L10a82

L10a81

L10a83

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a82 at Knotilus!

Link Presentations

[edit Notes on L10a82's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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