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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

q-15/2 - 2q-13/2 + 5q-11/2 - 9q-9/2 + 11q-7/2 - 14q-5/2 + 14q-3/2 - 13q-1/2 + 9q1/2 - 6q3/2 + 3q5/2 - q7/2

A2 (sl(3)) Invariant:

- q-22 - 2q-18 - q-16 + 2q-14 - q-12 + 4q-10 + q-8 + q-6 + 3q-4 - q-2 + 4 - q2 + q6 - q8 + q10

HOMFLY-PT Polynomial:

- 2a-1z - 3a-1z3 - a-1z5 - 2az-1 - 2az + 4az3 + 4az5 + az7 + 3a3z-1 + 9a3z + 10a3z3 + 5a3z5 + a3z7 - a5z-1 - 5a5z - 4a5z3 - a5z5

Kauffman Polynomial:

2a-3z3 - a-3z5 - a-2z2 + 6a-2z4 - 3a-2z6 + 4a-1z - 10a-1z3 + 12a-1z5 - 5a-1z7 + 3z2 - 10z4 + 11z6 - 5z8 + 2az-1 - 3az - 8az3 + 5az5 + 2az7 - 3az9 - 3a2 + 11a2z2 - 25a2z4 + 18a2z6 - 5a2z8 - a2z10 + 3a3z-1 - 17a3z + 35a3z3 - 35a3z5 + 19a3z7 - 6a3z9 - 3a4 + 18a4z2 - 23a4z4 + 14a4z6 - 3a4z8 - a4z10 + a5z-1 - 10a5z + 27a5z3 - 21a5z5 + 10a5z7 - 3a5z9 - a6 + 7a6z2 - 10a6z4 + 9a6z6 - 3a6z8 - 4a7z3 + 6a7z5 - 2a7z7 - 4a8z2 + 4a8z4 - a8z6

Khovanov Homology:

trqj r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 j = 8                       1 j = 6                     2   j = 4                   4 1   j = 2                 5 2     j = 0               8 4       j = -2             8 7         j = -4           6 6           j = -6         5 8             j = -8       4 6               j = -10     1 5                 j = -12   1 4                   j = -14   1                     j = -16 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, 296]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 296]]
Out[4]=	PD[X[10, 1, 11, 2], X[12, 3, 13, 4], X[18, 8, 19, 7], X[16, 6, 17, 5], > X[20, 13, 21, 14], X[22, 15, 9, 16], X[14, 21, 15, 22], X[6, 18, 7, 17], > X[4, 20, 5, 19], X[2, 9, 3, 10], X[8, 11, 1, 12]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -9, 4, -8, 3, -11}, > {10, -1, 11, -2, 5, -7, 6, -4, 8, -3, 9, -5, 7, -6}]
In[6]:=	Jones[L][q]
Out[6]=	-(15/2) 2 5 9 11 14 14 13 q - ----- + ----- - ---- + ---- - ---- + ---- - ------- + 9 Sqrt[q] - 13/2 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q q 3/2 5/2 7/2 > 6 q + 3 q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-22 2 -16 2 -12 4 -8 -6 3 -2 2 6 4 - q - --- - q + --- - q + --- + q + q + -- - q - q + q - 18 14 10 4 q q q q 8 10 > q + q
In[8]:=	HOMFLYPT[Link[11, Alternating, 296]][a, z]
Out[8]=	3 5 3 -2 a 3 a a 2 z 3 5 3 z 3 3 3 ---- + ---- - -- - --- - 2 a z + 9 a z - 5 a z - ---- + 4 a z + 10 a z - z z z a a 5 5 3 z 5 3 5 5 5 7 3 7 > 4 a z - -- + 4 a z + 5 a z - a z + a z + a z a
In[9]:=	Kauffman[Link[11, Alternating, 296]][a, z]
Out[9]=	3 5 2 4 6 2 a 3 a a 4 z 3 5 2 -3 a - 3 a - a + --- + ---- + -- + --- - 3 a z - 17 a z - 10 a z + 3 z - z z z a 2 3 3 z 2 2 4 2 6 2 8 2 2 z 10 z 3 > -- + 11 a z + 18 a z + 7 a z - 4 a z + ---- - ----- - 8 a z + 2 3 a a a 4 3 3 5 3 7 3 4 6 z 2 4 4 4 > 35 a z + 27 a z - 4 a z - 10 z + ---- - 25 a z - 23 a z - 2 a 5 5 6 4 8 4 z 12 z 5 3 5 5 5 7 5 > 10 a z + 4 a z - -- + ----- + 5 a z - 35 a z - 21 a z + 6 a z + 3 a a 6 7 6 3 z 2 6 4 6 6 6 8 6 5 z 7 > 11 z - ---- + 18 a z + 14 a z + 9 a z - a z - ---- + 2 a z + 2 a a 3 7 5 7 7 7 8 2 8 4 8 6 8 > 19 a z + 10 a z - 2 a z - 5 z - 5 a z - 3 a z - 3 a z - 9 3 9 5 9 2 10 4 10 > 3 a z - 6 a z - 3 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	7 1 1 1 4 1 5 4 6 8 + -- + ------ + ------ + ------ + ------ + ------ + ------ + ----- + ----- + 2 16 7 14 6 12 6 12 5 10 5 10 4 8 4 8 3 q q t q t q t q t q t q t q t q t 5 8 6 6 8 2 2 2 4 2 > ----- + ----- + ----- + ---- + ---- + 4 t + 5 q t + 2 q t + 4 q t + 6 3 6 2 4 2 4 2 q t q t q t q t q t 4 3 6 3 8 4 > q t + 2 q t + q t

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