L11n352

L11n351

L11n353

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Link Presentations

[edit Notes on L11n352's Link Presentations]

Planar diagram presentation	X6172 X14,3,15,4 X11,20,12,21 X7,18,8,19 X9,13,10,22 X21,17,22,16 X17,8,18,9 X15,11,16,10 X19,12,20,5 X2536 X4,13,1,14
Gauss code	{1, -10, 2, -11}, {10, -1, -4, 7, -5, 8, -3, 9}, {11, -2, -8, 6, -7, 4, -9, 3, -6, 5}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {uv^{2}w^{3}-2uv^{2}w^{2}+2uv^{2}w-uvw^{3}+2uvw^{2}-uvw+uw^{2}-uw+v^{3}w^{2}-v^{3}w+v^{2}w^{2}-2v^{2}w+v^{2}-2vw^{2}+2vw-v}{{\sqrt {u}}v^{3/2}w^{3/2}}}$ (db)
Jones polynomial	$q^{-8}-q^{-7}+q^{-6}+2q^{-5}-3q^{-4}+6q^{-3}-q^{2}-5q^{-2}+3q+6q^{-1}-5$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$a^{8}z^{-2}+a^{8}-2a^{6}z^{2}-2a^{6}z^{-2}-5a^{6}+2a^{4}z^{2}+a^{4}z^{-2}+3a^{4}+a^{2}z^{6}+4a^{2}z^{4}+5a^{2}z^{2}+2a^{2}-z^{4}-2z^{2}-1$ (db)
Kauffman polynomial	$a^{7}z^{9}+a^{5}z^{9}+a^{8}z^{8}+2a^{6}z^{8}+2a^{4}z^{8}+a^{2}z^{8}-7a^{7}z^{7}-7a^{5}z^{7}+3a^{3}z^{7}+3az^{7}-7a^{8}z^{6}-18a^{6}z^{6}-12a^{4}z^{6}+2a^{2}z^{6}+3z^{6}+12a^{7}z^{5}+11a^{5}z^{5}-7a^{3}z^{5}-5az^{5}+z^{5}a^{-1}+15a^{8}z^{4}+45a^{6}z^{4}+28a^{4}z^{4}-9a^{2}z^{4}-7z^{4}-3a^{7}z^{3}+5a^{5}z^{3}+7a^{3}z^{3}-3az^{3}-2z^{3}a^{-1}-14a^{8}z^{2}-38a^{6}z^{2}-23a^{4}z^{2}+4a^{2}z^{2}+3z^{2}-5a^{7}z-8a^{5}z-3a^{3}z+az+za^{-1}+6a^{8}+13a^{6}+9a^{4}-1+2a^{7}z^{-1}+2a^{5}z^{-1}-a^{8}z^{-2}-2a^{6}z^{-2}-a^{4}z^{-2}$ (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   \ -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 χ 5                       1 -1 3                     2   2 1                   3 1   -2 -1                 3 2     1 -3             1 4 4       1 -5             3 2         1 -7         1 2 4           3 -9       1 3 3             -1 -11       1 4               3 -13   1 1                   0 -15                         0 -17 1                       1

Integral Khovanov Homology (db, data source)

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

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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