L11n263

L11n262

L11n264

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Link Presentations

[edit Notes on L11n263's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X14,7,15,8 X8,13,5,14 X11,18,12,19 X22,20,9,19 X20,16,21,15 X16,22,17,21 X17,12,18,13 X2536 X4,9,1,10
Gauss code	{1, -10, 2, -11}, {10, -1, 3, -4}, {11, -2, -5, 9, 4, -3, 7, -8, -9, 5, 6, -7, 8, -6}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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