L10n35

L10n34

L10n36

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Link Presentations

[edit Notes on L10n35's Link Presentations]

Planar diagram presentation	X6172 X3,13,4,12 X13,17,14,16 X9,15,10,14 X15,11,16,10 X17,5,18,20 X7,19,8,18 X19,9,20,8 X2536 X11,1,12,4
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 $u$, $v$, $w$, ...)	$-{\frac {2(t(1)-1)(t(2)-1)\left(t(2)^{2}-t(2)+1\right)}{{\sqrt {t(1)}}t(2)^{3/2}}}$ (db)
Jones polynomial	$-q^{19/2}+3q^{17/2}-5q^{15/2}+8q^{13/2}-8q^{11/2}+8q^{9/2}-8q^{7/2}+4q^{5/2}-3q^{3/2}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$-za^{-9}-a^{-9}z^{-1}+3z^{3}a^{-7}+7za^{-7}+5a^{-7}z^{-1}-2z^{5}a^{-5}-8z^{3}a^{-5}-13za^{-5}-8a^{-5}z^{-1}+3z^{3}a^{-3}+7za^{-3}+4a^{-3}z^{-1}$ (db)
Kauffman polynomial	$z^{5}a^{-11}-2z^{3}a^{-11}+za^{-11}+3z^{6}a^{-10}-7z^{4}a^{-10}+5z^{2}a^{-10}-2a^{-10}+3z^{7}a^{-9}-3z^{5}a^{-9}-4z^{3}a^{-9}+za^{-9}+a^{-9}z^{-1}+z^{8}a^{-8}+8z^{6}a^{-8}-26z^{4}a^{-8}+24z^{2}a^{-8}-9a^{-8}+7z^{7}a^{-7}-14z^{5}a^{-7}+12z^{3}a^{-7}-9za^{-7}+5a^{-7}z^{-1}+z^{8}a^{-6}+8z^{6}a^{-6}-25z^{4}a^{-6}+31z^{2}a^{-6}-14a^{-6}+4z^{7}a^{-5}-10z^{5}a^{-5}+20z^{3}a^{-5}-19za^{-5}+8a^{-5}z^{-1}+3z^{6}a^{-4}-6z^{4}a^{-4}+12z^{2}a^{-4}-8a^{-4}+6z^{3}a^{-3}-10za^{-3}+4a^{-3}z^{-1}$ (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   \ 0 1 2 3 4 5 6 7 8 χ 20                 1 1 18               2   -2 16             3 1   2 14           5 2     -3 12         3 3       0 10       5 5         0 8     3 3           0 6   1 5             4 4 2 3               -1 2 3                 3

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=2$ $i=4$ $r=0$ ${\mathbb {Z} }^{3}$ ${\mathbb {Z} }^{2}$ $r=1$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=2$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=3$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=4$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=5$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=6$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=7$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=8$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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