L11a285

L11a284

L11a286

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Link Presentations

[edit Notes on L11a285's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-\frac{(u-1) (v-1) \left(u^2 v^4-u^2 v^3+u^2 v^2+2 u v^3-3 u v^2+2 u v+v^2-v+1\right)}{u^{3/2} v^{5/2}}$ (db)
Jones polynomial	$q^{9/2}-3 q^{7/2}+6 q^{5/2}-11 q^{3/2}+14 \sqrt{q}-\frac{17}{\sqrt{q}}+\frac{16}{q^{3/2}}-\frac{15}{q^{5/2}}+\frac{11}{q^{7/2}}-\frac{6}{q^{9/2}}+\frac{3}{q^{11/2}}-\frac{1}{q^{13/2}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$-a z^9+a^3 z^7-7 a z^7+z^7 a^{-1} +5 a^3 z^5-18 a z^5+5 z^5 a^{-1} +8 a^3 z^3-19 a z^3+8 z^3 a^{-1} +3 a^3 z-6 a z+3 z a^{-1} +a z^{-1} - a^{-1} z^{-1}$ (db)
Kauffman polynomial	$-2 a^2 z^{10}-2 z^{10}-5 a^3 z^9-9 a z^9-4 z^9 a^{-1} -6 a^4 z^8-2 a^2 z^8-4 z^8 a^{-2} -5 a^5 z^7+11 a^3 z^7+28 a z^7+9 z^7 a^{-1} -3 z^7 a^{-3} -3 a^6 z^6+13 a^4 z^6+11 a^2 z^6+9 z^6 a^{-2} -z^6 a^{-4} +5 z^6-a^7 z^5+10 a^5 z^5-15 a^3 z^5-44 a z^5-9 z^5 a^{-1} +9 z^5 a^{-3} +6 a^6 z^4-12 a^4 z^4-17 a^2 z^4-3 z^4 a^{-2} +3 z^4 a^{-4} -5 z^4+2 a^7 z^3-6 a^5 z^3+8 a^3 z^3+32 a z^3+9 z^3 a^{-1} -7 z^3 a^{-3} -2 a^6 z^2+2 a^4 z^2+7 a^2 z^2-z^2 a^{-2} -2 z^2 a^{-4} +4 z^2+a^5 z-2 a^3 z-6 a z-2 z a^{-1} +z a^{-3} +1-a z^{-1} - a^{-1} z^{-1}$ (db)

Khovanov Homology

The coefficients of the monomials $t^rq^j$ are shown, along with their alternating sums $\chi$ (fixed $j$, alternation over $r$).

\   r    \    j   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 10                       1 -1 8                     2   2 6                   4 1   -3 4                 7 2     5 2               7 4       -3 0             10 7         3 -2           8 9           1 -4         7 8             -1 -6       4 8               4 -8     2 7                 -5 -10   1 4                   3 -12   2                     -2 -14 1                       1

Integral Khovanov Homology (db, data source)

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-2$ $i=0$ $r=-6$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-3$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-2$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=-1$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ $r=0$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{10}$ $r=1$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=2$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=3$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=4$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=5$ ${\mathbb Z}_2$ ${\mathbb Z}$

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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