L11n348

L11n347

L11n349

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Link Presentations

[edit Notes on L11n348's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {uv^{2}w^{3}-2uv^{2}w^{2}-uvw^{3}+uvw+uw^{2}-uw+v^{3}w^{2}-v^{3}w-v^{2}w^{2}+v^{2}+2vw-v}{{\sqrt {u}}v^{3/2}w^{3/2}}}$ (db)
Jones polynomial	$1-2q^{-1}+3q^{-2}-2q^{-3}+2q^{-4}+q^{-6}+2q^{-7}-2q^{-8}+2q^{-9}-q^{-10}$ (db)
Signature	-4 (db)
HOMFLY-PT polynomial	$-a^{10}+2z^{2}a^{8}+a^{8}z^{-2}+2a^{8}-2a^{6}z^{-2}-3a^{6}-z^{6}a^{4}-4z^{4}a^{4}-2z^{2}a^{4}+a^{4}z^{-2}+a^{4}+z^{4}a^{2}+3z^{2}a^{2}+a^{2}$ (db)
Kauffman polynomial	$a^{11}z^{7}-5a^{11}z^{5}+6a^{11}z^{3}-2a^{11}z+2a^{10}z^{8}-11a^{10}z^{6}+16a^{10}z^{4}-10a^{10}z^{2}+3a^{10}+a^{9}z^{9}-4a^{9}z^{7}-3a^{9}z^{5}+14a^{9}z^{3}-7a^{9}z+3a^{8}z^{8}-21a^{8}z^{6}+40a^{8}z^{4}-29a^{8}z^{2}-a^{8}z^{-2}+11a^{8}+a^{7}z^{9}-5a^{7}z^{7}-a^{7}z^{5}+16a^{7}z^{3}-12a^{7}z+2a^{7}z^{-1}+2a^{6}z^{8}-14a^{6}z^{6}+27a^{6}z^{4}-24a^{6}z^{2}-2a^{6}z^{-2}+11a^{6}+2a^{5}z^{7}-11a^{5}z^{5}+14a^{5}z^{3}-8a^{5}z+2a^{5}z^{-1}+a^{4}z^{8}-3a^{4}z^{6}-a^{4}z^{4}-a^{4}z^{2}-a^{4}z^{-2}+3a^{4}+2a^{3}z^{7}-8a^{3}z^{5}+6a^{3}z^{3}-a^{3}z+a^{2}z^{6}-4a^{2}z^{4}+4a^{2}z^{2}-a^{2}$ (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   \ -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 χ 1                       1 1 -1                     1   -1 -3                   2 1   1 -5               1 2 2     1 -7             1 2 1       0 -9           2 2 2         2 -11         2 5 2           1 -13       1 1 3             3 -15     1 2 1               0 -17   1 1                   0 -19   1                     1 -21 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-5$ $i=-3$ $i=-1$ $r=-9$ ${\mathbb {Z} }$ $r=-8$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-7$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-6$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-4$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }^{2}$ $r=-3$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-2$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-1$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=0$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }^{2}$ $r=1$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=2$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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