L10a7

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L10a8

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a7 at Knotilus!

Link Presentations

[edit Notes on L10a7's Link Presentations]

Planar diagram presentation	X6172 X18,7,19,8 X4,19,1,20 X14,12,15,11 X8493 X12,5,13,6 X20,13,5,14 X16,10,17,9 X10,16,11,15 X2,18,3,17
Gauss code	{1, -10, 5, -3}, {6, -1, 2, -5, 8, -9, 4, -6, 7, -4, 9, -8, 10, -2, 3, -7}

A Braid Representative	{{{braid_table}}}
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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