L11n135

L11n134

L11n136

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Link Presentations

[edit Notes on L11n135's Link Presentations]

Planar diagram presentation	X8192 X11,19,12,18 X3,10,4,11 X17,3,18,2 X12,5,13,6 X6718 X16,10,17,9 X20,14,21,13 X22,16,7,15 X4,20,5,19 X14,22,15,21
Gauss code	{1, 4, -3, -10, 5, -6}, {6, -1, 7, 3, -2, -5, 8, -11, 9, -7, -4, 2, 10, -8, 11, -9}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {t(1)^{2}t(2)^{4}-t(1)^{2}t(2)^{3}+t(1)t(2)^{3}-t(2)^{3}+t(1)^{2}t(2)^{2}-t(1)t(2)^{2}+t(2)^{2}-t(1)^{2}t(2)+t(1)t(2)-t(2)+1}{t(1)t(2)^{2}}}$ (db)
Jones polynomial	$-3q^{9/2}+3q^{7/2}-4q^{5/2}+3q^{3/2}-{\frac {1}{q^{3/2}}}+q^{15/2}-q^{13/2}+2q^{11/2}-3{\sqrt {q}}+{\frac {1}{\sqrt {q}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$z^{5}a^{-5}+5z^{3}a^{-5}+7za^{-5}+2a^{-5}z^{-1}-z^{7}a^{-3}-7z^{5}a^{-3}-17z^{3}a^{-3}-16za^{-3}-5a^{-3}z^{-1}+z^{5}a^{-1}+5z^{3}a^{-1}+8za^{-1}+3a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-z^{9}a^{-3}-z^{9}a^{-5}-z^{8}a^{-2}-2z^{8}a^{-4}-z^{8}a^{-6}+7z^{7}a^{-3}+6z^{7}a^{-5}-z^{7}a^{-7}+6z^{6}a^{-2}+11z^{6}a^{-4}+4z^{6}a^{-6}-z^{6}a^{-8}-2z^{5}a^{-1}-21z^{5}a^{-3}-15z^{5}a^{-5}+4z^{5}a^{-7}-15z^{4}a^{-2}-23z^{4}a^{-4}-4z^{4}a^{-6}+5z^{4}a^{-8}-z^{4}-az^{3}+6z^{3}a^{-1}+29z^{3}a^{-3}+19z^{3}a^{-5}-3z^{3}a^{-7}+13z^{2}a^{-2}+20z^{2}a^{-4}+2z^{2}a^{-6}-6z^{2}a^{-8}+z^{2}+2az-9za^{-1}-20za^{-3}-9za^{-5}-5a^{-2}-5a^{-4}+a^{-8}+3a^{-1}z^{-1}+5a^{-3}z^{-1}+2a^{-5}z^{-1}$ (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 7 χ 16                   1 -1 14                     0 12               2 1   -1 10             1       1 8           2 2       0 6         2 1         1 4       1 2           1 2     2 2             0 0     2               2 -2 1 1                 0 -4 1                   1

Integral Khovanov Homology (db, data source)

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

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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