L11a84

L11a83

L11a85

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Link Presentations

[edit Notes on L11a84's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {3t(1)t(2)^{3}-5t(2)^{3}-9t(1)t(2)^{2}+10t(2)^{2}+10t(1)t(2)-9t(2)-5t(1)+3}{{\sqrt {t(1)}}t(2)^{3/2}}}$ (db)
Jones polynomial	$q^{15/2}-4q^{13/2}+8q^{11/2}-12q^{9/2}+16q^{7/2}-17q^{5/2}+17q^{3/2}-15{\sqrt {q}}+{\frac {9}{\sqrt {q}}}-{\frac {6}{q^{3/2}}}+{\frac {2}{q^{5/2}}}-{\frac {1}{q^{7/2}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$z^{5}a^{-1}+2z^{5}a^{-3}-2az^{3}-z^{3}a^{-1}+4z^{3}a^{-3}-3z^{3}a^{-5}+a^{3}z-2az-3za^{-1}+4za^{-3}-3za^{-5}+za^{-7}+a^{3}z^{-1}-2a^{-1}z^{-1}+a^{-3}z^{-1}$ (db)
Kauffman polynomial	$z^{6}a^{-8}-2z^{4}a^{-8}+z^{2}a^{-8}+4z^{7}a^{-7}-10z^{5}a^{-7}+6z^{3}a^{-7}-za^{-7}+6z^{8}a^{-6}-14z^{6}a^{-6}+7z^{4}a^{-6}-2z^{2}a^{-6}+4z^{9}a^{-5}-z^{7}a^{-5}-17z^{5}a^{-5}+14z^{3}a^{-5}-2za^{-5}+z^{10}a^{-4}+11z^{8}a^{-4}-33z^{6}a^{-4}+28z^{4}a^{-4}-7z^{2}a^{-4}-2a^{-4}+7z^{9}a^{-3}-9z^{7}a^{-3}+a^{3}z^{5}-3z^{5}a^{-3}-3a^{3}z^{3}+7z^{3}a^{-3}+3a^{3}z-za^{-3}-a^{3}z^{-1}+a^{-3}z^{-1}+z^{10}a^{-2}+8z^{8}a^{-2}+2a^{2}z^{6}-19z^{6}a^{-2}-3a^{2}z^{4}+14z^{4}a^{-2}+4z^{2}a^{-2}+a^{2}-5a^{-2}+3z^{9}a^{-1}+3az^{7}-z^{7}a^{-1}-3az^{5}+2z^{3}a^{-1}-3za^{-1}+2a^{-1}z^{-1}+3z^{8}+z^{6}-8z^{4}+8z^{2}-3$ (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 16                       1 -1 14                     3   3 12                   5 1   -4 10                 7 3     4 8               9 5       -4 6             8 7         1 4           9 9           0 2         6 8             -2 0       4 10               6 -2     2 5                 -3 -4     4                   4 -6 1 2                     -1 -8 1                       1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=0$ $i=2$ $r=-4$ ${\mathbb {Z} }$ ${\mathbb {Z} }$ $r=-3$ ${\mathbb {Z} }^{2}$ $r=-2$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-1$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=0$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{6}$ $r=1$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=2$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=3$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=4$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=5$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=6$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=7$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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