L11n266

L11n265

L11n267

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Link Presentations

[edit Notes on L11n266's Link Presentations]

Planar diagram presentation	X6172 X3,11,4,10 X7,15,8,14 X13,5,14,8 X18,11,19,12 X20,15,21,16 X22,17,9,18 X16,21,17,22 X12,19,13,20 X2536 X9,1,10,4
Gauss code	{1, -10, -2, 11}, {10, -1, -3, 4}, {-11, 2, 5, -9, -4, 3, 6, -8, 7, -5, 9, -6, 8, -7}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {-uvw^{3}+uvw^{2}-uv-uw^{4}+2uw^{3}-2uw^{2}+uw-vw^{4}+2vw^{3}-2vw^{2}+vw+w^{5}-w^{3}+w^{2}}{{\sqrt {u}}{\sqrt {v}}w^{5/2}}}$ (db)
Jones polynomial	$-q^{-8}+2q^{-7}-5q^{-6}+5q^{-5}-5q^{-4}+6q^{-3}+q^{2}-3q^{-2}+3q^{-1}+1$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$-z^{4}a^{6}-3z^{2}a^{6}-2a^{6}z^{-2}-5a^{6}+z^{6}a^{4}+6z^{4}a^{4}+16z^{2}a^{4}+7a^{4}z^{-2}+18a^{4}-z^{6}a^{2}-8z^{4}a^{2}-19z^{2}a^{2}-8a^{2}z^{-2}-20a^{2}+z^{4}+5z^{2}+3z^{-2}+7$ (db)
Kauffman polynomial	$z^{5}a^{9}-3z^{3}a^{9}+3za^{9}-a^{9}z^{-1}+2z^{6}a^{8}-4z^{4}a^{8}+a^{8}+2z^{7}a^{7}-2z^{5}a^{7}-5z^{3}a^{7}+3za^{7}-a^{7}z^{-1}+z^{8}a^{6}-z^{4}a^{6}-6z^{2}a^{6}-2a^{6}z^{-2}+7a^{6}+3z^{7}a^{5}-9z^{5}a^{5}+19z^{3}a^{5}-21za^{5}+7a^{5}z^{-1}+z^{8}a^{4}-5z^{6}a^{4}+18z^{4}a^{4}-27z^{2}a^{4}-7a^{4}z^{-2}+22a^{4}+2z^{7}a^{3}-16z^{5}a^{3}+47z^{3}a^{3}-45za^{3}+15a^{3}z^{-1}+z^{8}a^{2}-11z^{6}a^{2}+36z^{4}a^{2}-45z^{2}a^{2}-8a^{2}z^{-2}+28a^{2}+z^{7}a-10z^{5}a+26z^{3}a-24za+8az^{-1}+z^{8}-8z^{6}+21z^{4}-24z^{2}-3z^{-2}+13$ (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 3 4 χ 5                       1 1 3                       1 1 1                   1     1 -1               4         4 -3             3 4 1       0 -5           4 1           3 -7         1 3 1           1 -9       4 4               0 -11     1 1                 0 -13   1 4                   -3 -15   1                     1 -17 1                       -1

Integral Khovanov Homology (db, data source)

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

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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