L10a166

L10a165

L10a167

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Link Presentations

[edit Notes on L10a166's Link Presentations]

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

A Braid Representative	{{{braid_table}}}
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {-t(1)t(4)^{2}t(3)^{2}-t(2)t(4)^{2}t(3)^{2}+t(4)^{2}t(3)^{2}+t(1)t(4)t(3)^{2}-t(1)t(2)t(4)t(3)^{2}+2t(2)t(4)t(3)^{2}-t(4)t(3)^{2}+2t(1)t(4)^{2}t(3)-t(1)t(2)t(4)^{2}t(3)+t(2)t(4)^{2}t(3)-t(4)^{2}t(3)+t(1)t(3)-t(1)t(2)t(3)+2t(2)t(3)-3t(1)t(4)t(3)+2t(1)t(2)t(4)t(3)-3t(2)t(4)t(3)+2t(4)t(3)-t(3)-t(1)+t(1)t(2)-t(2)+2t(1)t(4)-t(1)t(2)t(4)+t(2)t(4)-t(4)}{{\sqrt {t(1)}}{\sqrt {t(2)}}t(3)t(4)}}$ (db)
Jones polynomial	$-{\frac {7}{q^{9/2}}}+{\frac {3}{q^{7/2}}}-{\frac {1}{q^{5/2}}}-{\frac {1}{q^{25/2}}}+{\frac {2}{q^{23/2}}}-{\frac {6}{q^{21/2}}}+{\frac {8}{q^{19/2}}}-{\frac {12}{q^{17/2}}}+{\frac {11}{q^{15/2}}}-{\frac {13}{q^{13/2}}}+{\frac {8}{q^{11/2}}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$a^{13}z^{-1}+a^{13}z^{-3}-4za^{11}-8a^{11}z^{-1}-3a^{11}z^{-3}+6z^{3}a^{9}+17za^{9}+13a^{9}z^{-1}+3a^{9}z^{-3}-3z^{5}a^{7}-11z^{3}a^{7}-13za^{7}-6a^{7}z^{-1}-a^{7}z^{-3}-z^{5}a^{5}-2z^{3}a^{5}$ (db)
Kauffman polynomial	$-z^{5}a^{15}+3z^{3}a^{15}-3za^{15}+a^{15}z^{-1}-2z^{6}a^{14}+3z^{4}a^{14}-a^{14}-3z^{7}a^{13}+4z^{5}a^{13}-2z^{3}a^{13}+3za^{13}-3a^{13}z^{-1}+a^{13}z^{-3}-2z^{8}a^{12}-4z^{6}a^{12}+15z^{4}a^{12}-16z^{2}a^{12}-3a^{12}z^{-2}+11a^{12}-z^{9}a^{11}-6z^{7}a^{11}+15z^{5}a^{11}-20z^{3}a^{11}+21za^{11}-12a^{11}z^{-1}+3a^{11}z^{-3}-6z^{8}a^{10}+5z^{6}a^{10}+14z^{4}a^{10}-33z^{2}a^{10}-6a^{10}z^{-2}+24a^{10}-z^{9}a^{9}-9z^{7}a^{9}+28z^{5}a^{9}-37z^{3}a^{9}+28za^{9}-14a^{9}z^{-1}+3a^{9}z^{-3}-4z^{8}a^{8}+4z^{6}a^{8}+7z^{4}a^{8}-17z^{2}a^{8}-3a^{8}z^{-2}+13a^{8}-6z^{7}a^{7}+17z^{5}a^{7}-20z^{3}a^{7}+13za^{7}-6a^{7}z^{-1}+a^{7}z^{-3}-3z^{6}a^{6}+5z^{4}a^{6}-z^{5}a^{5}+2z^{3}a^{5}$ (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   \ -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 χ -4                     1 1 -6                   3 1 -2 -8                 4     4 -10               4 3     -1 -12             9 4       5 -14           5 7         2 -16         7 6           1 -18       4 8             4 -20     2 4               -2 -22     4                 4 -24 1 2                   -1 -26 1                     1

Integral Khovanov Homology (db, data source)

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

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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