L11n171

L11n170

L11n172

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Link Presentations

[edit Notes on L11n171's Link Presentations]

Planar diagram presentation	X8192 X11,21,12,20 X10,4,11,3 X2,17,3,18 X14,5,15,6 X6718 X16,10,17,9 X13,19,14,18 X22,16,7,15 X19,13,20,12 X4,22,5,21
Gauss code	{1, -4, 3, -11, 5, -6}, {6, -1, 7, -3, -2, 10, -8, -5, 9, -7, 4, 8, -10, 2, 11, -9}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {t(1)^{2}t(2)^{4}-t(1)t(2)^{4}-3t(1)^{2}t(2)^{3}+6t(1)t(2)^{3}-t(2)^{3}+3t(1)^{2}t(2)^{2}-9t(1)t(2)^{2}+3t(2)^{2}-t(1)^{2}t(2)+6t(1)t(2)-3t(2)-t(1)+1}{t(1)t(2)^{2}}}$ (db)
Jones polynomial	$2q^{11/2}-5q^{9/2}+9q^{7/2}-12q^{5/2}+13q^{3/2}-14{\sqrt {q}}+{\frac {10}{\sqrt {q}}}-{\frac {8}{q^{3/2}}}+{\frac {4}{q^{5/2}}}-{\frac {1}{q^{7/2}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$za^{-5}+a^{-5}z^{-1}+z^{5}a^{-3}+z^{3}a^{-3}-2za^{-3}-2a^{-3}z^{-1}-z^{7}a^{-1}+az^{5}-4z^{5}a^{-1}+2az^{3}-5z^{3}a^{-1}+az-za^{-1}+az^{-1}$ (db)
Kauffman polynomial	$-2z^{9}a^{-1}-2z^{9}a^{-3}-9z^{8}a^{-2}-3z^{8}a^{-4}-6z^{8}-7az^{7}-6z^{7}a^{-1}-z^{7}a^{-5}-4a^{2}z^{6}+20z^{6}a^{-2}+6z^{6}a^{-4}+10z^{6}-a^{3}z^{5}+14az^{5}+18z^{5}a^{-1}-3z^{5}a^{-5}+6a^{2}z^{4}-22z^{4}a^{-2}-14z^{4}a^{-4}-3z^{4}a^{-6}-5z^{4}+a^{3}z^{3}-7az^{3}-12z^{3}a^{-1}+z^{3}a^{-3}+5z^{3}a^{-5}+12z^{2}a^{-2}+15z^{2}a^{-4}+5z^{2}a^{-6}+2z^{2}+2az-3za^{-3}-za^{-5}-3a^{-2}-5a^{-4}-2a^{-6}+1-az^{-1}+2a^{-3}z^{-1}+a^{-5}z^{-1}$ (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   \ -4 -3 -2 -1 0 1 2 3 4 5 χ 12                   2 -2 10                 3   3 8               6 2   -4 6             6 3     3 4           7 6       -1 2         7 6         1 0       4 8           4 -2     4 6             -2 -4   1 5               4 -6   3                 -3 -8 1                   1

Integral Khovanov Homology (db, data source)

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

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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