L11n304

L11n303

L11n305

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Link Presentations

[edit Notes on L11n304's Link Presentations]

Planar diagram presentation	X6172 X12,3,13,4 X20,13,21,14 X22,19,11,20 X15,5,16,10 X17,9,18,8 X7,17,8,16 X9,19,10,18 X14,21,15,22 X2536 X4,11,1,12
Gauss code	{1, -10, 2, -11}, {10, -1, -7, 6, -8, 5}, {11, -2, 3, -9, -5, 7, -6, 8, 4, -3, 9, -4}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {-uv^{2}w^{3}+uv^{2}w^{2}+uvw^{3}-2uvw^{2}+uvw+2uw^{2}-2uw+u-v^{2}w^{4}+2v^{2}w^{3}-2v^{2}w^{2}-vw^{3}+2vw^{2}-vw-w^{2}+w}{{\sqrt {u}}vw^{2}}}$ (db)
Jones polynomial	$-q^{-7}+2q^{-6}-4q^{-5}+6q^{-4}-6q^{-3}+2q^{2}+8q^{-2}-3q-6q^{-1}+6$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$-z^{2}a^{6}-a^{6}z^{-2}-3a^{6}+3z^{4}a^{4}+12z^{2}a^{4}+4a^{4}z^{-2}+13a^{4}-2z^{6}a^{2}-11z^{4}a^{2}-21z^{2}a^{2}-5a^{2}z^{-2}-17a^{2}+2z^{4}+7z^{2}+2z^{-2}+7$ (db)
Kauffman polynomial	$a^{7}z^{7}-5a^{7}z^{5}+8a^{7}z^{3}-5a^{7}z+a^{7}z^{-1}+2a^{6}z^{8}-9a^{6}z^{6}+12a^{6}z^{4}-7a^{6}z^{2}-a^{6}z^{-2}+4a^{6}+a^{5}z^{9}+a^{5}z^{7}-20a^{5}z^{5}+35a^{5}z^{3}-21a^{5}z+5a^{5}z^{-1}+6a^{4}z^{8}-26a^{4}z^{6}+38a^{4}z^{4}-32a^{4}z^{2}-4a^{4}z^{-2}+17a^{4}+a^{3}z^{9}+4a^{3}z^{7}-28a^{3}z^{5}+44a^{3}z^{3}-33a^{3}z+9a^{3}z^{-1}+4a^{2}z^{8}-15a^{2}z^{6}+26a^{2}z^{4}-32a^{2}z^{2}+3z^{2}a^{-2}-5a^{2}z^{-2}+20a^{2}-2a^{-2}+4az^{7}-12az^{5}+z^{5}a^{-1}+18az^{3}+z^{3}a^{-1}-16az+za^{-1}+5az^{-1}+2z^{6}-4z^{2}-2z^{-2}+6$ (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 5                   2 2 3                 3 2 -1 1               3     3 -1             4 4     0 -3           4 2       2 -5         2 4         2 -7       4 4           0 -9     1 3             2 -11   1 3               -2 -13   1                 1 -15 1                   -1

Integral Khovanov Homology (db, data source)

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

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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