L11a337

L11a336

L11a338

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Link Presentations

[edit Notes on L11a337's Link Presentations]

Planar diagram presentation	X10,1,11,2 X22,5,9,6 X14,3,15,4 X4,15,5,16 X16,7,17,8 X20,17,21,18 X18,11,19,12 X12,19,13,20 X2,9,3,10 X8,13,1,14 X6,21,7,22
Gauss code	{1, -9, 3, -4, 2, -11, 5, -10}, {9, -1, 7, -8, 10, -3, 4, -5, 6, -7, 8, -6, 11, -2}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {-t(1)t(2)^{5}-4t(1)^{2}t(2)^{4}+5t(1)t(2)^{4}-t(2)^{4}-2t(1)^{3}t(2)^{3}+9t(1)^{2}t(2)^{3}-10t(1)t(2)^{3}+2t(2)^{3}+2t(1)^{3}t(2)^{2}-10t(1)^{2}t(2)^{2}+9t(1)t(2)^{2}-2t(2)^{2}-t(1)^{3}t(2)+5t(1)^{2}t(2)-4t(1)t(2)-t(1)^{2}}{t(1)^{3/2}t(2)^{5/2}}}$ (db)
Jones polynomial	$-{\frac {8}{q^{9/2}}}+{\frac {3}{q^{7/2}}}-{\frac {1}{q^{5/2}}}+{\frac {1}{q^{27/2}}}-{\frac {4}{q^{25/2}}}+{\frac {9}{q^{23/2}}}-{\frac {15}{q^{21/2}}}+{\frac {19}{q^{19/2}}}-{\frac {22}{q^{17/2}}}+{\frac {22}{q^{15/2}}}-{\frac {19}{q^{13/2}}}+{\frac {13}{q^{11/2}}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$-za^{13}-a^{13}z^{-1}+4z^{3}a^{11}+7za^{11}+3a^{11}z^{-1}-3z^{5}a^{9}-7z^{3}a^{9}-5za^{9}-2a^{9}z^{-1}-3z^{5}a^{7}-7z^{3}a^{7}-4za^{7}-z^{5}a^{5}-2z^{3}a^{5}-za^{5}$ (db)
Kauffman polynomial	$-z^{6}a^{16}+2z^{4}a^{16}-z^{2}a^{16}-4z^{7}a^{15}+9z^{5}a^{15}-6z^{3}a^{15}+za^{15}-7z^{8}a^{14}+15z^{6}a^{14}-8z^{4}a^{14}+2z^{2}a^{14}-a^{14}-6z^{9}a^{13}+6z^{7}a^{13}+8z^{5}a^{13}-6z^{3}a^{13}+a^{13}z^{-1}-2z^{10}a^{12}-12z^{8}a^{12}+36z^{6}a^{12}-31z^{4}a^{12}+15z^{2}a^{12}-3a^{12}-12z^{9}a^{11}+21z^{7}a^{11}-16z^{5}a^{11}+19z^{3}a^{11}-14za^{11}+3a^{11}z^{-1}-2z^{10}a^{10}-12z^{8}a^{10}+31z^{6}a^{10}-28z^{4}a^{10}+12z^{2}a^{10}-3a^{10}-6z^{9}a^{9}+5z^{7}a^{9}-4z^{5}a^{9}+8z^{3}a^{9}-8za^{9}+2a^{9}z^{-1}-7z^{8}a^{8}+8z^{6}a^{8}-3z^{4}a^{8}-z^{2}a^{8}-6z^{7}a^{7}+10z^{5}a^{7}-9z^{3}a^{7}+4za^{7}-3z^{6}a^{6}+4z^{4}a^{6}-z^{2}a^{6}-z^{5}a^{5}+2z^{3}a^{5}-za^{5}$ (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   \ -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 χ -4                       1 1 -6                     3 1 -2 -8                   5     5 -10                 8 3     -5 -12               11 5       6 -14             11 8         -3 -16           11 11           0 -18         8 11             3 -20       7 11               -4 -22     3 9                 6 -24   1 6                   -5 -26   3                     3 -28 1                       -1

Integral Khovanov Homology (db, data source)

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

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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