L11a348

L11a347

L11a349

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Link Presentations

[edit Notes on L11a348's Link Presentations]

Planar diagram presentation	X12,1,13,2 X10,11,1,12 X14,5,15,6 X18,9,19,10 X2,17,3,18 X16,8,17,7 X8493 X20,16,21,15 X22,13,11,14 X4,20,5,19 X6,21,7,22
Gauss code	{1, -5, 7, -10, 3, -11, 6, -7, 4, -2}, {2, -1, 9, -3, 8, -6, 5, -4, 10, -8, 11, -9}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$\frac{(2 t(2) t(1)-t(1)-t(2)+2) \left(t(2)^2 t(1)^2-2 t(2) t(1)^2+t(1)^2-2 t(2)^2 t(1)+3 t(2) t(1)-2 t(1)+t(2)^2-2 t(2)+1\right)}{t(1)^{3/2} t(2)^{3/2}}$ (db)
Jones polynomial	$-q^{7/2}+4 q^{5/2}-9 q^{3/2}+17 \sqrt{q}-\frac{25}{\sqrt{q}}+\frac{28}{q^{3/2}}-\frac{30}{q^{5/2}}+\frac{26}{q^{7/2}}-\frac{20}{q^{9/2}}+\frac{13}{q^{11/2}}-\frac{6}{q^{13/2}}+\frac{1}{q^{15/2}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^3 z^7+a z^7-a^5 z^5+a^3 z^5+3 a z^5-z^5 a^{-1} -5 a^3 z^3+5 a z^3-2 z^3 a^{-1} +3 a^5 z-8 a^3 z+4 a z-z a^{-1} +a^5 z^{-1} -a^3 z^{-1}$ (db)
Kauffman polynomial	$-4 a^4 z^{10}-4 a^2 z^{10}-12 a^5 z^9-22 a^3 z^9-10 a z^9-13 a^6 z^8-20 a^4 z^8-18 a^2 z^8-11 z^8-6 a^7 z^7+16 a^5 z^7+36 a^3 z^7+6 a z^7-8 z^7 a^{-1} -a^8 z^6+26 a^6 z^6+59 a^4 z^6+49 a^2 z^6-4 z^6 a^{-2} +13 z^6+8 a^7 z^5+6 a^5 z^5-2 a^3 z^5+11 a z^5+10 z^5 a^{-1} -z^5 a^{-3} -10 a^6 z^4-33 a^4 z^4-33 a^2 z^4+5 z^4 a^{-2} -5 z^4-6 a^5 z^3-17 a^3 z^3-18 a z^3-6 z^3 a^{-1} +z^3 a^{-3} -2 a^6 z^2+a^4 z^2+3 a^2 z^2-2 z^2 a^{-2} -2 z^2-2 a^7 z+3 a^5 z+10 a^3 z+7 a z+2 z a^{-1} +a^4-a^5 z^{-1} -a^3 z^{-1}$ (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   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 χ 8                       1 1 6                     3   -3 4                   6 1   5 2                 11 3     -8 0               14 6       8 -2             15 12         -3 -4           15 13           2 -6         11 15             4 -8       9 15               -6 -10     5 12                 7 -12   1 8                   -7 -14   5                     5 -16 1                       -1

Integral Khovanov Homology (db, data source)

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-2$ $i=0$ $r=-7$ ${\mathbb Z}$ $r=-6$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=-4$ ${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{9}$ $r=-3$ ${\mathbb Z}^{15}\oplus{\mathbb Z}_2^{11}$ ${\mathbb Z}^{11}$ $r=-2$ ${\mathbb Z}^{15}\oplus{\mathbb Z}_2^{15}$ ${\mathbb Z}^{15}$ $r=-1$ ${\mathbb Z}^{13}\oplus{\mathbb Z}_2^{15}$ ${\mathbb Z}^{15}$ $r=0$ ${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{13}$ ${\mathbb Z}^{14}$ $r=1$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{11}$ ${\mathbb Z}^{11}$ $r=2$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=3$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=4$ ${\mathbb Z}_2$ ${\mathbb Z}$

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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