q^{-4}-2q^{-5}+q^{-6}$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$a^{6}z^{-2}+a^{6}-3z^{2}a^{4}-3a^{4}z^{-2}-5a^{4}+3z^{4}a^{2}+6z^{2}a^{2}+4a^{2}z^{-2}+8a^{2}-z^{6}-2z^{4}-5z^{2}-3z^{-2}-6+2z^{4}a^{-2}+2z^{2}a^{-2}+a^{-2}z^{-2}+2a^{-2}-z^{2}a^{-4}$ (db)
Kauffman polynomial	$a^{2}z^{10}+z^{10}+2a^{3}z^{9}+6az^{9}+4z^{9}a^{-1}+2a^{4}z^{8}+5a^{2}z^{8}+7z^{8}a^{-2}+10z^{8}+2a^{5}z^{7}+3a^{3}z^{7}-az^{7}+5z^{7}a^{-1}+7z^{7}a^{-3}+a^{6}z^{6}+2a^{4}z^{6}-a^{2}z^{6}-5z^{6}a^{-2}+4z^{6}a^{-4}-11z^{6}-5a^{5}z^{5}-12a^{3}z^{5}-11az^{5}-15z^{5}a^{-1}-10z^{5}a^{-3}+z^{5}a^{-5}-4a^{6}z^{4}-18a^{4}z^{4}-26a^{2}z^{4}-7z^{4}a^{-2}-6z^{4}a^{-4}-13z^{4}+3a^{5}z^{3}+8a^{3}z^{3}+5az^{3}+5z^{3}a^{-1}+4z^{3}a^{-3}-z^{3}a^{-5}+6a^{6}z^{2}+25a^{4}z^{2}+38a^{2}z^{2}+8z^{2}a^{-2}+3z^{2}a^{-4}+24z^{2}+a^{5}z+a^{3}z+az+za^{-1}-4a^{6}-14a^{4}-21a^{2}-4a^{-2}-14-a^{5}z^{-1}-a^{3}z^{-1}-az^{-1}-a^{-1}z^{-1}+a^{6}z^{-2}+3a^{4}z^{-2}+4a^{2}z^{-2}+a^{-2}z^{-2}+3z^{-2}$ (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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 11                       1 -1 9                     3   3 7                   5 1   -4 5                 9 3     6 3               8 5       -3 1             11 9         2 -1           10 11           1 -3         6 8             -2 -5       4 10               6 -7     2 6                 -4 -9   1 5                   4 -11   1                     -1 -13 1                       1

Integral Khovanov Homology (db, data source)

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

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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