L10n38

L10n37

L10n39

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Link Presentations

[edit Notes on L10n38's Link Presentations]

Planar diagram presentation	X6172 X12,3,13,4 X16,13,17,14 X14,9,15,10 X10,15,11,16 X17,5,18,20 X7,19,8,18 X19,9,20,8 X2536 X4,11,1,12
Gauss code	{1, -9, 2, -10}, {9, -1, -7, 8, 4, -5, 10, -2, 3, -4, 5, -3, -6, 7, -8, 6}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(t(1)+t(2))\left(t(2)^{2}-t(2)+1\right)^{2}}{{\sqrt {t(1)}}t(2)^{5/2}}}$ (db)
Jones polynomial	$-{\frac {6}{q^{9/2}}}+{\frac {5}{q^{7/2}}}-{\frac {7}{q^{5/2}}}+{\frac {5}{q^{3/2}}}+{\frac {1}{q^{15/2}}}-{\frac {2}{q^{13/2}}}+{\frac {4}{q^{11/2}}}+2{\sqrt {q}}-{\frac {4}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$-za^{7}-2a^{7}z^{-1}+3z^{3}a^{5}+9za^{5}+7a^{5}z^{-1}-2z^{5}a^{3}-9z^{3}a^{3}-14za^{3}-7a^{3}z^{-1}+2z^{3}a+4za+2az^{-1}$ (db)
Kauffman polynomial	$-z^{6}a^{8}+4z^{4}a^{8}-5z^{2}a^{8}+2a^{8}-2z^{7}a^{7}+7z^{5}a^{7}-7z^{3}a^{7}+4za^{7}-2a^{7}z^{-1}-z^{8}a^{6}-2z^{6}a^{6}+17z^{4}a^{6}-20z^{2}a^{6}+8a^{6}-6z^{7}a^{5}+20z^{5}a^{5}-22z^{3}a^{5}+17za^{5}-7a^{5}z^{-1}-z^{8}a^{4}-5z^{6}a^{4}+24z^{4}a^{4}-29z^{2}a^{4}+13a^{4}-4z^{7}a^{3}+12z^{5}a^{3}-18z^{3}a^{3}+16za^{3}-7a^{3}z^{-1}-4z^{6}a^{2}+11z^{4}a^{2}-17z^{2}a^{2}+8a^{2}-z^{5}a-3z^{3}a+3za-2az^{-1}-3z^{2}+2$ (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   \ -7 -6 -5 -4 -3 -2 -1 0 1 χ 2                 2 -2 0               2   2 -2             4 3   -1 -4           3 1     2 -6         2 4       2 -8       4 3         1 -10     1 3           2 -12   1 3             -2 -14   1               1 -16 1                 -1

Integral Khovanov Homology (db, data source)

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

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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