L11n289

L11n288

L11n290

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Link Presentations

[edit Notes on L11n289's Link Presentations]

Planar diagram presentation	X6172 X5,12,6,13 X3849 X2,14,3,13 X14,7,15,8 X11,18,12,19 X9,21,10,20 X19,5,20,10 X15,1,16,4 X17,22,18,11 X21,16,22,17
Gauss code	{1, -4, -3, 9}, {-2, -1, 5, 3, -7, 8}, {-6, 2, 4, -5, -9, 11, -10, 6, -8, 7, -11, 10}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {uvw^{4}-2uvw^{3}+2uvw^{2}-2uvw-uw^{4}+uw^{3}-uw^{2}+uw-v^{2}w^{3}+v^{2}w^{2}-v^{2}w+v^{2}+2vw^{3}-2vw^{2}+2vw-v}{{\sqrt {u}}vw^{2}}}$ (db)
Jones polynomial	$2q-2+6q^{-1}-6q^{-2}+8q^{-3}-7q^{-4}+6q^{-5}-4q^{-6}+2q^{-7}-q^{-8}$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$-z^{4}a^{6}-3z^{2}a^{6}-a^{6}z^{-2}-3a^{6}+z^{6}a^{4}+5z^{4}a^{4}+11z^{2}a^{4}+4a^{4}z^{-2}+11a^{4}-3z^{4}a^{2}-11z^{2}a^{2}-5a^{2}z^{-2}-13a^{2}+2z^{2}+2z^{-2}+5$ (db)
Kauffman polynomial	$z^{5}a^{9}-3z^{3}a^{9}+za^{9}+2z^{6}a^{8}-5z^{4}a^{8}+z^{2}a^{8}+3z^{7}a^{7}-9z^{5}a^{7}+8z^{3}a^{7}-4za^{7}+a^{7}z^{-1}+3z^{8}a^{6}-11z^{6}a^{6}+17z^{4}a^{6}-12z^{2}a^{6}-a^{6}z^{-2}+5a^{6}+z^{9}a^{5}+z^{7}a^{5}-14z^{5}a^{5}+31z^{3}a^{5}-21za^{5}+5a^{5}z^{-1}+5z^{8}a^{4}-23z^{6}a^{4}+47z^{4}a^{4}-40z^{2}a^{4}-4a^{4}z^{-2}+18a^{4}+z^{9}a^{3}-z^{7}a^{3}-7z^{5}a^{3}+28z^{3}a^{3}-29za^{3}+9a^{3}z^{-1}+2z^{8}a^{2}-10z^{6}a^{2}+28z^{4}a^{2}-37z^{2}a^{2}-5a^{2}z^{-2}+21a^{2}+z^{7}a-3z^{5}a+8z^{3}a-13za+5az^{-1}+3z^{4}-10z^{2}-2z^{-2}+9$ (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 3                   2 2 1                 1 1 0 -1               5 1   4 -3             3 3     0 -5           5 3       2 -7         3 4         1 -9       3 4           -1 -11     1 3             2 -13   1 3               -2 -15   1                 1 -17 1                   -1

Integral Khovanov Homology (db, data source)

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

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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