L11n329

L11n328

L11n330

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Link Presentations

[edit Notes on L11n329's Link Presentations]

Planar diagram presentation	X6172 X5,14,6,15 X3849 X15,2,16,3 X16,7,17,8 X13,18,14,19 X9,13,10,22 X11,21,12,20 X19,5,20,12 X21,11,22,10 X4,17,1,18
Gauss code	{1, 4, -3, -11}, {-2, -1, 5, 3, -7, 10, -8, 9}, {-6, 2, -4, -5, 11, 6, -9, 8, -10, 7}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {-t(1)t(3)^{2}t(2)^{3}+t(1)t(3)t(2)^{3}-t(1)t(3)^{3}t(2)^{2}+t(1)t(3)^{2}t(2)^{2}-t(2)^{2}+t(1)t(3)^{3}t(2)-t(3)t(2)+t(2)-t(3)^{2}+t(3)}{{\sqrt {t(1)}}t(2)^{3/2}t(3)^{3/2}}}$ (db)
Jones polynomial	$-q^{-7}+2q^{-6}-2q^{-5}+3q^{-4}-2q^{-3}+q^{2}+4q^{-2}-q-2q^{-1}+2$ (db)
Signature	-4 (db)
HOMFLY-PT polynomial	$-a^{8}+a^{6}z^{4}+4a^{6}z^{2}+2a^{6}-a^{4}z^{6}-4a^{4}z^{4}-2a^{4}z^{2}+a^{4}z^{-2}+a^{4}-a^{2}z^{6}-5a^{2}z^{4}-7a^{2}z^{2}-2a^{2}z^{-2}-5a^{2}+z^{4}+4z^{2}+z^{-2}+3$ (db)
Kauffman polynomial	$za^{9}+2z^{2}a^{8}-a^{8}+z^{5}a^{7}-2z^{3}a^{7}+za^{7}+2z^{6}a^{6}-7z^{4}a^{6}+5z^{2}a^{6}+2z^{7}a^{5}-8z^{5}a^{5}+8z^{3}a^{5}-3za^{5}+2z^{8}a^{4}-10z^{6}a^{4}+17z^{4}a^{4}-18z^{2}a^{4}-a^{4}z^{-2}+9a^{4}+z^{9}a^{3}-4z^{7}a^{3}+z^{5}a^{3}+8z^{3}a^{3}-8za^{3}+2a^{3}z^{-1}+3z^{8}a^{2}-19z^{6}a^{2}+40z^{4}a^{2}-35z^{2}a^{2}-2a^{2}z^{-2}+13a^{2}+z^{9}a-6z^{7}a+10z^{5}a-2z^{3}a-5za+2az^{-1}+z^{8}-7z^{6}+16z^{4}-14z^{2}-z^{-2}+6$ (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   \ -5 -4 -3 -2 -1 0 1 2 3 4 χ 5                   1 1 3                     0 1               2 1   1 -1           1 1       0 -3           4 2       2 -5       1 2 3         2 -7       2 1           1 -9   1 2 2             1 -11   2 2               0 -13   1                 1 -15 1                   -1

Integral Khovanov Homology (db, data source)

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

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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