L10a67

L10a66

L10a68

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Link Presentations

[edit Notes on L10a67's Link Presentations]

Planar diagram presentation	X8192 X2,9,3,10 X10,3,11,4 X14,5,15,6 X18,11,19,12 X20,16,7,15 X16,20,17,19 X12,17,13,18 X6718 X4,13,5,14
Gauss code	{1, -2, 3, -10, 4, -9}, {9, -1, 2, -3, 5, -8, 10, -4, 6, -7, 8, -5, 7, -6}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {t(1)^{2}t(2)^{4}-4t(1)^{2}t(2)^{3}+2t(1)t(2)^{3}+2t(1)^{2}t(2)^{2}-5t(1)t(2)^{2}+2t(2)^{2}+2t(1)t(2)-4t(2)+1}{t(1)t(2)^{2}}}$ (db)
Jones polynomial	$-{\frac {7}{q^{9/2}}}+{\frac {5}{q^{7/2}}}-{\frac {4}{q^{5/2}}}+{\frac {2}{q^{3/2}}}-{\frac {1}{q^{21/2}}}+{\frac {2}{q^{19/2}}}-{\frac {4}{q^{17/2}}}+{\frac {6}{q^{15/2}}}-{\frac {7}{q^{13/2}}}+{\frac {7}{q^{11/2}}}-{\frac {1}{\sqrt {q}}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$z^{3}a^{9}+3za^{9}+a^{9}z^{-1}-2z^{5}a^{7}-8z^{3}a^{7}-8za^{7}-2a^{7}z^{-1}+z^{7}a^{5}+5z^{5}a^{5}+8z^{3}a^{5}+6za^{5}+2a^{5}z^{-1}-z^{5}a^{3}-4z^{3}a^{3}-4za^{3}-a^{3}z^{-1}$ (db)
Kauffman polynomial	$a^{13}z^{3}-a^{13}z+2a^{12}z^{4}-a^{12}z^{2}+3a^{11}z^{5}-2a^{11}z^{3}+a^{11}z+4a^{10}z^{6}-6a^{10}z^{4}+4a^{10}z^{2}+4a^{9}z^{7}-8a^{9}z^{5}+6a^{9}z^{3}-4a^{9}z+a^{9}z^{-1}+3a^{8}z^{8}-6a^{8}z^{6}+a^{8}z^{4}+a^{7}z^{9}+3a^{7}z^{7}-20a^{7}z^{5}+24a^{7}z^{3}-12a^{7}z+2a^{7}z^{-1}+5a^{6}z^{8}-19a^{6}z^{6}+20a^{6}z^{4}-8a^{6}z^{2}+a^{6}+a^{5}z^{9}-14a^{5}z^{5}+23a^{5}z^{3}-11a^{5}z+2a^{5}z^{-1}+2a^{4}z^{8}-9a^{4}z^{6}+11a^{4}z^{4}-3a^{4}z^{2}+a^{3}z^{7}-5a^{3}z^{5}+8a^{3}z^{3}-5a^{3}z+a^{3}z^{-1}$ (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 0                     1 1 -2                   1   -1 -4                 3 1   2 -6               3 2     -1 -8             4 2       2 -10           3 3         0 -12         4 4           0 -14       2 3             1 -16     2 4               -2 -18   1 3                 2 -20   1                   -1 -22 1                     1

Integral Khovanov Homology (db, data source)

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

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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