L11n363

L11n362

L11n364

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Link Presentations

[edit Notes on L11n363's Link Presentations]

Planar diagram presentation	X6172 X12,7,13,8 X4,13,1,14 X5,18,6,19 X8493 X9,21,10,20 X19,11,20,10 X15,22,16,17 X17,16,18,5 X21,14,22,15 X2,12,3,11
Gauss code	{1, -11, 5, -3}, {-9, 4, -7, 6, -10, 8}, {-4, -1, 2, -5, -6, 7, 11, -2, 3, 10, -8, 9}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(t(1)-1)(t(3)-1)^{2}}{{\sqrt {t(1)}}t(3)}}$ (db)
Jones polynomial	$-q^{-8}+2q^{-7}-3q^{-6}+4q^{-5}-3q^{-4}+4q^{-3}+q^{2}-q^{-2}-q+q^{-1}+1$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$-z^{4}a^{6}-3z^{2}a^{6}-3a^{6}+z^{6}a^{4}+6z^{4}a^{4}+13z^{2}a^{4}+a^{4}z^{-2}+9a^{4}-z^{6}a^{2}-7z^{4}a^{2}-14z^{2}a^{2}-2a^{2}z^{-2}-10a^{2}+z^{4}+4z^{2}+z^{-2}+4$ (db)
Kauffman polynomial	$z^{5}a^{9}-3z^{3}a^{9}+za^{9}+2z^{6}a^{8}-6z^{4}a^{8}+3z^{2}a^{8}-a^{8}+2z^{7}a^{7}-6z^{5}a^{7}+3z^{3}a^{7}+z^{8}a^{6}-3z^{6}a^{6}+3z^{4}a^{6}-3z^{2}a^{6}+2a^{6}+2z^{7}a^{5}-9z^{5}a^{5}+14z^{3}a^{5}-6za^{5}+2z^{8}a^{4}-15z^{6}a^{4}+37z^{4}a^{4}-34z^{2}a^{4}-a^{4}z^{-2}+13a^{4}+z^{9}a^{3}-7z^{7}a^{3}+11z^{5}a^{3}+3z^{3}a^{3}-9za^{3}+2a^{3}z^{-1}+2z^{8}a^{2}-17z^{6}a^{2}+43z^{4}a^{2}-41z^{2}a^{2}-2a^{2}z^{-2}+15a^{2}+z^{9}a-7z^{7}a+13z^{5}a-5z^{3}a-4za+2az^{-1}+z^{8}-7z^{6}+15z^{4}-13z^{2}-z^{-2}+6$ (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   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 χ 5                       1 1 3                         0 1                   1 1   0 -1               3 1       2 -3             1 2 1       0 -5           4 2 1         3 -7         2 3             1 -9       2 2 1             1 -11     1 2                 1 -13   1 2                   -1 -15   1                     1 -17 1                       -1

Integral Khovanov Homology (db, data source)

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

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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