L10a170

L10a169

L10a171

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Link Presentations

[edit Notes on L10a170's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(u-1)(w-1)(x-1)(vwx+v(-w)+v-wx+x-1)}{{\sqrt {u}}{\sqrt {v}}wx}}$ (db)
Jones polynomial	$-4q^{9/2}+{\frac {1}{q^{9/2}}}+8q^{7/2}-{\frac {4}{q^{7/2}}}-13q^{5/2}+{\frac {7}{q^{5/2}}}+14q^{3/2}-{\frac {13}{q^{3/2}}}+q^{11/2}-18{\sqrt {q}}+{\frac {13}{\sqrt {q}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$z^{5}a^{-3}-a^{3}z^{3}+2z^{3}a^{-3}+a^{3}z^{-3}-a^{-3}z^{-3}-a^{3}z+za^{-3}+a^{3}z^{-1}-a^{-3}z^{-1}-z^{7}a^{-1}+2az^{5}-4z^{5}a^{-1}+5az^{3}-6z^{3}a^{-1}-3az^{-3}+3a^{-1}z^{-3}+2az-2za^{-1}-3az^{-1}+3a^{-1}z^{-1}$ (db)
Kauffman polynomial	$z^{4}a^{-6}+4z^{5}a^{-5}-2z^{3}a^{-5}+a^{4}z^{6}+8z^{6}a^{-4}-2a^{4}z^{4}-7z^{4}a^{-4}+a^{4}z^{2}+z^{2}a^{-4}+4a^{3}z^{7}+11z^{7}a^{-3}-11a^{3}z^{5}-17z^{5}a^{-3}+10a^{3}z^{3}+12z^{3}a^{-3}+a^{3}z^{-3}+a^{-3}z^{-3}-3a^{3}z-3za^{-3}-2a^{3}z^{-1}-2a^{-3}z^{-1}+5a^{2}z^{8}+8z^{8}a^{-2}-9a^{2}z^{6}-6z^{6}a^{-2}-a^{2}z^{4}-6z^{4}a^{-2}+3a^{2}z^{2}+3z^{2}a^{-2}-3a^{2}z^{-2}-3a^{-2}z^{-2}+4a^{2}+4a^{-2}+2az^{9}+2z^{9}a^{-1}+11az^{7}+18z^{7}a^{-1}-43az^{5}-53z^{5}a^{-1}+40az^{3}+44z^{3}a^{-1}+3az^{-3}+3a^{-1}z^{-3}-11az-11za^{-1}-3az^{-1}-3a^{-1}z^{-1}+13z^{8}-24z^{6}+3z^{4}+4z^{2}-6z^{-2}+7$ (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   \ -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 12                     1 -1 10                   3   3 8                 5 1   -4 6               8 3     5 4             8 7       -1 2           10 6         4 0         7 12           5 -2       6 6             0 -4     3 9               6 -6   1 4                 -3 -8   3                   3 -10 1                     -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=0$ $i=2$ $r=-5$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-3$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-2$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{6}$ $r=-1$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=0$ ${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{10}$ $r=1$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=2$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{8}$ $r=3$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=4$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=5$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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