L11n397

L11n396

L11n398

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Link Presentations

[edit Notes on L11n397's Link Presentations]

Planar diagram presentation	X6172 X10,4,11,3 X12,8,13,7 X16,10,17,9 X17,22,18,19 X13,20,14,21 X19,14,20,15 X21,18,22,5 X8,16,9,15 X2536 X4,12,1,11
Gauss code	{1, -10, 2, -11}, {-7, 6, -8, 5}, {10, -1, 3, -9, 4, -2, 11, -3, -6, 7, 9, -4, -5, 8}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(u-1)(w-1)^{2}(w+1)\left(v+w^{2}\right)}{{\sqrt {u}}{\sqrt {v}}w^{5/2}}}$ (db)
Jones polynomial	$-q^{5}+2q^{4}-2q^{3}+q^{2}-q+1+2q^{-1}-q^{-2}+3q^{-3}-q^{-4}+q^{-5}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$z^{6}-2a^{2}z^{4}+6z^{4}+a^{4}z^{2}-9a^{2}z^{2}-2z^{2}a^{-2}-z^{2}a^{-4}+11z^{2}+3a^{4}-11a^{2}-3a^{-2}+11+2a^{4}z^{-2}-5a^{2}z^{-2}-a^{-2}z^{-2}+4z^{-2}$ (db)
Kauffman polynomial	$a^{3}z^{9}+az^{9}+a^{4}z^{8}+4a^{2}z^{8}+3z^{8}-5a^{3}z^{7}-4az^{7}+2z^{7}a^{-1}+z^{7}a^{-3}-7a^{4}z^{6}-27a^{2}z^{6}+2z^{6}a^{-4}-22z^{6}+3a^{3}z^{5}-7az^{5}-15z^{5}a^{-1}-4z^{5}a^{-3}+z^{5}a^{-5}+17a^{4}z^{4}+56a^{2}z^{4}-7z^{4}a^{-4}+46z^{4}+11a^{3}z^{3}+32az^{3}+27z^{3}a^{-1}+3z^{3}a^{-3}-3z^{3}a^{-5}-19a^{4}z^{2}-49a^{2}z^{2}-5z^{2}a^{-2}+3z^{2}a^{-4}-38z^{2}-15a^{3}z-29az-17za^{-1}-2za^{-3}+za^{-5}+10a^{4}+22a^{2}+4a^{-2}+17+5a^{3}z^{-1}+9az^{-1}+5a^{-1}z^{-1}+a^{-3}z^{-1}-2a^{4}z^{-2}-5a^{2}z^{-2}-a^{-2}z^{-2}-4z^{-2}$ (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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 11                       1 -1 9                     1   1 7                   1 1   0 5               2 2 1     -1 3             1 2 1       0 1           2 4 2         0 -1         1 2 4           3 -3       1 2               1 -5     2 1 1               2 -7   1 3                   2 -9                         0 -11 1                       1

Integral Khovanov Homology (db, data source)

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

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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