L10n42

L10n41

L10n43

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Link Presentations

[edit Notes on L10n42's Link Presentations]

Planar diagram presentation	X8192 X10,4,11,3 X20,10,7,9 X2738 X15,5,16,4 X5,13,6,12 X16,12,17,11 X6,18,1,17 X19,15,20,14 X13,19,14,18
Gauss code	{1, -4, 2, 5, -6, -8}, {4, -1, 3, -2, 7, 6, -10, 9, -5, -7, 8, 10, -9, -3}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {u^{2}v^{4}-u^{2}v^{3}-uv^{4}+uv^{2}-u-v+1}{uv^{2}}}$ (db)
Jones polynomial	$-2q^{9/2}+q^{7/2}-2q^{5/2}+q^{3/2}+q^{15/2}-q^{13/2}+q^{11/2}-{\sqrt {q}}$ (db)
Signature	5 (db)
HOMFLY-PT polynomial	$-za^{-9}+z^{5}a^{-7}+5z^{3}a^{-7}+5za^{-7}+a^{-7}z^{-1}-z^{7}a^{-5}-6z^{5}a^{-5}-11z^{3}a^{-5}-9za^{-5}-3a^{-5}z^{-1}+z^{5}a^{-3}+5z^{3}a^{-3}+6za^{-3}+2a^{-3}z^{-1}$ (db)
Kauffman polynomial	$-z^{8}a^{-4}-z^{8}a^{-6}-z^{7}a^{-3}-3z^{7}a^{-5}-2z^{7}a^{-7}+5z^{6}a^{-4}+4z^{6}a^{-6}-z^{6}a^{-8}+6z^{5}a^{-3}+17z^{5}a^{-5}+11z^{5}a^{-7}-5z^{4}a^{-4}+5z^{4}a^{-8}-11z^{3}a^{-3}-27z^{3}a^{-5}-16z^{3}a^{-7}-2z^{2}a^{-4}-7z^{2}a^{-6}-5z^{2}a^{-8}+8za^{-3}+15za^{-5}+8za^{-7}+za^{-9}+3a^{-4}+3a^{-6}+a^{-8}-2a^{-3}z^{-1}-3a^{-5}z^{-1}-a^{-7}z^{-1}$ (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 χ 16               1 -1 14             1 1 0 12           1 1   0 10         1 1 1   1 8       1 2       1 6     1 1 1       1 4   1 2           1 2                 0 0 1               1

Integral Khovanov Homology (db, data source)

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

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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