L11a433

L11a432

L11a434

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Link Presentations

[edit Notes on L11a433's Link Presentations]

Planar diagram presentation	X6172 X14,6,15,5 X8493 X2,16,3,15 X16,7,17,8 X18,14,19,13 X22,9,13,10 X20,11,21,12 X12,19,5,20 X10,21,11,22 X4,17,1,18
Gauss code	{1, -4, 3, -11}, {2, -1, 5, -3, 7, -10, 8, -9}, {6, -2, 4, -5, 11, -6, 9, -8, 10, -7}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(t(1)-1)(t(2)-1)(t(3)-1)\left(t(3)t(2)^{2}-t(2)^{2}+t(3)^{2}t(2)-t(3)t(2)+t(2)-t(3)^{2}+t(3)\right)}{{\sqrt {t(1)}}t(2)^{3/2}t(3)^{3/2}}}$ (db)
Jones polynomial	$-q^{-7}+4q^{-6}-6q^{-5}+q^{4}+11q^{-4}-4q^{3}-14q^{-3}+8q^{2}+18q^{-2}-12q-17q^{-1}+16$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$-z^{2}a^{6}+a^{6}z^{-2}+2z^{4}a^{4}+3z^{2}a^{4}-2a^{4}z^{-2}-a^{4}-z^{6}a^{2}-2z^{4}a^{2}-2z^{2}a^{2}+a^{2}z^{-2}-z^{6}-2z^{4}-z^{2}+1+z^{4}a^{-2}+z^{2}a^{-2}$ (db)
Kauffman polynomial	$a^{7}z^{7}-3a^{7}z^{5}+a^{7}z^{3}+4a^{6}z^{8}-16a^{6}z^{6}+17a^{6}z^{4}-3a^{6}z^{2}+a^{6}z^{-2}-3a^{6}+5a^{5}z^{9}-18a^{5}z^{7}+17a^{5}z^{5}-5a^{5}z^{3}+4a^{5}z-2a^{5}z^{-1}+2a^{4}z^{10}+4a^{4}z^{8}-35a^{4}z^{6}+45a^{4}z^{4}+z^{4}a^{-4}-13a^{4}z^{2}+2a^{4}z^{-2}-4a^{4}+11a^{3}z^{9}-34a^{3}z^{7}+30a^{3}z^{5}+4z^{5}a^{-3}-9a^{3}z^{3}-2z^{3}a^{-3}+4a^{3}z-2a^{3}z^{-1}+2a^{2}z^{10}+9a^{2}z^{8}-38a^{2}z^{6}+8z^{6}a^{-2}+39a^{2}z^{4}-8z^{4}a^{-2}-14a^{2}z^{2}+2z^{2}a^{-2}+a^{2}z^{-2}-a^{2}+6az^{9}-5az^{7}+10z^{7}a^{-1}-6az^{5}-12z^{5}a^{-1}+2az^{3}+3z^{3}a^{-1}+9z^{8}-11z^{6}+2z^{4}-2z^{2}+1$ (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   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 χ 9                       1 1 7                     3   -3 5                   5 1   4 3                 7 3     -4 1               9 5       4 -1             10 9         -1 -3           8 7           1 -5         6 10             4 -7       5 8               -3 -9     3 8                 5 -11   1 3                   -2 -13   3                     3 -15 1                       -1

Integral Khovanov Homology (db, data source)

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

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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