L10a123

L10a122

L10a124

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Link Presentations

[edit Notes on L10a123's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {t(1)t(3)^{3}+t(2)t(3)^{3}-t(3)^{3}-5t(1)t(3)^{2}+2t(1)t(2)t(3)^{2}-5t(2)t(3)^{2}+4t(3)^{2}+5t(1)t(3)-4t(1)t(2)t(3)+5t(2)t(3)-2t(3)-t(1)+t(1)t(2)-t(2)}{{\sqrt {t(1)}}{\sqrt {t(2)}}t(3)^{3/2}}}$ (db)
Jones polynomial	$q^{-9}-2q^{-8}+6q^{-7}-9q^{-6}+12q^{-5}-12q^{-4}+13q^{-3}-10q^{-2}+q+7q^{-1}-3$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$a^{10}z^{-2}-3a^{8}z^{-2}-4a^{8}+6a^{6}z^{2}+4a^{6}z^{-2}+8a^{6}-3a^{4}z^{4}-6a^{4}z^{2}-3a^{4}z^{-2}-7a^{4}-a^{2}z^{4}+2a^{2}z^{2}+a^{2}z^{-2}+3a^{2}+z^{2}$ (db)
Kauffman polynomial	$z^{6}a^{10}-4z^{4}a^{10}+6z^{2}a^{10}+a^{10}z^{-2}-4a^{10}+2z^{7}a^{9}-5z^{5}a^{9}+3z^{3}a^{9}+za^{9}-a^{9}z^{-1}+2z^{8}a^{8}+z^{6}a^{8}-17z^{4}a^{8}+25z^{2}a^{8}+3a^{8}z^{-2}-14a^{8}+z^{9}a^{7}+6z^{7}a^{7}-17z^{5}a^{7}+7z^{3}a^{7}+za^{7}-a^{7}z^{-1}+6z^{8}a^{6}-z^{6}a^{6}-29z^{4}a^{6}+38z^{2}a^{6}+4a^{6}z^{-2}-21a^{6}+z^{9}a^{5}+11z^{7}a^{5}-24z^{5}a^{5}+11z^{3}a^{5}+za^{5}-a^{5}z^{-1}+4z^{8}a^{4}+5z^{6}a^{4}-25z^{4}a^{4}+28z^{2}a^{4}+3a^{4}z^{-2}-14a^{4}+7z^{7}a^{3}-9z^{5}a^{3}+5z^{3}a^{3}+za^{3}-a^{3}z^{-1}+6z^{6}a^{2}-8z^{4}a^{2}+8z^{2}a^{2}+a^{2}z^{-2}-4a^{2}+3z^{5}a-2z^{3}a+z^{4}-z^{2}$ (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   \ -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 χ 3                     1 1 1                   2   -2 -1                 5 1   4 -3               6 3     -3 -5             7 4       3 -7           5 6         1 -9         7 7           0 -11       5 8             3 -13     1 4               -3 -15   1 5                 4 -17   1                   -1 -19 1                     1

Integral Khovanov Homology (db, data source)

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

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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