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DrawMorseLink

PD Presentation:

X8192 X10,3,11,4 X14,17,15,18 X16,5,17,6 X4,15,5,16 X18,13,19,14 X22,20,7,19 X20,12,21,11 X12,22,13,21 X2738 X6,9,1,10

Gauss Code:

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

Jones Polynomial:

- q-17/2 + 3q-15/2 - 6q-13/2 + 9q-11/2 - 13q-9/2 + 14q-7/2 - 15q-5/2 + 13q-3/2 - 10q-1/2 + 6q1/2 - 3q3/2 + q5/2

A2 (sl(3)) Invariant:

q-26 - q-24 + 2q-20 - 2q-18 + 3q-16 + 3q-14 + q-12 + 4q-10 - q-8 + q-6 - q-4 - 2q-2 + 3 - 2q2 + q6 - q8

HOMFLY-PT Polynomial:

a-1z + a-1z3 + az-1 - az3 - az5 - 3a3z-1 - 6a3z - 5a3z3 - 2a3z5 + 2a5z-1 + a5z - a5z3 - a5z5 + a7z + a7z3

Kauffman Polynomial:

a-2z2 - a-2z4 - a-1z + 3a-1z3 - 3a-1z5 + 1 - 3z2 + 5z4 - 5z6 - az-1 + az - 3az3 + 6az5 - 6az7 + 3a2 - 12a2z2 + 11a2z4 + a2z6 - 5a2z8 - 3a3z-1 + 12a3z - 20a3z3 + 20a3z5 - 4a3z7 - 3a3z9 + 3a4 - 6a4z2 - 4a4z4 + 18a4z6 - 7a4z8 - a4z10 - 2a5z-1 + 9a5z - 15a5z3 + 6a5z5 + 11a5z7 - 6a5z9 + 6a6z2 - 23a6z4 + 24a6z6 - 5a6z8 - a6z10 + a7z - 6a7z3 - a7z5 + 8a7z7 - 3a7z9 + 4a8z2 - 14a8z4 + 12a8z6 - 3a8z8 + 2a9z - 5a9z3 + 4a9z5 - a9z7

Khovanov Homology:

trqj r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 j = 6                       1 j = 4                     2   j = 2                   4 1   j = 0                 6 2     j = -2               8 5       j = -4             7 5         j = -6           7 8           j = -8         6 7             j = -10       3 7               j = -12     3 6                 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, Alternating, 190]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 190]]
Out[4]=	PD[X[8, 1, 9, 2], X[10, 3, 11, 4], X[14, 17, 15, 18], X[16, 5, 17, 6], > X[4, 15, 5, 16], X[18, 13, 19, 14], X[22, 20, 7, 19], X[20, 12, 21, 11], > X[12, 22, 13, 21], X[2, 7, 3, 8], X[6, 9, 1, 10]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -5, 4, -11}, > {10, -1, 11, -2, 8, -9, 6, -3, 5, -4, 3, -6, 7, -8, 9, -7}]
In[6]:=	Jones[L][q]
Out[6]=	-(17/2) 3 6 9 13 14 15 13 10 -q + ----- - ----- + ----- - ---- + ---- - ---- + ---- - ------- + 15/2 13/2 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q q q 3/2 5/2 > 6 Sqrt[q] - 3 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-26 -24 2 2 3 3 -12 4 -8 -6 -4 2 3 + q - q + --- - --- + --- + --- + q + --- - q + q - q - -- - 20 18 16 14 10 2 q q q q q q 2 6 8 > 2 q + q - q
In[8]:=	HOMFLYPT[Link[11, Alternating, 190]][a, z]
Out[8]=	3 5 3 a 3 a 2 a z 3 5 7 z 3 3 3 5 3 - - ---- + ---- + - - 6 a z + a z + a z + -- - a z - 5 a z - a z + z z z a a 7 3 5 3 5 5 5 > a z - a z - 2 a z - a z
In[9]:=	Kauffman[Link[11, Alternating, 190]][a, z]
Out[9]=	3 5 2 4 a 3 a 2 a z 3 5 7 1 + 3 a + 3 a - - - ---- - ---- - - + a z + 12 a z + 9 a z + a z + z z z a 2 3 9 2 z 2 2 4 2 6 2 8 2 3 z > 2 a z - 3 z + -- - 12 a z - 6 a z + 6 a z + 4 a z + ---- - 2 a a 4 3 3 3 5 3 7 3 9 3 4 z 2 4 > 3 a z - 20 a z - 15 a z - 6 a z - 5 a z + 5 z - -- + 11 a z - 2 a 5 4 4 6 4 8 4 3 z 5 3 5 5 5 > 4 a z - 23 a z - 14 a z - ---- + 6 a z + 20 a z + 6 a z - a 7 5 9 5 6 2 6 4 6 6 6 8 6 7 > a z + 4 a z - 5 z + a z + 18 a z + 24 a z + 12 a z - 6 a z - 3 7 5 7 7 7 9 7 2 8 4 8 6 8 > 4 a z + 11 a z + 8 a z - a z - 5 a z - 7 a z - 5 a z - 8 8 3 9 5 9 7 9 4 10 6 10 > 3 a z - 3 a z - 6 a z - 3 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	5 1 2 1 4 3 6 3 6 + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 2 18 8 16 7 14 7 14 6 12 6 12 5 10 5 q q t q t q t q t q t q t q t 7 6 7 7 8 7 5 8 > ------ + ----- + ----- + ----- + ----- + ----- + ---- + ---- + 2 t + 10 4 8 4 8 3 6 3 6 2 4 2 4 2 q t q t q t q t q t q t q t q t 2 2 2 4 2 6 3 > 4 q t + q t + 2 q t + q t

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