L11n271

L11n270

L11n272

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Link Presentations

[edit Notes on L11n271's Link Presentations]

Planar diagram presentation	X6172 X3,11,4,10 X7,17,8,16 X15,5,16,8 X11,19,12,18 X17,9,18,22 X21,13,22,12 X13,21,14,20 X19,15,20,14 X2536 X9,1,10,4
Gauss code	{1, -10, -2, 11}, {10, -1, -3, 4}, {-11, 2, -5, 7, -8, 9, -4, 3, -6, 5, -9, 8, -7, 6}

A Braid Representative	{{{braid_table}}}
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {-4uvw^{2}+5uvw-2uv+2uw^{2}-2uw+2vw^{2}-2vw+2w^{3}-5w^{2}+4w}{{\sqrt {u}}{\sqrt {v}}w^{3/2}}}$ (db)
Jones polynomial	$-q^{10}+2q^{9}-5q^{8}+7q^{7}-9q^{6}+11q^{5}-8q^{4}+9q^{3}-5q^{2}+3q$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$-a^{-10}z^{-2}-a^{-10}+3z^{2}a^{-8}+3a^{-8}z^{-2}+5a^{-8}-2z^{4}a^{-6}-4z^{2}a^{-6}-2a^{-6}z^{-2}-5a^{-6}-2z^{4}a^{-4}-3z^{2}a^{-4}-a^{-4}z^{-2}-2a^{-4}+3z^{2}a^{-2}+a^{-2}z^{-2}+3a^{-2}$ (db)
Kauffman polynomial	$z^{7}a^{-11}-5z^{5}a^{-11}+9z^{3}a^{-11}-7za^{-11}+2a^{-11}z^{-1}+2z^{8}a^{-10}-8z^{6}a^{-10}+9z^{4}a^{-10}-2z^{2}a^{-10}-a^{-10}z^{-2}+a^{-10}+z^{9}a^{-9}+3z^{7}a^{-9}-26z^{5}a^{-9}+43z^{3}a^{-9}-27za^{-9}+8a^{-9}z^{-1}+7z^{8}a^{-8}-23z^{6}a^{-8}+20z^{4}a^{-8}-8z^{2}a^{-8}-3a^{-8}z^{-2}+5a^{-8}+z^{9}a^{-7}+10z^{7}a^{-7}-43z^{5}a^{-7}+52z^{3}a^{-7}-34za^{-7}+10a^{-7}z^{-1}+5z^{8}a^{-6}-8z^{6}a^{-6}-z^{4}a^{-6}-3z^{2}a^{-6}-2a^{-6}z^{-2}+4a^{-6}+8z^{7}a^{-5}-19z^{5}a^{-5}+18z^{3}a^{-5}-10za^{-5}+2a^{-5}z^{-1}+7z^{6}a^{-4}-12z^{4}a^{-4}+9z^{2}a^{-4}+a^{-4}z^{-2}-3a^{-4}+3z^{5}a^{-3}+4za^{-3}-2a^{-3}z^{-1}+6z^{2}a^{-2}+a^{-2}z^{-2}-4a^{-2}$ (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   \ 0 1 2 3 4 5 6 7 8 9 χ 21                   1 -1 19                 1   1 17               4 1   -3 15             3 1     2 13           6 4       -2 11         5 3         2 9       4 7           3 7     5 4             1 5     4               4 3 3 5                 -2 1 3                   3

Integral Khovanov Homology (db, data source)

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

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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