L10n39

L10n38

L10n40

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Link Presentations

[edit Notes on L10n39's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(t(1)-1)(t(2)-1)\left(t(2)^{2}-t(2)+1\right)\left(t(2)^{2}+t(2)+1\right)}{{\sqrt {t(1)}}t(2)^{5/2}}}$ (db)
Jones polynomial	$-4q^{9/2}+3q^{7/2}-4q^{5/2}+q^{3/2}-q^{17/2}+3q^{15/2}-3q^{13/2}+4q^{11/2}-{\sqrt {q}}$ (db)
Signature	5 (db)
HOMFLY-PT polynomial	$-za^{-9}-a^{-9}z^{-1}+z^{5}a^{-7}+5z^{3}a^{-7}+8za^{-7}+5a^{-7}z^{-1}-z^{7}a^{-5}-6z^{5}a^{-5}-13z^{3}a^{-5}-15za^{-5}-8a^{-5}z^{-1}+z^{5}a^{-3}+5z^{3}a^{-3}+8za^{-3}+4a^{-3}z^{-1}$ (db)
Kauffman polynomial	$-z^{8}a^{-4}-z^{8}a^{-6}-z^{7}a^{-3}-5z^{7}a^{-5}-4z^{7}a^{-7}+3z^{6}a^{-4}-2z^{6}a^{-6}-5z^{6}a^{-8}+6z^{5}a^{-3}+24z^{5}a^{-5}+16z^{5}a^{-7}-2z^{5}a^{-9}+3z^{4}a^{-4}+23z^{4}a^{-6}+20z^{4}a^{-8}-13z^{3}a^{-3}-35z^{3}a^{-5}-18z^{3}a^{-7}+4z^{3}a^{-9}-13z^{2}a^{-4}-33z^{2}a^{-6}-23z^{2}a^{-8}-3z^{2}a^{-10}+12za^{-3}+23za^{-5}+11za^{-7}-za^{-9}-za^{-11}+8a^{-4}+14a^{-6}+9a^{-8}+2a^{-10}-4a^{-3}z^{-1}-8a^{-5}z^{-1}-5a^{-7}z^{-1}-a^{-9}z^{-1}$ (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   \ -2 -1 0 1 2 3 4 5 6 χ 18                 1 1 16               2   -2 14             2 2   0 12           3 2     -1 10         1 2 1     0 8       2 3         1 6     2 1           1 4   1 4             3 2                   0 0 1                 1

Integral Khovanov Homology (db, data source)

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

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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