L11n413

L11n412

L11n414

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Link Presentations

[edit Notes on L11n413's Link Presentations]

Planar diagram presentation	X8192 X5,15,6,14 X10,3,11,4 X13,5,14,4 X2738 X6,9,1,10 X11,18,12,19 X17,12,18,7 X20,16,21,15 X22,20,13,19 X16,22,17,21
Gauss code	{1, -5, 3, 4, -2, -6}, {5, -1, 6, -3, -7, 8}, {-4, 2, 9, -11, -8, 7, 10, -9, 11, -10}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {t(1)^{2}t(3)^{3}-t(1)t(2)^{2}t(3)^{2}-t(1)t(3)^{2}-t(1)^{2}t(2)t(3)^{2}+2t(1)t(2)t(3)^{2}-t(2)t(3)^{2}+t(1)t(2)^{2}t(3)+t(1)t(3)+t(1)^{2}t(2)t(3)-2t(1)t(2)t(3)+t(2)t(3)-t(2)^{2}}{t(1)t(2)t(3)^{3/2}}}$ (db)
Jones polynomial	$-q^{5}+2q^{4}-3q^{3}+4q^{2}-4q+4-2q^{-1}+2q^{-2}+q^{-3}+q^{-5}$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$z^{2}a^{4}+2a^{4}z^{-2}+3a^{4}-z^{4}a^{2}-6z^{2}a^{2}-5a^{2}z^{-2}-9a^{2}+z^{4}+3z^{2}+4z^{-2}+6+z^{4}a^{-2}+2z^{2}a^{-2}-a^{-2}z^{-2}+a^{-2}-z^{2}a^{-4}-a^{-4}$ (db)
Kauffman polynomial	$a^{4}z^{8}+a^{2}z^{8}+z^{8}a^{-2}+z^{8}+a^{3}z^{7}+2az^{7}+3z^{7}a^{-1}+2z^{7}a^{-3}-8a^{4}z^{6}-10a^{2}z^{6}-2z^{6}a^{-2}+2z^{6}a^{-4}-6z^{6}-9a^{3}z^{5}-16az^{5}-14z^{5}a^{-1}-6z^{5}a^{-3}+z^{5}a^{-5}+21a^{4}z^{4}+31a^{2}z^{4}-z^{4}a^{-2}-6z^{4}a^{-4}+15z^{4}+22a^{3}z^{3}+41az^{3}+26z^{3}a^{-1}+4z^{3}a^{-3}-3z^{3}a^{-5}-23a^{4}z^{2}-39a^{2}z^{2}+3z^{2}a^{-2}+3z^{2}a^{-4}-16z^{2}-19a^{3}z-35az-19za^{-1}-2za^{-3}+za^{-5}+11a^{4}+22a^{2}-a^{-4}+13+5a^{3}z^{-1}+9az^{-1}+5a^{-1}z^{-1}+a^{-3}z^{-1}-2a^{4}z^{-2}-5a^{2}z^{-2}-a^{-2}z^{-2}-4z^{-2}$ (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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 11                       1 -1 9                     1   1 7                   2 1   -1 5                 2 1     1 3               2 2       0 1           1 3 2         0 -1           2 4           2 -3       1 2 1             0 -5         3               3 -7     1                   1 -9 1                       1 -11 1                       1

Integral Khovanov Homology (db, data source)

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

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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