L10a61

L10a60

L10a62

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Link Presentations

[edit Notes on L10a61's Link Presentations]

Planar diagram presentation	X8192 X10,4,11,3 X20,10,7,9 X2738 X18,14,19,13 X6,12,1,11 X16,20,17,19 X4,16,5,15 X14,6,15,5 X12,18,13,17
Gauss code	{1, -4, 2, -8, 9, -6}, {4, -1, 3, -2, 6, -10, 5, -9, 8, -7, 10, -5, 7, -3}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {t(1)t(2)^{4}-t(2)^{4}+2t(1)^{2}t(2)^{3}-6t(1)t(2)^{3}+3t(2)^{3}-4t(1)^{2}t(2)^{2}+9t(1)t(2)^{2}-4t(2)^{2}+3t(1)^{2}t(2)-6t(1)t(2)+2t(2)-t(1)^{2}+t(1)}{t(1)t(2)^{2}}}$ (db)
Jones polynomial	$-q^{19/2}+4q^{17/2}-7q^{15/2}+11q^{13/2}-14q^{11/2}+14q^{9/2}-14q^{7/2}+10q^{5/2}-7q^{3/2}+3{\sqrt {q}}-{\frac {1}{\sqrt {q}}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$-z^{5}a^{-3}-2z^{5}a^{-5}+z^{3}a^{-1}-5z^{3}a^{-5}+3z^{3}a^{-7}+za^{-1}+3za^{-3}-6za^{-5}+4za^{-7}-za^{-9}+2a^{-3}z^{-1}-3a^{-5}z^{-1}+a^{-7}z^{-1}$ (db)
Kauffman polynomial	$z^{5}a^{-11}-z^{3}a^{-11}+4z^{6}a^{-10}-7z^{4}a^{-10}+3z^{2}a^{-10}+6z^{7}a^{-9}-10z^{5}a^{-9}+5z^{3}a^{-9}-2za^{-9}+4z^{8}a^{-8}+3z^{6}a^{-8}-18z^{4}a^{-8}+12z^{2}a^{-8}-a^{-8}+z^{9}a^{-7}+14z^{7}a^{-7}-34z^{5}a^{-7}+28z^{3}a^{-7}-10za^{-7}+a^{-7}z^{-1}+8z^{8}a^{-6}-5z^{6}a^{-6}-13z^{4}a^{-6}+14z^{2}a^{-6}-3a^{-6}+z^{9}a^{-5}+13z^{7}a^{-5}-33z^{5}a^{-5}+32z^{3}a^{-5}-15za^{-5}+3a^{-5}z^{-1}+4z^{8}a^{-4}-z^{6}a^{-4}-7z^{4}a^{-4}+7z^{2}a^{-4}-3a^{-4}+5z^{7}a^{-3}-9z^{5}a^{-3}+8z^{3}a^{-3}-6za^{-3}+2a^{-3}z^{-1}+3z^{6}a^{-2}-5z^{4}a^{-2}+2z^{2}a^{-2}+z^{5}a^{-1}-2z^{3}a^{-1}+za^{-1}$ (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   \ -2 -1 0 1 2 3 4 5 6 7 8 χ 20                     1 1 18                   3   -3 16                 4 1   3 14               7 3     -4 12             7 4       3 10           7 7         0 8         7 7           0 6       4 8             4 4     3 6               -3 2   1 5                 4 0   2                   -2 -2 1                     1

Integral Khovanov Homology (db, data source)

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

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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