L11a396

L11a395

L11a397

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Link Presentations

[edit Notes on L11a396's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X22,14,9,13 X20,12,21,11 X12,22,13,21 X18,16,19,15 X8,18,5,17 X16,8,17,7 X14,20,15,19 X2536 X4,9,1,10
Gauss code	{1, -10, 2, -11}, {10, -1, 8, -7}, {11, -2, 4, -5, 3, -9, 6, -8, 7, -6, 9, -4, 5, -3}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {2uvw^{3}-3uvw^{2}+3uvw-2uv-2uw^{3}+4uw^{2}-4uw+3u-3vw^{3}+4vw^{2}-4vw+2v+2w^{3}-3w^{2}+3w-2}{{\sqrt {u}}{\sqrt {v}}w^{3/2}}}$ (db)
Jones polynomial	$-q^{8}+4q^{7}-8q^{6}+11q^{5}-14q^{4}+15q^{3}-14q^{2}+12q-6+5q^{-1}-q^{-2}+q^{-3}$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$-z^{4}a^{-6}-z^{2}a^{-6}+z^{6}a^{-4}+2z^{4}a^{-4}-a^{-4}z^{-2}-2a^{-4}+z^{6}a^{-2}+3z^{4}a^{-2}+a^{2}z^{2}+6z^{2}a^{-2}+2a^{2}z^{-2}+4a^{-2}z^{-2}+3a^{2}+8a^{-2}-2z^{4}-7z^{2}-5z^{-2}-9$ (db)
Kauffman polynomial	$z^{10}a^{-2}+z^{10}a^{-4}+z^{9}a^{-1}+5z^{9}a^{-3}+4z^{9}a^{-5}-z^{8}a^{-2}+5z^{8}a^{-4}+7z^{8}a^{-6}+z^{8}+az^{7}+z^{7}a^{-1}-12z^{7}a^{-3}-5z^{7}a^{-5}+7z^{7}a^{-7}+a^{2}z^{6}+8z^{6}a^{-2}-11z^{6}a^{-4}-14z^{6}a^{-6}+4z^{6}a^{-8}+2z^{6}-az^{5}+20z^{5}a^{-3}+5z^{5}a^{-5}-13z^{5}a^{-7}+z^{5}a^{-9}-5a^{2}z^{4}-17z^{4}a^{-2}+8z^{4}a^{-4}+10z^{4}a^{-6}-6z^{4}a^{-8}-14z^{4}-6az^{3}-15z^{3}a^{-1}-19z^{3}a^{-3}-5z^{3}a^{-5}+4z^{3}a^{-7}-z^{3}a^{-9}+9a^{2}z^{2}+15z^{2}a^{-2}+z^{2}a^{-4}-3z^{2}a^{-6}+20z^{2}+11az+21za^{-1}+13za^{-3}+3za^{-5}-7a^{2}-10a^{-2}-2a^{-4}-14-5az^{-1}-9a^{-1}z^{-1}-5a^{-3}z^{-1}-a^{-5}z^{-1}+2a^{2}z^{-2}+4a^{-2}z^{-2}+a^{-4}z^{-2}+5z^{-2}$ (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                     3   3 13                   5 1   -4 11                 6 3     3 9               8 5       -3 7             7 6         1 5           7 8           1 3         5 7             -2 1       4 10               6 -1     1 2                 -1 -3     4                   4 -5 1 1                     0 -7 1                       1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=1$ $i=3$ $r=-4$ ${\mathbb {Z} }$ ${\mathbb {Z} }$ $r=-3$ ${\mathbb {Z} }$ $r=-2$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-1$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=0$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{5}$ $r=1$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=2$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=3$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=4$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=5$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=6$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=7$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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