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DrawMorseLink

PD Presentation:

X10,1,11,2 X3,12,4,13 X16,9,17,10 X11,20,12,21 X22,15,9,16 X5,14,6,15 X13,4,14,5 X6,20,7,19 X18,8,19,7 X8,18,1,17 X21,3,22,2

Gauss Code:

{{1, 11, -2, 7, -6, -8, 9, -10}, {3, -1, -4, 2, -7, 6, 5, -3, 10, -9, 8, 4, -11, -5}}

Jones Polynomial:

q-17/2 - 3q-15/2 + 6q-13/2 - 8q-11/2 + 9q-9/2 - 10q-7/2 + 8q-5/2 - 6q-3/2 + 3q-1/2 - 2q1/2

A2 (sl(3)) Invariant:

- q-26 + q-24 - 2q-20 + q-18 - 2q-16 + q-14 + q-12 + 3q-8 - q-6 + 3q-4 + 2q-2 + 1 + 2q2

HOMFLY-PT Polynomial:

- 2az-1 - 5az - 2az3 + 3a3z-1 + 8a3z + 7a3z3 + 2a3z5 - a5z-1 - 2a5z + a5z3 + a5z5 - a7z - a7z3

Kauffman Polynomial:

2az-1 - 9az + 10az3 - 3az5 - 3a2 + 4a2z2 - 2a2z4 + 2a2z6 - a2z8 + 3a3z-1 - 15a3z + 24a3z3 - 12a3z5 + 3a3z7 - a3z9 - 3a4 + 12a4z2 - 14a4z4 + 10a4z6 - 4a4z8 + a5z-1 - 4a5z + 4a5z3 + 2a5z5 - 2a5z7 - a5z9 - a6 + 4a6z2 - 4a6z4 + 3a6z6 - 3a6z8 + 2a7z - 7a7z3 + 8a7z5 - 5a7z7 - 3a8z2 + 7a8z4 - 5a8z6 + 3a9z3 - 3a9z5 + a10z2 - a10z4

Khovanov Homology:

trqj r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 j = 2                   2 j = 0                 1   j = -2               5 2   j = -4             5 3     j = -6           5 3       j = -8         4 5         j = -10       4 5           j = -12     2 4             j = -14   1 4               j = -16   2                 j = -18 1

Computer Talk. The data above can be recomputed by Mathematica using the package KnotTheory`. Following setup, the sample Mathematica session below reproduces most of the above data (Mathematica system prompts in blue, human input in red, Mathematica output in black):

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 30, 2005, 10:15:35)...
In[2]:=	Length[Skeleton[L]]
Out[2]=	2
In[3]:=	Show[DrawMorseLink[Link[11, NonAlternating, 225]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 225]]
Out[4]=	PD[X[10, 1, 11, 2], X[3, 12, 4, 13], X[16, 9, 17, 10], X[11, 20, 12, 21], > X[22, 15, 9, 16], X[5, 14, 6, 15], X[13, 4, 14, 5], X[6, 20, 7, 19], > X[18, 8, 19, 7], X[8, 18, 1, 17], X[21, 3, 22, 2]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, 11, -2, 7, -6, -8, 9, -10}, > {3, -1, -4, 2, -7, 6, 5, -3, 10, -9, 8, 4, -11, -5}]
In[6]:=	Jones[L][q]
Out[6]=	-(17/2) 3 6 8 9 10 8 6 3 q - ----- + ----- - ----- + ---- - ---- + ---- - ---- + ------- - 15/2 13/2 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q q q > 2 Sqrt[q]
In[7]:=	A2Invariant[L][q]
Out[7]=	-26 -24 2 -18 2 -14 -12 3 -6 3 2 2 1 - q + q - --- + q - --- + q + q + -- - q + -- + -- + 2 q 20 16 8 4 2 q q q q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 225]][a, z]
Out[8]=	3 5 -2 a 3 a a 3 5 7 3 3 3 5 3 ---- + ---- - -- - 5 a z + 8 a z - 2 a z - a z - 2 a z + 7 a z + a z - z z z 7 3 3 5 5 5 > a z + 2 a z + a z
In[9]:=	Kauffman[Link[11, NonAlternating, 225]][a, z]
Out[9]=	3 5 2 4 6 2 a 3 a a 3 5 7 -3 a - 3 a - a + --- + ---- + -- - 9 a z - 15 a z - 4 a z + 2 a z + z z z 2 2 4 2 6 2 8 2 10 2 3 3 3 > 4 a z + 12 a z + 4 a z - 3 a z + a z + 10 a z + 24 a z + 5 3 7 3 9 3 2 4 4 4 6 4 8 4 > 4 a z - 7 a z + 3 a z - 2 a z - 14 a z - 4 a z + 7 a z - 10 4 5 3 5 5 5 7 5 9 5 2 6 > a z - 3 a z - 12 a z + 2 a z + 8 a z - 3 a z + 2 a z + 4 6 6 6 8 6 3 7 5 7 7 7 2 8 > 10 a z + 3 a z - 5 a z + 3 a z - 2 a z - 5 a z - a z - 4 8 6 8 3 9 5 9 > 4 a z - 3 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	3 5 1 2 1 4 2 4 4 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 4 2 18 7 16 6 14 6 14 5 12 5 12 4 10 4 q q q t q t q t q t q t q t q t 5 4 5 5 3 5 2 t 2 2 > ------ + ----- + ----- + ----- + ---- + ---- + t + --- + 2 q t 10 3 8 3 8 2 6 2 6 4 2 q t q t q t q t q t q t q

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