L11n401

L11n400

L11n402

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Link Presentations

[edit Notes on L11n401's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X22,16,19,15 X7,20,8,21 X19,8,20,9 X18,14,5,13 X14,12,15,11 X12,18,13,17 X16,22,17,21 X2536 X4,9,1,10
Gauss code	{1, -10, 2, -11}, {-5, 4, 9, -3}, {10, -1, -4, 5, 11, -2, 7, -8, 6, -7, 3, -9, 8, -6}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(v-1)(w-1)\left(uw^{2}-2uw+2u-2w^{2}+2w-1\right)}{{\sqrt {u}}{\sqrt {v}}w^{3/2}}}$ (db)
Jones polynomial	$-q^{5}+3q^{4}+3q^{-4}-7q^{3}-5q^{-3}+11q^{2}+11q^{-2}-13q-12q^{-1}+14$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$-z^{2}a^{-4}+2a^{4}z^{-2}+2a^{4}-a^{-4}+a^{2}z^{4}+2z^{4}a^{-2}-2a^{2}z^{2}+3z^{2}a^{-2}-5a^{2}z^{-2}-a^{-2}z^{-2}-7a^{2}+a^{-2}-z^{6}-2z^{4}+4z^{-2}+5$ (db)
Kauffman polynomial	$az^{9}+z^{9}a^{-1}+4a^{2}z^{8}+4z^{8}a^{-2}+8z^{8}+3a^{3}z^{7}+11az^{7}+13z^{7}a^{-1}+5z^{7}a^{-3}-8a^{2}z^{6}+3z^{6}a^{-4}-11z^{6}-6a^{3}z^{5}-32az^{5}-35z^{5}a^{-1}-8z^{5}a^{-3}+z^{5}a^{-5}+6a^{4}z^{4}+18a^{2}z^{4}-10z^{4}a^{-2}-5z^{4}a^{-4}+7z^{4}+14a^{3}z^{3}+43az^{3}+36z^{3}a^{-1}+5z^{3}a^{-3}-2z^{3}a^{-5}-13a^{4}z^{2}-26a^{2}z^{2}+6z^{2}a^{-2}+3z^{2}a^{-4}-10z^{2}-16a^{3}z-32az-20za^{-1}-3za^{-3}+za^{-5}+9a^{4}+18a^{2}-a^{-4}+11+5a^{3}z^{-1}+9az^{-1}+5a^{-1}z^{-1}+a^{-3}z^{-1}-2a^{4}z^{-2}-5a^{2}z^{-2}-a^{-2}z^{-2}-4z^{-2}$ (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 11                   1 -1 9                 2   2 7               5 1   -4 5             6 2     4 3           7 5       -2 1         7 6         1 -1       7 9           2 -3     4 5             -1 -5   1 7               6 -7 2 4                 -2 -9 3                   3

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-1$ $i=1$ $r=-4$ ${\mathbb {Z} }^{3}$ ${\mathbb {Z} }^{2}$ $r=-3$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-2$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=-1$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=0$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{7}$ $r=1$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=2$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=3$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=4$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=5$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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