L11n303

L11n302

L11n304

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Link Presentations

[edit Notes on L11n303's Link Presentations]

Planar diagram presentation	X6172 X12,3,13,4 X13,21,14,20 X19,11,20,22 X15,5,16,10 X17,9,18,8 X7,17,8,16 X9,19,10,18 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 $u$, $v$, $w$, ...)	${\frac {-t(2)^{2}t(3)^{4}+t(1)t(2)^{2}t(3)^{3}-t(1)t(2)t(3)^{3}+t(2)t(3)^{3}-t(1)t(2)^{2}t(3)^{2}+2t(1)t(2)t(3)^{2}-2t(2)t(3)^{2}+t(3)^{2}-t(1)t(2)t(3)+t(2)t(3)-t(3)+t(1)}{{\sqrt {t(1)}}t(2)t(3)^{2}}}$ (db)
Jones polynomial	$-q^{8}+2q^{7}-3q^{6}+4q^{5}-4q^{4}+4q^{3}-2q^{2}+2q+1+q^{-2}$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$-z^{6}a^{-2}+z^{6}a^{-4}-8z^{4}a^{-2}+6z^{4}a^{-4}-z^{4}a^{-6}+z^{4}-19z^{2}a^{-2}+14z^{2}a^{-4}-3z^{2}a^{-6}+5z^{2}-17a^{-2}+13a^{-4}-3a^{-6}+7-5a^{-2}z^{-2}+4a^{-4}z^{-2}-a^{-6}z^{-2}+2z^{-2}$ (db)
Kauffman polynomial	$z^{8}a^{-2}+z^{8}a^{-4}+z^{8}a^{-6}+z^{8}+z^{7}a^{-1}+2z^{7}a^{-3}+3z^{7}a^{-5}+2z^{7}a^{-7}-10z^{6}a^{-2}-6z^{6}a^{-4}-2z^{6}a^{-6}+2z^{6}a^{-8}-8z^{6}-9z^{5}a^{-1}-16z^{5}a^{-3}-14z^{5}a^{-5}-6z^{5}a^{-7}+z^{5}a^{-9}+31z^{4}a^{-2}+15z^{4}a^{-4}-z^{4}a^{-6}-6z^{4}a^{-8}+21z^{4}+22z^{3}a^{-1}+41z^{3}a^{-3}+26z^{3}a^{-5}+4z^{3}a^{-7}-3z^{3}a^{-9}-39z^{2}a^{-2}-16z^{2}a^{-4}+3z^{2}a^{-6}+3z^{2}a^{-8}-23z^{2}-19za^{-1}-35za^{-3}-19za^{-5}-2za^{-7}+za^{-9}+22a^{-2}+13a^{-4}-a^{-8}+11+5a^{-1}z^{-1}+9a^{-3}z^{-1}+5a^{-5}z^{-1}+a^{-7}z^{-1}-5a^{-2}z^{-2}-4a^{-4}z^{-2}-a^{-6}z^{-2}-2z^{-2}$ (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   \ -4 -3 -2 -1 0 1 2 3 4 5 6 7 χ 17                       1 -1 15                     1   1 13                   2 1   -1 11                 2 1     1 9               2 2       0 7           1 3 2         0 5           2 4           2 3       1 2 1             0 1         3               3 -1     1                   1 -3 1                       1 -5 1                       1

Integral Khovanov Homology (db, data source)

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

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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