L11a476

L11a475

L11a477

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Link Presentations

[edit Notes on L11a476's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(t(1)-1)(t(3)-1)^{2}(t(3)t(2)-t(2)+1)(t(2)t(3)-t(3)+1)}{{\sqrt {t(1)}}t(2)t(3)^{2}}}$ (db)
Jones polynomial	$-q^{5}+4q^{4}-10q^{3}+16q^{2}-20q+25-22q^{-1}+20q^{-2}-14q^{-3}+8q^{-4}-3q^{-5}+q^{-6}$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$z^{8}-2a^{2}z^{6}-z^{6}a^{-2}+5z^{6}+a^{4}z^{4}-8a^{2}z^{4}-3z^{4}a^{-2}+11z^{4}+3a^{4}z^{2}-13a^{2}z^{2}-4z^{2}a^{-2}+14z^{2}+3a^{4}-10a^{2}-3a^{-2}+10+a^{4}z^{-2}-2a^{2}z^{-2}+z^{-2}$ (db)
Kauffman polynomial	$a^{6}z^{6}-3a^{6}z^{4}+3a^{6}z^{2}-a^{6}+3a^{5}z^{7}-7a^{5}z^{5}+z^{5}a^{-5}+5a^{5}z^{3}-z^{3}a^{-5}-a^{5}z+5a^{4}z^{8}-9a^{4}z^{6}+4z^{6}a^{-4}+4a^{4}z^{4}-4z^{4}a^{-4}-2a^{4}z^{2}-a^{4}z^{-2}+3a^{4}+5a^{3}z^{9}-4a^{3}z^{7}+9z^{7}a^{-3}-7a^{3}z^{5}-14z^{5}a^{-3}+9a^{3}z^{3}+8z^{3}a^{-3}-7a^{3}z-3za^{-3}+2a^{3}z^{-1}+2a^{2}z^{10}+12a^{2}z^{8}+12z^{8}a^{-2}-38a^{2}z^{6}-23z^{6}a^{-2}+43a^{2}z^{4}+21z^{4}a^{-2}-31a^{2}z^{2}-12z^{2}a^{-2}-2a^{2}z^{-2}+13a^{2}+5a^{-2}+13az^{9}+8z^{9}a^{-1}-20az^{7}-4z^{7}a^{-1}+az^{5}-14z^{5}a^{-1}+15az^{3}+20z^{3}a^{-1}-13az-10za^{-1}+2az^{-1}+2z^{10}+19z^{8}-55z^{6}+61z^{4}-38z^{2}-z^{-2}+15$ (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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 11                       1 -1 9                     3   3 7                   7 1   -6 5                 9 3     6 3               11 7       -4 1             14 9         5 -1           10 13           3 -3         10 12             -2 -5       6 12               6 -7     2 8                 -6 -9   1 6                   5 -11   2                     -2 -13 1                       1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-1$ $i=1$ $r=-6$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-3$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=-2$ ${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{10}$ $r=-1$ ${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=0$ ${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{12}$ ${\mathbb {Z} }^{14}$ $r=1$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{11}$ $r=2$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=3$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=4$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=5$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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