L11a445

L11a444

L11a446

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Link Presentations

[edit Notes on L11a445's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {t(1)t(3)^{2}t(2)^{3}-t(3)^{2}t(2)^{3}-t(1)t(3)t(2)^{3}+2t(3)t(2)^{3}-t(2)^{3}+t(1)t(3)^{3}t(2)^{2}-t(3)^{3}t(2)^{2}-5t(1)t(3)^{2}t(2)^{2}+5t(3)^{2}t(2)^{2}-t(1)t(2)^{2}+4t(1)t(3)t(2)^{2}-6t(3)t(2)^{2}+2t(2)^{2}-2t(1)t(3)^{3}t(2)+t(3)^{3}t(2)+6t(1)t(3)^{2}t(2)-4t(3)^{2}t(2)+t(1)t(2)-5t(1)t(3)t(2)+5t(3)t(2)-t(2)+t(1)t(3)^{3}-2t(1)t(3)^{2}+t(3)^{2}+t(1)t(3)-t(3)}{{\sqrt {t(1)}}t(2)^{3/2}t(3)^{3/2}}}$ (db)
Jones polynomial	$-q^{8}+4q^{7}-9q^{6}+14q^{5}-17q^{4}+21q^{3}+q^{-3}-19q^{2}-3q^{-2}+17q+7q^{-1}-11$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$-z^{4}a^{-6}-z^{2}a^{-6}-a^{-6}+z^{6}a^{-4}+2z^{4}a^{-4}+3z^{2}a^{-4}+a^{-4}z^{-2}+3a^{-4}+z^{6}a^{-2}+z^{4}a^{-2}+a^{2}z^{2}-z^{2}a^{-2}-2a^{-2}z^{-2}+a^{2}-3a^{-2}-2z^{4}-3z^{2}+z^{-2}$ (db)
Kauffman polynomial	$z^{5}a^{-9}-z^{3}a^{-9}+4z^{6}a^{-8}-5z^{4}a^{-8}+z^{2}a^{-8}+8z^{7}a^{-7}-13z^{5}a^{-7}+6z^{3}a^{-7}-2za^{-7}+9z^{8}a^{-6}-15z^{6}a^{-6}+11z^{4}a^{-6}-7z^{2}a^{-6}+3a^{-6}+5z^{9}a^{-5}+3z^{7}a^{-5}-20z^{5}a^{-5}+20z^{3}a^{-5}-7za^{-5}+z^{10}a^{-4}+16z^{8}a^{-4}-42z^{6}a^{-4}+47z^{4}a^{-4}-31z^{2}a^{-4}-a^{-4}z^{-2}+11a^{-4}+8z^{9}a^{-3}-6z^{7}a^{-3}-15z^{5}a^{-3}+24z^{3}a^{-3}-12za^{-3}+2a^{-3}z^{-1}+z^{10}a^{-2}+11z^{8}a^{-2}+a^{2}z^{6}-31z^{6}a^{-2}-3a^{2}z^{4}+35z^{4}a^{-2}+3a^{2}z^{2}-26z^{2}a^{-2}-2a^{-2}z^{-2}-a^{2}+11a^{-2}+3z^{9}a^{-1}+3az^{7}+2z^{7}a^{-1}-8az^{5}-17z^{5}a^{-1}+6az^{3}+17z^{3}a^{-1}-az-8za^{-1}+2a^{-1}z^{-1}+4z^{8}-7z^{6}+z^{4}-z^{-2}+3$ (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   \ -4 -3 -2 -1 0 1 2 3 4 5 6 7 χ 17                       1 -1 15                     3   3 13                   6 1   -5 11                 8 3     5 9               10 7       -3 7             11 7         4 5           8 10           2 3         9 11             -2 1       5 11               6 -1     2 6                 -4 -3   1 5                   4 -5   2                     -2 -7 1                       1

Integral Khovanov Homology (db, data source)

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

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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