L11n297

L11n296

L11n298

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Link Presentations

[edit Notes on L11n297's Link Presentations]

Planar diagram presentation	X6172 X12,4,13,3 X15,20,16,21 X14,8,15,7 X21,10,22,5 X18,11,19,12 X9,17,10,16 X22,17,11,18 X8,19,9,20 X2536 X4,14,1,13
Gauss code	{1, -10, 2, -11}, {10, -1, 4, -9, -7, 5}, {6, -2, 11, -4, -3, 7, 8, -6, 9, 3, -5, -8}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {t(1)t(3)^{2}t(2)^{2}-t(3)^{2}t(2)^{2}-t(1)t(3)t(2)^{2}+2t(3)t(2)^{2}-t(2)^{2}-t(1)t(3)^{2}t(2)+2t(1)t(3)t(2)-2t(3)t(2)+t(2)+t(1)t(3)^{2}+t(1)-2t(1)t(3)+t(3)-1}{{\sqrt {t(1)}}t(2)t(3)}}$ (db)
Jones polynomial	$q^{2}-2q+5-4q^{-1}+7q^{-2}-5q^{-3}+5q^{-4}-4q^{-5}+2q^{-6}-q^{-7}$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$-z^{2}a^{6}-a^{6}z^{-2}-2a^{6}+2z^{4}a^{4}+7z^{2}a^{4}+4a^{4}z^{-2}+8a^{4}-z^{6}a^{2}-5z^{4}a^{2}-10z^{2}a^{2}-5a^{2}z^{-2}-10a^{2}+z^{4}+3z^{2}+2z^{-2}+4$ (db)
Kauffman polynomial	$a^{5}z^{9}+a^{3}z^{9}+2a^{6}z^{8}+5a^{4}z^{8}+3a^{2}z^{8}+a^{7}z^{7}+a^{3}z^{7}+2az^{7}-9a^{6}z^{6}-22a^{4}z^{6}-13a^{2}z^{6}-5a^{7}z^{5}-16a^{5}z^{5}-17a^{3}z^{5}-6az^{5}+11a^{6}z^{4}+30a^{4}z^{4}+24a^{2}z^{4}+5z^{4}+8a^{7}z^{3}+29a^{5}z^{3}+29a^{3}z^{3}+10az^{3}+2z^{3}a^{-1}-5a^{6}z^{2}-22a^{4}z^{2}-27a^{2}z^{2}+z^{2}a^{-2}-9z^{2}-5a^{7}z-18a^{5}z-24a^{3}z-11az+3a^{6}+12a^{4}+15a^{2}+7+a^{7}z^{-1}+5a^{5}z^{-1}+9a^{3}z^{-1}+5az^{-1}-a^{6}z^{-2}-4a^{4}z^{-2}-5a^{2}z^{-2}-2z^{-2}$ (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   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 χ 5                   1 1 3                 2 1 -1 1               3     3 -1             3 4     1 -3           4 1       3 -5         2 4         2 -7       3 3           0 -9     1 2             1 -11   1 3               -2 -13   1                 1 -15 1                   -1

Integral Khovanov Homology (db, data source)

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

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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