L11n305

L11n304

L11n306

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Link Presentations

[edit Notes on L11n305's Link Presentations]

Planar diagram presentation	X6172 X12,3,13,4 X13,21,14,20 X19,11,20,22 X10,15,5,16 X8,17,9,18 X16,7,17,8 X18,9,19,10 X21,15,22,14 X2536 X4,11,1,12
Gauss code	{1, -10, 2, -11}, {10, -1, 7, -6, 8, -5}, {11, -2, -3, 9, 5, -7, 6, -8, -4, 3, -9, 4}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {t(2)^{2}t(3)^{4}+t(1)t(2)t(3)^{4}-t(2)t(3)^{4}-t(2)^{2}t(3)^{3}+t(1)t(3)^{3}-2t(1)t(2)t(3)^{3}+2t(2)t(3)^{3}-t(3)^{3}+t(2)^{2}t(3)^{2}-t(1)t(3)^{2}+2t(1)t(2)t(3)^{2}-2t(2)t(3)^{2}+t(1)t(2)^{2}t(3)-t(2)^{2}t(3)+t(1)t(3)-2t(1)t(2)t(3)+2t(2)t(3)-t(1)+t(1)t(2)-t(2)}{{\sqrt {t(1)}}t(2)t(3)^{2}}}$ (db)
Jones polynomial	$-2+4q^{-1}-6q^{-2}+9q^{-3}-8q^{-4}+9q^{-5}-6q^{-6}+5q^{-7}-2q^{-8}+q^{-9}$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$z^{2}a^{8}+2a^{8}z^{-2}+3a^{8}-3z^{4}a^{6}-12z^{2}a^{6}-5a^{6}z^{-2}-15a^{6}+2z^{6}a^{4}+11z^{4}a^{4}+22z^{2}a^{4}+4a^{4}z^{-2}+17a^{4}-2z^{4}a^{2}-6z^{2}a^{2}-a^{2}z^{-2}-5a^{2}$ (db)
Kauffman polynomial	$z^{6}a^{10}-4z^{4}a^{10}+5z^{2}a^{10}-2a^{10}+2z^{7}a^{9}-6z^{5}a^{9}+3z^{3}a^{9}+za^{9}+2z^{8}a^{8}-4z^{6}a^{8}-4z^{2}a^{8}-2a^{8}z^{-2}+6a^{8}+z^{9}a^{7}+2z^{7}a^{7}-14z^{5}a^{7}+19z^{3}a^{7}-16za^{7}+5a^{7}z^{-1}+6z^{8}a^{6}-22z^{6}a^{6}+36z^{4}a^{6}-36z^{2}a^{6}-5a^{6}z^{-2}+20a^{6}+z^{9}a^{5}+4z^{7}a^{5}-23z^{5}a^{5}+43z^{3}a^{5}-33za^{5}+9a^{5}z^{-1}+4z^{8}a^{4}-16z^{6}a^{4}+34z^{4}a^{4}-32z^{2}a^{4}-4a^{4}z^{-2}+17a^{4}+4z^{7}a^{3}-15z^{5}a^{3}+30z^{3}a^{3}-21za^{3}+5a^{3}z^{-1}+z^{6}a^{2}+2z^{4}a^{2}-5z^{2}a^{2}-a^{2}z^{-2}+4a^{2}+3z^{3}a-5za+az^{-1}$ (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   \ -8 -7 -6 -5 -4 -3 -2 -1 0 1 χ 1                   2 -2 -1                 2   2 -3               5 3   -2 -5             4 1     3 -7           4 5       1 -9         5 4         1 -11       2 5           3 -13     3 4             -1 -15   1 4               3 -17   1                 -1 -19 1                   1

Integral Khovanov Homology (db, data source)

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

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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