L10a66

L10a65

L10a67

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Link Presentations

[edit Notes on L10a66's Link Presentations]

Planar diagram presentation	X8192 X2,9,3,10 X10,3,11,4 X6718 X16,11,17,12 X14,6,15,5 X4,16,5,15 X20,17,7,18 X18,14,19,13 X12,20,13,19
Gauss code	{1, -2, 3, -7, 6, -4}, {4, -1, 2, -3, 5, -10, 9, -6, 7, -5, 8, -9, 10, -8}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {-t(1)^{2}t(2)^{4}+t(1)t(2)^{4}+3t(1)^{2}t(2)^{3}-3t(1)t(2)^{3}+t(2)^{3}-3t(1)^{2}t(2)^{2}+3t(1)t(2)^{2}-3t(2)^{2}+t(1)^{2}t(2)-3t(1)t(2)+3t(2)+t(1)-1}{t(1)t(2)^{2}}}$ (db)
Jones polynomial	$-q^{5/2}+2q^{3/2}-4{\sqrt {q}}+{\frac {6}{\sqrt {q}}}-{\frac {8}{q^{3/2}}}+{\frac {8}{q^{5/2}}}-{\frac {9}{q^{7/2}}}+{\frac {7}{q^{9/2}}}-{\frac {5}{q^{11/2}}}+{\frac {3}{q^{13/2}}}-{\frac {1}{q^{15/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a^{5}z^{5}+3a^{5}z^{3}+2a^{5}z+a^{5}z^{-1}-a^{3}z^{7}-5a^{3}z^{5}-9a^{3}z^{3}-8a^{3}z-2a^{3}z^{-1}+2az^{5}+8az^{3}-z^{3}a^{-1}+8az-3za^{-1}+2az^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-z^{3}a^{9}-3z^{4}a^{8}+z^{2}a^{8}-5z^{5}a^{7}+4z^{3}a^{7}-za^{7}-6z^{6}a^{6}+7z^{4}a^{6}-z^{2}a^{6}-6z^{7}a^{5}+11z^{5}a^{5}-5z^{3}a^{5}+2za^{5}-a^{5}z^{-1}-4z^{8}a^{4}+7z^{6}a^{4}-z^{9}a^{3}-6z^{7}a^{3}+29z^{5}a^{3}-31z^{3}a^{3}+12za^{3}-2a^{3}z^{-1}-6z^{8}a^{2}+22z^{6}a^{2}-22z^{4}a^{2}+7z^{2}a^{2}-a^{2}-z^{9}a-z^{7}a+18z^{5}a-29z^{3}a+14za-2az^{-1}-2z^{8}+9z^{6}-12z^{4}+5z^{2}-z^{7}a^{-1}+5z^{5}a^{-1}-8z^{3}a^{-1}+5za^{-1}-a^{-1}z^{-1}$ (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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 χ 6                     1 1 4                   1   -1 2                 3 1   2 0               3 1     -2 -2             5 3       2 -4           4 4         0 -6         5 4           1 -8       3 5             2 -10     2 4               -2 -12   1 3                 2 -14   2                   -2 -16 1                     1

Integral Khovanov Homology (db, data source)

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

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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