L10a83

L10a82

L10a84

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a83 at Knotilus!

Link Presentations

[edit Notes on L10a83's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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