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DrawMorseLink

PD Presentation:

X6172 X18,7,19,8 X20,9,21,10 X8,19,9,20 X4,21,1,22 X14,6,15,5 X10,4,11,3 X16,12,17,11 X12,16,13,15 X22,14,5,13 X2,18,3,17

Gauss Code:

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

Jones Polynomial:

q-11/2 - 3q-9/2 + 7q-7/2 - 12q-5/2 + 15q-3/2 - 18q-1/2 + 17q1/2 - 16q3/2 + 11q5/2 - 7q7/2 + 4q9/2 - q11/2

A2 (sl(3)) Invariant:

- q-16 + q-14 - 2q-12 - q-10 + 2q-8 - 2q-6 + 5q-4 + q-2 + 3 + 4q2 - q4 + 4q6 - 2q8 - q10 - 2q14 + q16

HOMFLY-PT Polynomial:

a-3z-1 + a-3z - 2a-3z3 - a-3z5 - 4a-1z-1 - 6a-1z + 3a-1z5 + a-1z7 + 4az-1 + 8az + 7az3 + 4az5 + az7 - a3z-1 - 3a3z - 3a3z3 - a3z5

Kauffman Polynomial:

- 2a-5z3 + 3a-5z5 - a-5z7 + a-4 + a-4z2 - 15a-4z4 + 15a-4z6 - 4a-4z8 - a-3z-1 + 2a-3z + 3a-3z3 - 16a-3z5 + 17a-3z7 - 5a-3z9 + 4a-2 - 3a-2z2 - 18a-2z4 + 23a-2z6 - 3a-2z8 - 2a-2z10 - 4a-1z-1 + 11a-1z - 7a-1z3 - 11a-1z5 + 25a-1z7 - 10a-1z9 + 7 - 11z2 - 4z4 + 22z6 - 7z8 - 2z10 - 4az-1 + 15az - 29az3 + 26az5 - 2az7 - 5az9 + 4a2 - 12a2z2 + 6a2z4 + 8a2z6 - 8a2z8 - a3z-1 + 6a3z - 15a3z3 + 15a3z5 - 9a3z7 + a4 - 4a4z2 + 6a4z4 - 6a4z6 + 2a5z3 - 3a5z5 + a6z2 - a6z4

Khovanov Homology:

trqj r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 j = 12                       1 j = 10                     3   j = 8                   4 1   j = 6                 7 3     j = 4               9 4       j = 2             8 7         j = 0           10 9           j = -2         7 10             j = -4       5 8               j = -6     2 7                 j = -8   1 5                   j = -10   2                     j = -12 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, 53]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 53]]
Out[4]=	PD[X[6, 1, 7, 2], X[18, 7, 19, 8], X[20, 9, 21, 10], X[8, 19, 9, 20], > X[4, 21, 1, 22], X[14, 6, 15, 5], X[10, 4, 11, 3], X[16, 12, 17, 11], > X[12, 16, 13, 15], X[22, 14, 5, 13], X[2, 18, 3, 17]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -11, 7, -5}, {6, -1, 2, -4, 3, -7, 8, -9, 10, -6, 9, -8, 11, -2, > 4, -3, 5, -10}]
In[6]:=	Jones[L][q]
Out[6]=	-(11/2) 3 7 12 15 18 3/2 q - ---- + ---- - ---- + ---- - ------- + 17 Sqrt[q] - 16 q + 9/2 7/2 5/2 3/2 Sqrt[q] q q q q 5/2 7/2 9/2 11/2 > 11 q - 7 q + 4 q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-16 -14 2 -10 2 2 5 -2 2 4 6 8 3 - q + q - --- - q + -- - -- + -- + q + 4 q - q + 4 q - 2 q - 12 8 6 4 q q q q 10 14 16 > q - 2 q + q
In[8]:=	HOMFLYPT[Link[11, Alternating, 53]][a, z]
Out[8]=	3 3 1 4 4 a a z 6 z 3 2 z 3 3 3 ---- - --- + --- - -- + -- - --- + 8 a z - 3 a z - ---- + 7 a z - 3 a z - 3 a z z z 3 a 3 a z a a 5 5 7 z 3 z 5 3 5 z 7 > -- + ---- + 4 a z - a z + -- + a z 3 a a a
In[9]:=	Kauffman[Link[11, Alternating, 53]][a, z]
Out[9]=	3 -4 4 2 4 1 4 4 a a 2 z 11 z 7 + a + -- + 4 a + a - ---- - --- - --- - -- + --- + ---- + 15 a z + 2 3 a z z z 3 a a a z a 2 2 3 3 3 2 z 3 z 2 2 4 2 6 2 2 z 3 z > 6 a z - 11 z + -- - ---- - 12 a z - 4 a z + a z - ---- + ---- - 4 2 5 3 a a a a 3 4 4 7 z 3 3 3 5 3 4 15 z 18 z 2 4 > ---- - 29 a z - 15 a z + 2 a z - 4 z - ----- - ----- + 6 a z + a 4 2 a a 5 5 5 4 4 6 4 3 z 16 z 11 z 5 3 5 5 5 > 6 a z - a z + ---- - ----- - ----- + 26 a z + 15 a z - 3 a z + 5 3 a a a 6 6 7 7 7 6 15 z 23 z 2 6 4 6 z 17 z 25 z 7 > 22 z + ----- + ----- + 8 a z - 6 a z - -- + ----- + ----- - 2 a z - 4 2 5 3 a a a a a 8 8 9 9 3 7 8 4 z 3 z 2 8 5 z 10 z 9 10 > 9 a z - 7 z - ---- - ---- - 8 a z - ---- - ----- - 5 a z - 2 z - 4 2 3 a a a a 10 2 z > ----- 2 a
In[10]:=	Kh[L][q, t]
Out[10]=	10 1 2 1 5 2 7 5 8 10 + -- + ------ + ------ + ----- + ----- + ----- + ----- + ----- + ---- + 2 12 5 10 4 8 4 8 3 6 3 6 2 4 2 4 q q t q t q t q t q t q t q t q t 7 2 2 2 4 2 4 3 6 3 6 4 > ---- + 9 t + 8 q t + 7 q t + 9 q t + 4 q t + 7 q t + 3 q t + 2 q t 8 4 8 5 10 5 12 6 > 4 q t + q t + 3 q t + q t

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