L11a462

L11a461

L11a463

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Link Presentations

[edit Notes on L11a462's Link Presentations]

Planar diagram presentation	X6172 X14,4,15,3 X16,5,17,6 X18,14,19,13 X12,15,5,16 X10,19,11,20 X8,21,9,22 X20,7,21,8 X22,9,13,10 X2,11,3,12 X4,18,1,17
Gauss code	{1, -10, 2, -11}, {3, -1, 8, -7, 9, -6, 10, -5}, {4, -2, 5, -3, 11, -4, 6, -8, 7, -9}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {t(1)t(3)^{3}t(2)^{3}-t(3)^{3}t(2)^{3}-2t(1)t(3)^{2}t(2)^{3}+2t(3)^{2}t(2)^{3}+t(1)t(3)t(2)^{3}-2t(1)t(3)^{3}t(2)^{2}+2t(3)^{3}t(2)^{2}+6t(1)t(3)^{2}t(2)^{2}-6t(3)^{2}t(2)^{2}+t(1)t(2)^{2}-5t(1)t(3)t(2)^{2}+4t(3)t(2)^{2}-t(3)^{3}t(2)-4t(1)t(3)^{2}t(2)+5t(3)^{2}t(2)-2t(1)t(2)+6t(1)t(3)t(2)-6t(3)t(2)+2t(2)-t(3)^{2}+t(1)-2t(1)t(3)+2t(3)-1}{{\sqrt {t(1)}}t(2)^{3/2}t(3)^{3/2}}}$ (db)
Jones polynomial	$-q+4-7q^{-1}+13q^{-2}-17q^{-3}+21q^{-4}-21q^{-5}+19q^{-6}-14q^{-7}+10q^{-8}-4q^{-9}+q^{-10}$ (db)
Signature	-4 (db)
HOMFLY-PT polynomial	$a^{8}z^{4}+2a^{8}z^{2}+a^{8}z^{-2}+2a^{8}-2a^{6}z^{6}-7a^{6}z^{4}-8a^{6}z^{2}-2a^{6}z^{-2}-5a^{6}+a^{4}z^{8}+5a^{4}z^{6}+9a^{4}z^{4}+6a^{4}z^{2}+a^{4}z^{-2}+a^{4}-a^{2}z^{6}-3a^{2}z^{4}-a^{2}z^{2}+2a^{2}$ (db)
Kauffman polynomial	$a^{12}z^{4}+4a^{11}z^{5}+10a^{10}z^{6}-10a^{10}z^{4}+6a^{10}z^{2}-2a^{10}+14a^{9}z^{7}-18a^{9}z^{5}+6a^{9}z^{3}+a^{9}z+13a^{8}z^{8}-17a^{8}z^{6}+6a^{8}z^{4}-6a^{8}z^{2}-a^{8}z^{-2}+3a^{8}+7a^{7}z^{9}+4a^{7}z^{7}-32a^{7}z^{5}+22a^{7}z^{3}-8a^{7}z+2a^{7}z^{-1}+2a^{6}z^{10}+16a^{6}z^{8}-50a^{6}z^{6}+42a^{6}z^{4}-22a^{6}z^{2}-2a^{6}z^{-2}+9a^{6}+12a^{5}z^{9}-26a^{5}z^{7}+4a^{5}z^{5}+14a^{5}z^{3}-8a^{5}z+2a^{5}z^{-1}+2a^{4}z^{10}+7a^{4}z^{8}-38a^{4}z^{6}+42a^{4}z^{4}-14a^{4}z^{2}-a^{4}z^{-2}+3a^{4}+5a^{3}z^{9}-15a^{3}z^{7}+11a^{3}z^{5}+a^{3}z+4a^{2}z^{8}-15a^{2}z^{6}+17a^{2}z^{4}-4a^{2}z^{2}-2a^{2}+az^{7}-3az^{5}+2az^{3}$ (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   \ -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 χ 3                       1 -1 1                     3   3 -1                   4 1   -3 -3                 9 3     6 -5               10 6       -4 -7             11 7         4 -9           10 10           0 -11         9 11             -2 -13       6 11               5 -15     4 8                 -4 -17     6                   6 -19 1 4                     -3 -21 1                       1

Integral Khovanov Homology (db, data source)

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

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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