L11n435

L11n434

L11n436

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Link Presentations

[edit Notes on L11n435's Link Presentations]

Planar diagram presentation	X8192 X14,4,15,3 X12,14,7,13 X2738 X22,10,13,9 X6,22,1,21 X20,16,21,15 X16,5,17,6 X11,19,12,18 X17,11,18,10 X4,19,5,20
Gauss code	{1, -4, 2, -11, 8, -6}, {4, -1, 5, 10, -9, -3}, {3, -2, 7, -8, -10, 9, 11, -7, 6, -5}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {-t(1)^{2}t(3)^{3}+t(1)t(3)^{3}+t(1)^{2}t(2)t(3)^{3}-2t(1)t(2)t(3)^{3}+t(2)t(3)^{3}+2t(1)^{2}t(3)^{2}-2t(1)t(2)^{2}t(3)^{2}+t(2)^{2}t(3)^{2}-2t(1)t(3)^{2}-2t(1)^{2}t(2)t(3)^{2}+5t(1)t(2)t(3)^{2}-2t(2)t(3)^{2}-t(1)^{2}t(3)+2t(1)t(2)^{2}t(3)-2t(2)^{2}t(3)+2t(1)t(3)+2t(1)^{2}t(2)t(3)-5t(1)t(2)t(3)+2t(2)t(3)-t(1)t(2)^{2}+t(2)^{2}-t(1)^{2}t(2)+2t(1)t(2)-t(2)}{t(1)t(2)t(3)^{3/2}}}$ (db)
Jones polynomial	$-q^{8}+4q^{7}-8q^{6}+12q^{5}-14q^{4}+16q^{3}-13q^{2}+11q+3q^{-1}-6$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$z^{6}a^{-4}-4z^{4}a^{-2}+3z^{4}a^{-4}-z^{4}a^{-6}-10z^{2}a^{-2}+6z^{2}a^{-4}-z^{2}a^{-6}+3z^{2}-8a^{-2}+5a^{-4}-a^{-6}+4-2a^{-2}z^{-2}+a^{-4}z^{-2}+z^{-2}$ (db)
Kauffman polynomial	$z^{5}a^{-9}-z^{3}a^{-9}+4z^{6}a^{-8}-6z^{4}a^{-8}+z^{2}a^{-8}+7z^{7}a^{-7}-12z^{5}a^{-7}+4z^{3}a^{-7}-za^{-7}+7z^{8}a^{-6}-12z^{6}a^{-6}+7z^{4}a^{-6}-4z^{2}a^{-6}+a^{-6}+3z^{9}a^{-5}+4z^{7}a^{-5}-17z^{5}a^{-5}+14z^{3}a^{-5}-3za^{-5}+13z^{8}a^{-4}-34z^{6}a^{-4}+41z^{4}a^{-4}-24z^{2}a^{-4}-a^{-4}z^{-2}+8a^{-4}+3z^{9}a^{-3}-7z^{5}a^{-3}+12z^{3}a^{-3}-8za^{-3}+2a^{-3}z^{-1}+6z^{8}a^{-2}-18z^{6}a^{-2}+34z^{4}a^{-2}-31z^{2}a^{-2}-2a^{-2}z^{-2}+12a^{-2}+3z^{7}a^{-1}-3z^{5}a^{-1}+3z^{3}a^{-1}-6za^{-1}+2a^{-1}z^{-1}+6z^{4}-12z^{2}-z^{-2}+6$ (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   \ -2 -1 0 1 2 3 4 5 6 7 χ 17                   1 -1 15                 3   3 13               5 1   -4 11             7 3     4 9           8 6       -2 7         8 6         2 5       6 9           3 3     5 7             -2 1   2 7               5 -1 1 4                 -3 -3 3                   3

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=1$ $i=3$ $r=-2$ ${\mathbb {Z} }^{3}$ ${\mathbb {Z} }$ $r=-1$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=0$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{5}$ $r=1$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=2$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{8}$ $r=3$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=4$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{7}$ $r=5$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=6$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=7$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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