L11n310

L11n309

L11n311

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Link Presentations

[edit Notes on L11n310's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {-t(2)t(3)^{4}+t(3)^{4}+t(1)t(2)^{2}t(3)^{3}+t(1)t(3)^{3}-3t(1)t(2)t(3)^{3}+3t(2)t(3)^{3}-2t(3)^{3}-2t(1)t(2)^{2}t(3)^{2}+t(2)^{2}t(3)^{2}-t(1)t(3)^{2}+4t(1)t(2)t(3)^{2}-4t(2)t(3)^{2}+2t(3)^{2}+2t(1)t(2)^{2}t(3)-t(2)^{2}t(3)-3t(1)t(2)t(3)+3t(2)t(3)-t(3)-t(1)t(2)^{2}+t(1)t(2)}{{\sqrt {t(1)}}t(2)t(3)^{2}}}$ (db)
Jones polynomial	$-q^{8}+3q^{7}-7q^{6}+10q^{5}-12q^{4}+14q^{3}-11q^{2}+10q-5+3q^{-1}$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$-z^{4}a^{-6}-2z^{2}a^{-6}-a^{-6}z^{-2}-3a^{-6}+z^{6}a^{-4}+4z^{4}a^{-4}+10z^{2}a^{-4}+4a^{-4}z^{-2}+11a^{-4}-4z^{4}a^{-2}-12z^{2}a^{-2}-5a^{-2}z^{-2}-13a^{-2}+3z^{2}+2z^{-2}+5$ (db)
Kauffman polynomial	$z^{5}a^{-9}-2z^{3}a^{-9}+za^{-9}+3z^{6}a^{-8}-5z^{4}a^{-8}+z^{2}a^{-8}+5z^{7}a^{-7}-9z^{5}a^{-7}+5z^{3}a^{-7}-4za^{-7}+a^{-7}z^{-1}+5z^{8}a^{-6}-9z^{6}a^{-6}+8z^{4}a^{-6}-8z^{2}a^{-6}-a^{-6}z^{-2}+5a^{-6}+2z^{9}a^{-5}+5z^{7}a^{-5}-22z^{5}a^{-5}+32z^{3}a^{-5}-21za^{-5}+5a^{-5}z^{-1}+10z^{8}a^{-4}-29z^{6}a^{-4}+44z^{4}a^{-4}-36z^{2}a^{-4}-4a^{-4}z^{-2}+18a^{-4}+2z^{9}a^{-3}+3z^{7}a^{-3}-18z^{5}a^{-3}+35z^{3}a^{-3}-29za^{-3}+9a^{-3}z^{-1}+5z^{8}a^{-2}-17z^{6}a^{-2}+37z^{4}a^{-2}-41z^{2}a^{-2}-5a^{-2}z^{-2}+21a^{-2}+3z^{7}a^{-1}-6z^{5}a^{-1}+10z^{3}a^{-1}-13za^{-1}+5a^{-1}z^{-1}+6z^{4}-14z^{2}-2z^{-2}+9$ (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   \ -2 -1 0 1 2 3 4 5 6 7 χ 17                   1 -1 15                 2   2 13               5 1   -4 11             5 2     3 9           7 5       -2 7         7 5         2 5       5 8           3 3     5 6             -1 1   2 7               5 -1 1 3                 -2 -3 3                   3

Integral Khovanov Homology (db, data source)

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

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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