L11n414

L11n413

L11n415

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Link Presentations

[edit Notes on L11n414's Link Presentations]

Planar diagram presentation	X8192 X5,15,6,14 X10,3,11,4 X13,5,14,4 X2738 X6,9,1,10 X18,12,19,11 X12,18,7,17 X15,20,16,21 X19,22,20,13 X21,16,22,17
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 $u$, $v$, $w$, ...)	$-{\frac {-t(1)t(3)^{3}+t(1)t(2)t(3)^{3}-t(2)t(3)^{3}+t(1)^{2}t(3)^{2}+t(2)^{2}t(3)^{2}-t(1)^{2}t(3)-t(2)^{2}t(3)+t(1)t(2)^{2}+t(1)^{2}t(2)-t(1)t(2)}{t(1)t(2)t(3)^{3/2}}}$ (db)
Jones polynomial	$q^{3}-q^{2}+2q-1+2q^{-1}+q^{-3}+q^{-4}-q^{-5}+q^{-6}-q^{-7}$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$a^{6}\left(-z^{2}\right)-a^{6}z^{-2}-2a^{6}+a^{4}z^{4}+5a^{4}z^{2}+4a^{4}z^{-2}+8a^{4}-a^{2}z^{4}-5a^{2}z^{2}-5a^{2}z^{-2}+z^{2}a^{-2}-8a^{2}+2a^{-2}-z^{4}-3z^{2}+2z^{-2}$ (db)
Kauffman polynomial	$a^{7}z^{7}-6a^{7}z^{5}+10a^{7}z^{3}-5a^{7}z+a^{7}z^{-1}+a^{6}z^{8}-6a^{6}z^{6}+10a^{6}z^{4}-7a^{6}z^{2}-a^{6}z^{-2}+4a^{6}+2a^{5}z^{7}-14a^{5}z^{5}+28a^{5}z^{3}-21a^{5}z+5a^{5}z^{-1}+a^{4}z^{8}-8a^{4}z^{6}+20a^{4}z^{4}-25a^{4}z^{2}-4a^{4}z^{-2}+17a^{4}+2a^{3}z^{7}-16a^{3}z^{5}+37a^{3}z^{3}-33a^{3}z+9a^{3}z^{-1}+a^{2}z^{8}-8a^{2}z^{6}+z^{6}a^{-2}+22a^{2}z^{4}-5z^{4}a^{-2}-31a^{2}z^{2}+6z^{2}a^{-2}-5a^{2}z^{-2}+20a^{2}-2a^{-2}+2az^{7}+z^{7}a^{-1}-12az^{5}-4z^{5}a^{-1}+21az^{3}+2z^{3}a^{-1}-16az+za^{-1}+5az^{-1}+z^{8}-5z^{6}+7z^{4}-7z^{2}-2z^{-2}+6$ (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 3 4 χ 7                       1 1 5                     1 1 0 3                   1     1 1               1 1 1     1 -1             1 3 1       1 -3           1   1         2 -5         1 4 2           1 -7       1   1             2 -9       1 1               0 -11   1 1                   0 -13                         0 -15 1                       -1

Integral Khovanov Homology (db, data source)

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

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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