L11n121

L11n120

L11n122

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Link Presentations

[edit Notes on L11n121's Link Presentations]

Planar diagram presentation	X6172 X14,3,15,4 X16,10,17,9 X11,21,12,20 X21,9,22,8 X7,19,8,18 X19,13,20,12 X10,16,11,15 X17,5,18,22 X2536 X4,13,1,14
Gauss code	{1, -10, 2, -11}, {10, -1, -6, 5, 3, -8, -4, 7, 11, -2, 8, -3, -9, 6, -7, 4, -5, 9}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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