L11n59

L11n58

L11n60

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Link Presentations

[edit Notes on L11n59's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X14,8,15,7 X11,19,12,18 X19,5,20,22 X15,21,16,20 X21,17,22,16 X17,13,18,12 X8,14,9,13 X2536 X4,9,1,10
Gauss code	{1, -10, 2, -11}, {10, -1, 3, -9, 11, -2, -4, 8, 9, -3, -6, 7, -8, 4, -5, 6, -7, 5}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {t(2)^{5}-t(1)t(2)^{4}+t(2)^{4}+4t(1)t(2)^{3}-4t(2)^{3}-4t(1)t(2)^{2}+4t(2)^{2}+t(1)t(2)-t(2)+t(1)}{{\sqrt {t(1)}}t(2)^{5/2}}}$ (db)
Jones polynomial	$-6q^{9/2}+7q^{7/2}-6q^{5/2}-{\frac {1}{q^{5/2}}}+5q^{3/2}+q^{15/2}-3q^{13/2}+5q^{11/2}-4{\sqrt {q}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$-z^{5}a^{-1}+z^{5}a^{-3}+az^{3}-7z^{3}a^{-1}+3z^{3}a^{-3}-2z^{3}a^{-5}+4az-10za^{-1}+5za^{-3}-2za^{-5}+za^{-7}+3az^{-1}-5a^{-1}z^{-1}+2a^{-3}z^{-1}$ (db)
Kauffman polynomial	$z^{6}a^{-8}-3z^{4}a^{-8}+2z^{2}a^{-8}+3z^{7}a^{-7}-10z^{5}a^{-7}+7z^{3}a^{-7}-za^{-7}+3z^{8}a^{-6}-9z^{6}a^{-6}+5z^{4}a^{-6}-3z^{2}a^{-6}+a^{-6}+z^{9}a^{-5}+z^{7}a^{-5}-11z^{5}a^{-5}+8z^{3}a^{-5}-za^{-5}+4z^{8}a^{-4}-15z^{6}a^{-4}+19z^{4}a^{-4}-10z^{2}a^{-4}+z^{9}a^{-3}-3z^{7}a^{-3}+2z^{5}a^{-3}+3z^{3}a^{-3}-3za^{-3}+2a^{-3}z^{-1}+z^{8}a^{-2}-5z^{6}a^{-2}+7z^{4}a^{-2}+5z^{2}a^{-2}-5a^{-2}+az^{7}-7az^{5}-4z^{5}a^{-1}+14az^{3}+16z^{3}a^{-1}-11az-14za^{-1}+3az^{-1}+5a^{-1}z^{-1}-4z^{4}+10z^{2}-5$ (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   \ -4 -3 -2 -1 0 1 2 3 4 5 6 7 χ 16                       1 -1 14                     2   2 12                   3 1   -2 10                 3 2     1 8               4 3       -1 6           1 3 3         -1 4           3 4           1 2       1 3 3             -1 0         4               4 -2     1 1                 0 -4 1                       1 -6 1                       1

Integral Khovanov Homology (db, data source)

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

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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