L11a460

L11a459

L11a461

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Link Presentations

[edit Notes on L11a460's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {t(1)t(3)^{2}t(2)^{3}-t(3)^{2}t(2)^{3}-2t(1)t(3)t(2)^{3}+2t(3)t(2)^{3}-t(2)^{3}+t(1)t(3)^{3}t(2)^{2}-t(3)^{3}t(2)^{2}-5t(1)t(3)^{2}t(2)^{2}+5t(3)^{2}t(2)^{2}-2t(1)t(2)^{2}+6t(1)t(3)t(2)^{2}-6t(3)t(2)^{2}+2t(2)^{2}-2t(1)t(3)^{3}t(2)+2t(3)^{3}t(2)+6t(1)t(3)^{2}t(2)-6t(3)^{2}t(2)+t(1)t(2)-5t(1)t(3)t(2)+5t(3)t(2)-t(2)+t(1)t(3)^{3}-2t(1)t(3)^{2}+2t(3)^{2}+t(1)t(3)-t(3)}{{\sqrt {t(1)}}t(2)^{3/2}t(3)^{3/2}}}$ (db)
Jones polynomial	$q^{9}-3q^{8}+8q^{7}-12q^{6}+19q^{5}-22q^{4}+23q^{3}-20q^{2}-q^{-2}+16q+5q^{-1}-10$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$z^{6}a^{-2}+z^{6}a^{-4}+z^{4}a^{-2}+z^{4}a^{-4}-2z^{4}a^{-6}-z^{4}+z^{2}a^{-4}-3z^{2}a^{-6}+z^{2}a^{-8}+a^{-4}-3a^{-6}+a^{-8}+1+a^{-4}z^{-2}-2a^{-6}z^{-2}+a^{-8}z^{-2}$ (db)
Kauffman polynomial	$z^{6}a^{-10}-3z^{4}a^{-10}+2z^{2}a^{-10}+3z^{7}a^{-9}-7z^{5}a^{-9}+3z^{3}a^{-9}+6z^{8}a^{-8}-18z^{6}a^{-8}+24z^{4}a^{-8}-20z^{2}a^{-8}-a^{-8}z^{-2}+8a^{-8}+5z^{9}a^{-7}-6z^{7}a^{-7}-7z^{5}a^{-7}+17z^{3}a^{-7}-11za^{-7}+2a^{-7}z^{-1}+2z^{10}a^{-6}+10z^{8}a^{-6}-36z^{6}a^{-6}+46z^{4}a^{-6}-31z^{2}a^{-6}-2a^{-6}z^{-2}+13a^{-6}+12z^{9}a^{-5}-19z^{7}a^{-5}+z^{5}a^{-5}+17z^{3}a^{-5}-11za^{-5}+2a^{-5}z^{-1}+2z^{10}a^{-4}+15z^{8}a^{-4}-36z^{6}a^{-4}+26z^{4}a^{-4}-10z^{2}a^{-4}-a^{-4}z^{-2}+5a^{-4}+7z^{9}a^{-3}-14z^{5}a^{-3}+6z^{3}a^{-3}+11z^{8}a^{-2}-14z^{6}a^{-2}+2z^{4}a^{-2}-z^{2}a^{-2}+10z^{7}a^{-1}+az^{5}-14z^{5}a^{-1}+3z^{3}a^{-1}+5z^{6}-5z^{4}+1$ (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   \ -3 -2 -1 0 1 2 3 4 5 6 7 8 χ 19                       1 1 17                     3 1 -2 15                   5     5 13                 7 3     -4 11               12 5       7 9             11 8         -3 7           12 11           1 5         8 11             3 3       8 12               -4 1     4 10                 6 -1   1 6                   -5 -3   4                     4 -5 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=1$ $i=3$ $r=-3$ ${\mathbb {Z} }$ $r=-2$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-1$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=0$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{8}$ $r=1$ ${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=2$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{12}$ ${\mathbb {Z} }^{12}$ $r=3$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{11}$ $r=4$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{12}$ $r=5$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=6$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=7$ ${\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=8$ ${\mathbb {Z} }$ ${\mathbb {Z} }$

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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