L11n323

L11n322

L11n324

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Link Presentations

[edit Notes on L11n323's Link Presentations]

Planar diagram presentation	X6172 X5,14,6,15 X3849 X15,2,16,3 X16,7,17,8 X9,18,10,19 X11,21,12,20 X19,22,20,13 X13,12,14,5 X4,17,1,18 X21,11,22,10
Gauss code	{1, 4, -3, -10}, {-2, -1, 5, 3, -6, 11, -7, 9}, {-9, 2, -4, -5, 10, 6, -8, 7, -11, 8}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {t(1)t(3)^{3}t(2)^{3}-t(1)t(3)^{2}t(2)^{3}-t(1)t(3)^{3}t(2)^{2}+t(2)+t(3)-1}{{\sqrt {t(1)}}t(2)^{3/2}t(3)^{3/2}}}$ (db)
Jones polynomial	$-q^{-10}+q^{-9}+2q^{-7}-q^{-6}+2q^{-5}-q^{-4}+2q^{-3}-q^{-2}+q^{-1}$ (db)
Signature	-6 (db)
HOMFLY-PT polynomial	$a^{10}\left(-z^{2}\right)-2a^{10}+a^{8}z^{6}+6a^{8}z^{4}+10a^{8}z^{2}+a^{8}z^{-2}+6a^{8}-a^{6}z^{8}-7a^{6}z^{6}-16a^{6}z^{4}-16a^{6}z^{2}-2a^{6}z^{-2}-9a^{6}+a^{4}z^{6}+6a^{4}z^{4}+10a^{4}z^{2}+a^{4}z^{-2}+5a^{4}$ (db)
Kauffman polynomial	$za^{13}+z^{2}a^{12}-a^{12}+za^{11}+z^{4}a^{10}-3z^{2}a^{10}+z^{7}a^{9}-5z^{5}a^{9}+6z^{3}a^{9}-3za^{9}+2z^{8}a^{8}-13z^{6}a^{8}+27z^{4}a^{8}-24z^{2}a^{8}-a^{8}z^{-2}+9a^{8}+z^{9}a^{7}-5z^{7}a^{7}+4z^{5}a^{7}+6z^{3}a^{7}-8za^{7}+2a^{7}z^{-1}+3z^{8}a^{6}-20z^{6}a^{6}+42z^{4}a^{6}-35z^{2}a^{6}-2a^{6}z^{-2}+13a^{6}+z^{9}a^{5}-6z^{7}a^{5}+9z^{5}a^{5}-5za^{5}+2a^{5}z^{-1}+z^{8}a^{4}-7z^{6}a^{4}+16z^{4}a^{4}-15z^{2}a^{4}-a^{4}z^{-2}+6a^{4}$ (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 2 χ -1                   1 1 -3                     0 -5               2 1   1 -7           1 1 1     1 -9           2 1       1 -11       2 2 1         1 -13     1 4 2           1 -15   1 1 2             2 -17   2 1               1 -19 1 1                 0 -21 1                   -1

Integral Khovanov Homology (db, data source)

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

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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