L11a544

L11a543

L11a545

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Link Presentations

[edit Notes on L11a544's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {-t(1)t(3)^{2}t(2)^{2}+t(3)^{2}t(2)^{2}+t(1)t(3)t(2)^{2}-2t(3)t(2)^{2}+2t(1)t(3)^{2}t(4)t(2)^{2}-2t(3)^{2}t(4)t(2)^{2}-t(1)t(3)t(4)t(2)^{2}+3t(3)t(4)t(2)^{2}-t(4)t(2)^{2}+t(2)^{2}+2t(1)t(3)^{2}t(2)-t(3)^{2}t(2)+t(1)t(2)-3t(1)t(3)t(2)+3t(3)t(2)-3t(1)t(3)^{2}t(4)t(2)+t(3)^{2}t(4)t(2)-t(1)t(4)t(2)+3t(1)t(3)t(4)t(2)-3t(3)t(4)t(2)+2t(4)t(2)-3t(2)-t(1)t(3)^{2}-2t(1)+3t(1)t(3)-t(3)+t(1)t(3)^{2}t(4)+t(1)t(4)-2t(1)t(3)t(4)+t(3)t(4)-t(4)+2}{{\sqrt {t(1)}}t(2)t(3){\sqrt {t(4)}}}}$ (db)
Jones polynomial	$q^{23/2}-4q^{21/2}+8q^{19/2}-13q^{17/2}+16q^{15/2}-19q^{13/2}+16q^{11/2}-16q^{9/2}+9q^{7/2}-7q^{5/2}+2q^{3/2}-{\sqrt {q}}$ (db)
Signature	5 (db)
HOMFLY-PT polynomial	$z^{5}a^{-9}+2z^{3}a^{-9}-a^{-9}z^{-3}-a^{-9}z^{-1}-z^{7}a^{-7}-3z^{5}a^{-7}-z^{3}a^{-7}+3a^{-7}z^{-3}+4za^{-7}+6a^{-7}z^{-1}-z^{7}a^{-5}-4z^{5}a^{-5}-7z^{3}a^{-5}-3a^{-5}z^{-3}-10za^{-5}-9a^{-5}z^{-1}+z^{5}a^{-3}+4z^{3}a^{-3}+a^{-3}z^{-3}+6za^{-3}+4a^{-3}z^{-1}$ (db)
Kauffman polynomial	$z^{4}a^{-14}+4z^{5}a^{-13}-2z^{3}a^{-13}+8z^{6}a^{-12}-7z^{4}a^{-12}+z^{2}a^{-12}+11z^{7}a^{-11}-15z^{5}a^{-11}+8z^{3}a^{-11}-2za^{-11}+10z^{8}a^{-10}-13z^{6}a^{-10}+2z^{4}a^{-10}+2z^{2}a^{-10}+5z^{9}a^{-9}+5z^{7}a^{-9}-32z^{5}a^{-9}+32z^{3}a^{-9}-a^{-9}z^{-3}-16za^{-9}+5a^{-9}z^{-1}+z^{10}a^{-8}+14z^{8}a^{-8}-37z^{6}a^{-8}+17z^{4}a^{-8}+10z^{2}a^{-8}+3a^{-8}z^{-2}-10a^{-8}+7z^{9}a^{-7}-7z^{7}a^{-7}-28z^{5}a^{-7}+48z^{3}a^{-7}-3a^{-7}z^{-3}-31za^{-7}+12a^{-7}z^{-1}+z^{10}a^{-6}+6z^{8}a^{-6}-22z^{6}a^{-6}+9z^{4}a^{-6}+18z^{2}a^{-6}+6a^{-6}z^{-2}-19a^{-6}+2z^{9}a^{-5}-20z^{5}a^{-5}+36z^{3}a^{-5}-3a^{-5}z^{-3}-27za^{-5}+12a^{-5}z^{-1}+2z^{8}a^{-4}-6z^{6}a^{-4}+2z^{4}a^{-4}+9z^{2}a^{-4}+3a^{-4}z^{-2}-10a^{-4}+z^{7}a^{-3}-5z^{5}a^{-3}+10z^{3}a^{-3}-a^{-3}z^{-3}-10za^{-3}+5a^{-3}z^{-1}$ (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   \ -2 -1 0 1 2 3 4 5 6 7 8 9 χ 24                       1 -1 22                     3   3 20                   5 1   -4 18                 8 3     5 16               10 7       -3 14             9 6         3 12           9 12           3 10         7 7             0 8       4 11               7 6     3 5                 -2 4   1 6                   5 2   1                     -1 0 1                       1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=4$ $i=6$ $r=-2$ ${\mathbb {Z} }$ $r=-1$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=0$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }^{3}$ $r=1$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=2$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{7}$ $r=3$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=4$ ${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{9}$ $r=5$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=6$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{8}$ $r=7$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=8$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=9$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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