L10a162

L10a161

L10a163

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Link Presentations

[edit Notes on L10a162's Link Presentations]

Planar diagram presentation	X8192 X16,6,17,5 X10,20,11,19 X18,10,19,9 X20,12,13,11 X2,13,3,14 X14,3,15,4 X4758 X12,16,7,15 X6,18,1,17
Gauss code	{1, -6, 7, -8, 2, -10}, {8, -1, 4, -3, 5, -9}, {6, -7, 9, -2, 10, -4, 3, -5}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(t(2)-1)(t(3)t(1)-t(1)+1)(t(1)t(3)-t(3)+1)(t(2)t(3)+1)}{t(1)t(2)t(3)^{3/2}}}$ (db)
Jones polynomial	$q^{7}-3q^{6}+6q^{5}-9q^{4}+12q^{3}-11q^{2}+12q-8+6q^{-1}-3q^{-2}+q^{-3}$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$z^{6}a^{-4}+4z^{4}a^{-4}+5z^{2}a^{-4}+a^{-4}z^{-2}+2a^{-4}-z^{8}a^{-2}-6z^{6}a^{-2}-13z^{4}a^{-2}-12z^{2}a^{-2}-2a^{-2}z^{-2}-5a^{-2}+z^{6}+4z^{4}+5z^{2}+z^{-2}+3$ (db)
Kauffman polynomial	$z^{4}a^{-8}-z^{2}a^{-8}+3z^{5}a^{-7}-3z^{3}a^{-7}+5z^{6}a^{-6}-6z^{4}a^{-6}+3z^{2}a^{-6}-a^{-6}+6z^{7}a^{-5}-9z^{5}a^{-5}+6z^{3}a^{-5}+5z^{8}a^{-4}-8z^{6}a^{-4}+7z^{4}a^{-4}-3z^{2}a^{-4}-a^{-4}z^{-2}+2a^{-4}+2z^{9}a^{-3}+3z^{7}a^{-3}-13z^{5}a^{-3}+12z^{3}a^{-3}-5za^{-3}+2a^{-3}z^{-1}+9z^{8}a^{-2}+a^{2}z^{6}-26z^{6}a^{-2}-3a^{2}z^{4}+29z^{4}a^{-2}+2a^{2}z^{2}-18z^{2}a^{-2}-2a^{-2}z^{-2}+6a^{-2}+2z^{9}a^{-1}+3az^{7}-9az^{5}-10z^{5}a^{-1}+6az^{3}+9z^{3}a^{-1}-5za^{-1}+2a^{-1}z^{-1}+4z^{8}-12z^{6}+12z^{4}-9z^{2}-z^{-2}+4$ (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 15                     1 1 13                   2   -2 11                 4 1   3 9               6 3     -3 7             6 3       3 5           6 7         1 3         6 5           1 1       3 7             4 -1     3 5               -2 -3   1 4                 3 -5   2                   -2 -7 1                     1

Integral Khovanov Homology (db, data source)

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

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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